
    UNITED STATES of America, Appellee, v. Raj RAJARATNAM, Defendant-Appellant.
    Docket No. 11-4416-cr.
    United States Court of Appeals, Second Circuit.
    Argued: Oct. 25, 2012.
    Decided: June 24, 2013.
    
      Patricia Ann Millett (Terence Joseph Lynam, Samidh Guha, Hyland Hunt, James Eamonn Sherry, on the brief), Akin Gump Strauss Hauer & Feld LLP, New York, NY, Washington, DC, and Dallas, TX, for Raj Rajaratnam.
    Andrew L. Fish (Reed Brodsky, on the brief), Assistant United States Attorneys, for Preet Bharara, United States Attorney for the Southern District of New York, New York, NY, for the United States of America.
    Lawrence S. Lustberg, Alicia L. Ban-non, Gibbons P.C., Newark, NJ, for Amici Curiae Retired Federal Judges.
    Yinoo P. Varghese, Varghese & Associates, P.C., New York, NY, for Amici Curiae National Legal Aid & Defender Association and the Bronx Defenders.
    Tai H. Park, Park & Jensen LLP, New York, NY; G. Robert Blakey, Notre Dame Law School, Notre Dame, IN, for Amicus Curiae Professor G. Robert Blakey.
    Before: CABRANES, SACK, and CARNEY, Circuit Judges.
   JOSÉ A. CABRANES, Circuit Judge:

In this “insider information” securities fraud case, we consider two issues on appeal raised by defendant-appellant Raj Rajaratnam. The first issue is whether the United States District Court for the Southern District of New York (Richard J. Holwell, Judge) should have suppressed the evidence obtained by the government’s wiretap of Rajaratnam’s cell phone. Specifically, Rajaratnam argues that the District Court erred by applying the analytical framework set forth in Franks v. Delaware, 438 U.S. 154, 98 S.Ct. 2674, 57 L.Ed.2d 667 (1978), to determine whether suppression was warranted, and by concluding that the alleged misstatements and omissions in the government’s wiretap application did not require suppression.

The second issue concerns the District Court’s instruction to the jury that it could convict Rajaratnam of securities fraud if the “material non-public information given to the defendant was a factor, however small, in the defendant’s decision to purchase or sell stock.” Rajaratnam contends that this instruction was in error and requires us to vacate the substantive counts of conviction for securities fraud (Counts 6 through 14).

Rajaratnam’s arguments are not persuasive. In affirming the judgment of conviction, we conclude that: (1) the District Court properly analyzed the alleged misstatements and omissions in the government’s wiretap application under the analytical framework prescribed by the Supreme Court in Franks; (2) the alleged misstatements and omissions in the wiretap application did not require suppression, both because, contrary to the District Court’s conclusion, the government did not omit information abput the SEC investigation of Rajaratnam with “reckless disregard for the truth,” and because, as the District Court correctly concluded, all of the alleged misstatements and omissions were not “material”; and (3) the jury instructions on the use of inside information satisfy the “knowing possession” standard that is the law of this Circuit.

BACKGROUND

Rajaratnam founded and managed the Galleon Group (“Galleon”), a family of hedge funds. When Galleon was at its pinnacle, the fund employed dozens of portfolio managers, analysts, and traders, and invested billions of dollars of client funds.

In 2011, Rajaratnam was indicted on five counts of conspiracy to commit securities fraud, in violation of 18 U.S.C. § 371, and nine counts of securities fraud, in violation of 15 U.S.C. § 78j(b) and 78ff, 17 C.F.R. §§ 240.10b-5 and 240.10b5-2, and 18 U.S.C. § 2. The conduct underlying the five charged conspiracies took place between 2003 and 2009 and consisted of Ra-jaratnam trading securities based on inside information he received from certain individuals about various publicly-traded companies. The alleged conspiracies involved inside information passed unlawfully to Rajaratnam from: (1) Anil Kumar, a senior partner at McKinsey & Company, Inc. (Counts 4 and 13); (2) Rajiv Goel, an executive of Intel Corporation (Counts 3, 6, 7, and 14); (3) Danielle Chiesi, a portfolio manager at another hedge fund (Counts 5, 8, 9, and 10); (4) Roomy Khan, a former Galleon employee (Count 2); and (5) other former and current Galleon employees, including one by the name of Adam Smith (Count 1). Joint App’x 268-97.

A. The Wiretap Application

Beginning in 2007, the United States Attorney’s Office for the Southern District of New York (“USAO”) and the Federal Bureau of Investigation (“FBI”) began investigating Rajaratnam based on suspicions that he was using inside information in executing certain securities transactions. On March 7, 2008, the government sought authorization to wiretap Rajaratnam’s cell phone. The wiretap application was submitted to then-United States District Judge Gerard E. Lynch and sworn to by then-Assistant United States Attorney (“AUSA”) Lauren Goldberg. It included a 53-page affidavit sworn to by FBI Special Agent B.J. Kang. The wiretap application stated that its purpose was to identify Rajaratnam’s network of alleged inside sources, to learn how the asserted conspirators operated, and to provide admissible evidence for possible criminal prosecutions. See id. at 72.

Title III of the Omnibus Crime Control and Safe Streets Act of 1968 (“Title III”), 18 U.S.C. §§ 2510-2522, requires that wiretap applications provide “a full and complete statement of the facts and circumstance relied upon by the applicant” to establish probable cause, id. § 2518(l)(b), and a “full and complete statement as to whether or not other investigative procedures have been tried and failed or why they reasonably appear to be unlikely to succeed if tried or to be too dangerous,” id. § 2518(l)(c). Accordingly, the wiretap application submitted by the government to Judge Lynch addressed (1) why “probable cause” existed to wiretap Rajaratnam’s cell phone; and (2) why the proposed wiretap was “necessary.”

To establish “probable cause,” the wiretap application set forth, inter alia, state-' ments made by Rajaratnam to Roomy Khan (identified as “CS-1”), as well as summaries of conversations between Khan and Rajaratnam that Khan had recorded, which indicated that Rajaratnam and Khan were exchanging material, non-public information used to trade securities. See Joint App’x 77-81. To establish “necessity,” the wiretap application stated, inter alia, that “normal investigative techniques,” such as physical surveillance, federal grand jury subpoenas for witness testimony, review of trading records, witness interviews, use of confidential informants, and placement of undercover agents, had been tried and had “failed or reasonably appealed] unlikely to succeed if tried.” Id. at 58,102-12.

On the basis of these representations, Judge Lynch authorized the wiretap of Rajaratnam’s cell phone on March 7, 2008. Seven subsequent wiretap applications were also approved. See note 1, ante. On October 16, 2009, based in large part on evidence obtained from the wiretap of Ra-jaratnam’s cell phone, Rajaratnam was arrested and charged with multiple counts of securities fraud. He was indicted two months later. A Superseding Indictment was returned on February 9, 2010, and a Second Superseding Indictment was returned on January 20, 2011.

B. Rajaratnam’s Suppression Motion

On May 7, 2010, Rajaratnam filed a motion to suppress the evidence obtained through the wiretap of his cell phone, claiming that the wiretap application contained certain misstatements and omissions. As relevant here, Rajaratnam took issue with the statements supplied on the government’s wiretap application regarding both “probable cause” and “necessity.”

On the question of “probable cause,” Rajaratnam argued that the government made misstatements and omissions regarding the reliability of Roomy Khan. In particular, he observed that the wiretap application stated that Khan “ha[d] not yet been charged with any crimes,” Joint App’x 77, and “ha[d] been cooperating with the FBI since approximately November 2007,” id. at 77 n. 4. In fact, in 2001, Khan was indicted and pleaded guilty to felony wire fraud and, in 2002, she began cooperating with the government in an earlier investigation involving Rajaratnam. Ra-jaratnam also asserted that the wiretap application included two paraphrased summaries of recorded conversations between Khan and Rajaratnam that mischaracter-ized the actual recorded conversations, as we describe in detail below. See Background Part C.ii.a., post.

On the question of “necessity,” Rajarat-nam argued that the wiretap application improperly omitted the fact that Rajarat-nam and Galleon had been the subject of an ongoing SEC investigation, which led to, inter alia, depositions of Rajaratnam and several other Galleon employees and production for the SEC of approximately four million documents—documents that had thereafter been conveyed to the USAO.

C. The Franks Hearing

i. The Analytical Framework of Franks v. Delaware

The District Court then decided whether to hold a hearing for the purpose of considering Rajaratnam’s suppression motion. In doing so, it noted that “[wjhere a defendant makes a preliminary showing that the government’s affidavit misstated or omitted material information, Franks instructs a district court to hold a hearing to determine” whether the alleged misstatements or omissions in the warrant or wiretap application were made intentionally or with “reckless disregard for the truth” and, if so, whether any such misstatements or omissions were “material.” United States v. Rajaratnam, No. 09 Cr. 1184(RJH), 2010 WL 4867402, at *7-8 (S.D.N.Y. Nov. 24, 2010); see United States v. Falso, 544 F.3d 110, 125 (2d Cir.2008). In other words, “[t]o suppress evidence obtained pursuant to an affidavit containing erroneous information, the defendant must show that: (1) the claimed inaccuracies or omissions are the result of the affiant’s deliberate falsehood or reckless disregard for the truth; and (2) the alleged falsehoods or omissions were necessary to the [issuing] judge’s probable cause [or necessity] finding.” United States v. Canfield, 212 F.3d 713, 717-18 (2d Cir.2000) (internal quotation marks omitted); see also United States v. Awadallah, 349 F.3d 42, 64 (2d Cir.2003) (noting that “[i]n order to invoke the Franks doctrine, [a defendant] must show that there were intentional and material misrepresentations or omissions in [the] warrant affidavit.” (emphases supplied)).

To determine whether misstatements are “material,” a court must “set[ ] aside the falsehoods” in the application, United States v. Coreas, 419 F.3d 151, 155 (2d Cir.2005), and determine “[w]hether the untainted portions [of the application] suffice to support a probable cause [or necessity] finding,” United States v. Nanni, 59 F.3d 1425, 1433 (2d Cir.1995). If the untainted portions of the application are sufficient to support the probable cause or necessity findings, then the misstatements are not “material” and suppression is not required.

Although omissions “are governed by the same rules” as misstatements, United States v. Ferguson, 758 F.2d 843, 848 (2d Cir.1985), “the literal Franks approach [does not] seem[] adequate because, by their nature, omissions cannot be deleted”; therefore “[a] better approach ... would be to ... insert the omitted truths revealed at the suppression hearing,” United States v. Ippolito, 774 F.2d 1482, 1487 n. 1 (9th Cir.1985). Accordingly, we have held that “[t]he ultimate inquiry is whether, after putting aside erroneous information and [correcting] material omissions, there remains a residue of independent and lawful information sufficient to support [a finding of] probable cause [or necessity].” Canfield, 212 F.3d at 718 (internal quotation marks omitted); see also United States v. Martin, 615 F.2d 318, 328 (5th Cir.1980) (“[W]e [are] required to determine whether, if the omitted material had been included in the affidavit, the affidavit would still establish probable cause [or necessity].... If it would not, we would be required to void the warrant and suppress the evidence seized pursuant to it.”).

ii. The Hearing

On August 12, 2010, the District Court found that Rajaratnam had “made a substantial preliminary showing” that the government recklessly omitted “several key facts” relating to .the “necessity” of wiretapping. United States v. Rajaratnam, No. 09 Cr. 1184(RJH), 2010 WL 3219333, at *2 (S.D.N.Y. Aug. 12, 2010). The Court therefore ordered a Franks hearing on the “necessity” issue. Id. However, the District Court rejected Rajaratnam’s argument that any misstatements or omissions were material to the existence of probable cause. Id. at *1 n. 2. Accordingly, the Court did not hold a Franks hearing on the “probable cause” issue.

At the Franks hearing, which began on October 4, 2010, the District Court heard testimony from (1) Linda Beaudreault, counsel to Galleon and Rajaratnam; (2) Andrew Michaelson, an SEC staff attorney who, after the wiretap was authorized, became a Special United States Attorney in order to participate in the investigation by the USAO; (3) FBI Special Agent Kang, the wiretap application affiant; and (4) former AUSA Goldberg, who filed the March 7, 2008 wiretap application.

The Franks hearing focused on the alleged misstatements and omissions in the wiretap application. Accordingly, we briefly describe those asserted misstatements and omissions as well as the evidence about the states of the mind of the government agents who filed the wiretap application.

a. Misstatements and Omissions Involving Roomy Khan (CS-1): “Probable Cause”

As noted, Khan served as a cooperating witness for the government and recorded various phone conversations with Rajarat-nam, some of which were summarized in the wiretap application and cited as evidence of probable cause.

The District Court determined that the government’s wiretap application made two misstatements with regard to Khan’s background. First, the wiretap application stated that Khan “has not yet been charged with any crimes,” Joint App’x 77, when, in fact, she had a prior felony fraud conviction, see Rajaratnam, 2010 WL 4867402, at *10. Second, the application stated that Khan “has been cooperating with the FBI since approximately November 2007,” Joint App’x 77 n.4, when, in fact, she had cooperated in an earlier insider trading investigation of Rajaratnam which began in the late 1990s, see Rajarat-nam, 2010 WL 4867402, at *10.

Moreover, the District Court found to be misleading two statements that the government had paraphrased from recorded conversations between Khan and Rajaratnam. With regard to the first paraphrased conversation, the wiretap application stated that, when Khan asked Rajaratnam whether he was “getting anything on Intel,” Rajaratnam said “that Intel would be up 9 to 10% and then guide [sic] down 8% and that margins would be good.” Joint App’x 79. In fact, Rajarat-nam had qualified his predictions about Intel’s margins by saying, “I think.” Rajaratnam, 2010 WL 4867402, at *11. The wiretap application also paraphrased a second conversation between Khan and Rajaratnam as follows:

During this call, CS-1 \i.e., Khan] asked whether RAJARATNAM had heard anything on Xilinx. RAJARATNAM responded that he thought this quarter would be okay, but next quarter would not be so good.... RAJARATNAM then said that he expected Xilinx to be “below the street.” CS-1 asked whether he got “it” from someone at the company and RAJARATNAM said yes, somebody who knows.

Joint App’x 80-81 (emphasis supplied). This paraphrase, however, differed from Rajaratnam’s actual answer to Khan’s question; instead of saying “somebody who knows,” Rajaratnam had in fact said, “Yeah I mean, somebody who knows his stuff.” Rajaratnam, 2010 WL 4867402, at *11. The District Court found this actual response to be “more equivocal than the government’s paraphrase....” Id.

b. Omissions Involving the Earlier SEC Investigation: “Necessity”

In addition to the misstatements and omissions involving Khan, the wiretap application also omitted certain information, which was relevant to the issue of “necessity,” regarding the ongoing investigation of Rajaratnam being conducted by the SEC. Judge Holwell noted that the Franks hearing established that the wiretap application did not disclose “that the SEC had for several years been conducting an extensive investigation into the very same activity the wiretap was intended to expose[,] using many of the same techniques the affidavit casually affirmed had been or were unlikely to be successful.” Id. at *15. Judge Holwell called this a “glaring omission,” and stated that he was “at a loss to understand how the government could have ever believed that Judge Lynch could determine whether a wiretap was necessary ... without knowing about the most important part of th[e] investigation — the millions of documents, witness interviews, and the actual deposition of Rajaratnam himself....” Id.

Specifically, the SEC investigation had consisted of the following. In September 2006, the SEC began an investigation of Sedna Capital Management LLC, a hedge fund managed by Rajaratnam’s brother. As a result of that investigation, the SEC began focusing on Galleon and Rajarat-nam. Beginning in early 2007, the SEC started an on-site investigation of Galleon, through which the SEC: (1) received four million pages of documents and subpoenaed records; (2) interviewed Rajaratnam twice and eighteen Galleon employees as well; and (3) formally deposed Rajaratnam under oath. See id. at *15-16. In March 2007, the SEC briefed the USAO and the FBI about the investigation and gave these entities access to its investigation files.

The District Court was troubled not only by the fact that the wiretap application did not disclose the existence of the SEC investigation, but also by the apparent consequence that this omission made other statements in the application misleading. See id. at *17-18. For example, the wiretap application asserted that interviewing or arresting Rajaratnam or other target subjects “is too risky at the present time,” Joint App’x 108-09, despite the fact that the SEC had already interviewed and deposed Rajaratnam. Similarly, the application asserted that requesting additional trading records “would jeopardize the investigation” because “clearing firms ... sometimes alert the traders to the requests,” id. at 108, even though the SEC had obtained more than four million documents from Galleon.

c. The States of Mind of the Wiretap Applicants

Finally, at the Franks hearing, the government presented testimony designed to demonstrate that the alleged “omission” in the wiretap application regarding the SEC investigation, such as it may have been, was not made with “reckless disregard for the truth.” In particular, former AUSA Goldberg testified that “[njobody tried to hide” the existence of the SEC investigation. Franks Tr. 773. Goldberg also expressed her view that “it would be obvious to anyone reading the affidavit that the SEC was” giving certain information to prosecutors and agents investigating criminal charges. Id.; see also Gov’t’s Br. 23 (noting that “Special Agent Kang’s affidavit referenced” in seven different places the USAO’s and FBI’s “reliance on documents collected or information provided by the SEC”). Moreover, the government asserts that, because of recent court decisions arising out of the improper use of civil SEC investigations for criminal prosecutions, the USAO “took pains not to direct the SEC’s investigative actions” and, in a similar vein, “did not view the SEC Staff investigation as an alternative law enforcement means to investigate Rajarat-nam and his associates.” Id. at 24 (citations omitted).

iii. The District Court’s Decision on Rajaratnam’s Suppression Motion

On November 24, 2010, the District Court denied Rajaratnam’s suppression motion. See Rajaratnam, 2010 WL 4867402, at *28. In analyzing the government’s wiretap application under Franks, the District Court made three central findings.

First, on the issue of “probable cause,” the District Court held that a Franks hearing was unnecessary because the alleged misstatements and omissions in the wiretap application regarding Khan’s prior conviction and cooperation as well as the assertedly misleading paraphrased conversations between Khan and Rajaratnam were not material. Although the District Court found “[p]articularly disturbing ... the omission of highly-relevant information regarding Khan’s prior criminal record,” it held that other indicia of Khan’s reliability, along with the accurate statements in the wiretap application, “suffice[d] for probable cause” purposes. Id. at *11-13.

Second, on the issue of “necessity,” the District Court held that the omission of the SEC’s investigation of Rajaratnam was made with “reckless disregard for the truth.” See id. at *19. The District Court summarized its standard for determining whether “reckless disregard” existed as follows: “Rajaratnam must prove that the drafters of the affidavit either intentionally omitted the information or that the omitted information was clearly critical to the affidavit, thereby raising an inference of recklessness.” Id. at *9 (relying on United States v. Reilly, 76 F.3d 1271, 1280 (2d Cir.1996) (noting that recklessness “may be inferred when omitted information was clearly critical to assessing the legality of the search” (internal quotation marks omitted))).

Third, although the District Court determined that information regarding the SEC investigation was omitted with “reckless disregard for the truth,” it concluded that suppression was not warranted because Rajaratnam had failed to show that the omission was “material” to the Court’s determination of “necessity.” In particular, the District Court held that,

while the SEC investigation ... was the bedrock of the prosecutor’s own criminal investigation, the SEC investigation had nevertheless failed to fully uncover the scope of Rajaratnam’s alleged insider trading ring and was reasonably unlikely to do so because evidence suggested that Rajaratnam and others conducted their scheme by telephone. Accordingly, disclosure of all the details of the SEC’s investigation ... would ultimately have shown that a wiretap was necessary and appropriate.

Id. at *1.

D. Other Procedural History

Rajaratnam’s trial began on March 8, 2011. On May 11, 2011 — after a seven-week trial and twelve days of deliberation — the jury returned a verdict convicting Rajaratnam on all nine counts of securities fraud and all five counts of conspiracy to commit securities fraud. On October 13, 2011, Judge Holwell sentenced Rajaratnam to a term of 132 months’ imprisonment, followed by 2 years of supervised released. Judge Holwell also ordered Rajaratnam to forfeit $53,816,434 and pay a $10,000,000 fine.

This appeal followed.

DISCUSSION

I. Applying the Analytical Framework of Franks to a Title III Wiretap Application

Rajaratnam first argues that the District Court erred by using the analytical framework set forth in Franks v. Delaware, 438 U.S. 154, 98 S.Ct. 2674, 57 L.Ed.2d 667 (1978) — which involved a warrant application for a physical search, not a wiretap^ — to determine whether the alleged misstatements and omissions in the government’s wiretap application required suppression. In particular, he takes issue with the “post hoc factual justification,” Rajaratnam’s Br. 30, that the Franks framework allows — ie., (1) removing misstatements from the application, see Co-reas, 419 F.3d at 155; and (2) “inserting] the omitted truths revealed at the suppression hearing” after the fact, Ippolito, 774 F.2d at 1487 n. 1, to determine whether the application would have been granted in any event. Simply put, he asserts that the statute authorizing Title III wiretaps requires suppression because the government’s wiretap application did not provide the “full and complete statement” regarding probable cause and necessity, as required by 18 U.S.C. § 2518(l)(b)-(e).

Rajaratnam’s argument is foreclosed by settled precedent. In United States v. Bianco, 998 F.2d 1112 (2d Cir.1993), abrogated on other grounds by Groh v. Ramirez, 540 U.S. 551, 124 S.Ct. 1284, 157 L.Ed.2d 1068 (2004), we noted our agreement “with the district court’s application of Franks and with its findings” where the government submitted a Title III application for a “roving bug” but omitted information concerning the location where the communications were to be intercepted, as required under Title III. Id. at 1126. Like Rajaratnam, the defendant in Bianco specifically asserted that Franks was inapplicable and that its application would vitiate Title Ill’s “full and complete” statement requirement. Id. at 1125-26. Despite this argument, we held that the “[u]se of the Franks standard is consistent with the purposes of [Title III],” and “[i]f anything, Franks enhances the protection of ... defendants, by applying to the wiretap statute an important constitutional principle that has been accepted by all courts.” Id. at 1126. And in United States v. Miller, 116 F.3d 641 (2d Cir.1997), we applied the analytical framework of Franks to a Title III wiretap application that “omitted material information that had been provided by informants who were cooperating with the State.” Id. at 664. In particular, we held that “[a] challenge to the veracity of such an affidavit will succeed only when it establishes intentional or reckless omissions or false statements that are ‘necessary to the finding of probable cause’ supporting the wiretap authorization.” Id. (quoting Franks, 438 U.S. at 156, 98 S.Ct. 2674).

Finally, we note that the cases relied on by Rajaratnam — United States v. Giordano, 416 U.S. 505, 94 S.Ct. 1820, 40 L.Ed.2d 341 (1974), and United States v. Gigante, 538 F.2d 502 (2d Cir.1976) — are not to the contrary. Both cases were decided before the Supreme Court’s decision in Franks and “[a]t that time there was no good-faith or other exception to the judicially crafted exclusionary rule for violations of the fourth amendment.” Bianco, 998 F.2d at 1126. When Title III was enacted, it was not intended “generally to press the scope of the suppression role beyond [then current] search and seizure law.” S.Rep. No. 90-1097, at 96 (1968), reprinted in 1968 U.S.C.C.A.N. 2112, 2185. But thereafter, Franks and other cases, including United States v. Leon, 468 U.S. 897, 104 S.Ct. 3405, 82 L.Ed.2d 677 (1984), “narrowed the circumstances in which ... [courts] apply the exclusionary rule.” Bianco, 998 F.2d at 1126. Although courts were once thought to face a “dilemma of whether [or not] to apply the Franks standard to Title III cases,” id., that supposed dilemma has been definitively resolved, and every Court of Appeals to consider the issue has concluded that the analytical framework of Franks is an appropriate standard against which to review allegedly deficient Title III wiretap applications.

In light of these precedents of our Court and our sister Circuits, we hold that the District Court did not err by applying the analytical framework of Franks to determine whether the government’s wiretap application required suppression.

II. Applying Franks to the Government’s Wiretap Application

As noted, Title III requires government agents who file a wiretap application to provide “a full and complete statement of the facts and circumstances relied upon by the application” to establish probable cause, 18 U.S.C. § 2518(l)(b), and a “full and complete statement as to whether or not other investigative procedures have been tried and failed or why they reasonably appear to be unlikely to succeed if tried or to be too dangerous,” id. § 2518(l)(c). We consider the relevant standard of review and apply it in turn to the District Court’s “necessity” and “probable cause” determinations.

A. Standards of Review

It is an axiom of appellate procedure that we review legal questions de novo and questions of fact for clear error. See Pierce v. Underwood, 487 U.S. 552, 558, 108 S.Ct. 2541, 101 L.Ed.2d 490 (1988). That axiom holds true in the context of Franks hearings, see Awadallah, 349 F.3d at 65; United States v. Moore, 968 F.2d 216, 220-21 (2d Cir.1992), and therefore our review is similar for each of the issues in this appeal. For instance, whether a person acted with “reckless disregard for the truth” is “a factual question of intent, and we therefore review the court’s decision for clear error,” United States v. Trzaska, 111 F.3d 1019, 1028 (2d Cir.1997), but a district court’s understanding of the “reckless disregard” standard is reviewed de novo. Similarly, we review for “clear error” the factual findings that underpin a district court’s assessment of probable cause, but we review de novo whether a set of facts satisfies the probable cause standard. See Ornelas v. United States, 517 U.S. 690, 699, 116 S.Ct. 1657, 134 L.Ed.2d 911 (1996). Along the same lines, whether a misstatement or omission is “material” — ie., “[w]hether the untainted portions [of the affidavit] suffice to support a probable cause [or necessity] finding,” Canfield, 212 F.3d at 717 (citation omitted) — is a mixed question of law and fact reviewed de novo, see Awadallah, 349 F.3d at 65, but any underlying factual findings are reviewed for “clear error.” An appellate court recognizes “clear error” only when it “is left with a definite and firm conviction that a mistake has been committed.” Brown v. Plata, — U.S. -, 131 S.Ct. 1910, 1930, 179 L.Ed.2d 969 (2011) (internal quotation marks omitted).

B. “Necessity”: Did the District Court Err in Concluding that the Wiretap Application Omitted Information About the SEC Investigation with “Reckless Disregard for the Truth”?

Rajaratnam maintains that the District Court correctly concluded that government agents omitted information about the SEC investigation of Rajarat-nam from the wiretap application with “reckless disregard for the truth.” In turn, the government argues that the District Court incorrectly applied the “reckless disregard” standard.

The Supreme Court in Franks held that misstatements or omissions caused by “negligence or innocent mistake[s]” do not warrant suppression. 438 U.S. at 171, 98 S.Ct. 2674. This inquiry, which looks to the mental states of mind of government officials, is said to be a “subjective” test rather than an “objective” one. See, e.g., Farmer v. Brennan, 511 U.S. 825, 838-40, 114 S.Ct. 1970, 128 L.Ed.2d 811 (1994) (discussing the difference between “subjective” and “objective” tests). Whether an individual had a particular mental state “is a question of fact subject to demonstration in the usual ways, including inference from circumstantial evidence,” id. at 842, 114 S.Ct. 1970, but courts must not “confus[e] a mental state with the proof of its existence,” id. (quotation marks omitted).

Relying on our decision in United States v. Reilly, 76 F.3d 1271, 1280 (2d Cir.1996), the District Court stated that “with respect to material omissions from the March 7, 2008 affidavit, Rajaratnam must prove that the drafters of the affidavit either intentionally omitted the information or that the omitted information was clearly critical to the affidavit, thereby raising an inference of recklessness.” Rajaratnam, 2010 WL 4867402, at *9. Based on our review of the record, we conclude that the District Court erred in applying the “reckless disregard” standard because the District Court failed to consider the actual states of mind of the wiretap applicants.

A wiretap applicant does not necessarily act with “reckless disregard for the truth” simply because he or she omits certain evidence that a reviewing court, in its judgment, considers to be “clearly critical.” Rather, the reviewing court must be presented with credible and probative evidence that the omission of information in a wiretap application was “designed to mislead” or was “made in reckless disregard of whether [it] would mislead.” Awadallah, 349 F.3d at 68 (emphasis and internal quotation marks omitted). As we have said:

“An affiant cannot be expected to include in an affidavit every piece of information gathered in the course of an investigation. However, every decision not to include certain information in the affidavit is ‘intentional’ insofar as it is made knowingly. If ... this type of ‘intentional’ omission is all that Franks requires, the Franks intent prerequisite would be satisfied in almost every case.... [Rather,] Franks protects against omissions that are designed to mislead, or that are made in reckless disregard of whether they would mislead, the magistrate.”

Id. at 67-68 (quoting United States v. Colkley, 899 F.2d 297, 300-01 (4th Cir.1990) (alterations in Awadallah; emphases in Colkley)). In a similar vein, the Seventh Circuit has explained:

To prove reckless disregard for the truth, the defendants [must] prove that the affiant in fact entertained serious doubts as to the truth of his allegations. Because states of mind must be proved circumstantially, a factfinder may infer reckless disregard from circumstances evincing obvious reasons to doubt the veracity of the allegations.

United States v. Whitley, 249 F.3d 614, 621 (7th Cir.2001) (internal quotation marks and alterations omitted); see also United States v. Williams, 718 F.3d 644, 649-50, No. 11-3129, 2013 WL 2149897, at *5 (7th Cir. May 20, 2013) (applying the subjective standard for recklessness to omissions from an affidavit). But see Wilson v. Russo, 212 F.3d 781, 788 (3d Cir.2000) (“[0]missions are made with reckless disregard if an officer withholds a fact in his ken that ‘[a]ny reasonable person would have known ... was the kind of thing the judge would wish to know.’ ” (quoting United States v. Jacobs, 986 F.2d 1231, 1235 (8th Cir.1993))).

Of course, the “reckless disregard” aspect of a Franks inquiry can sometimes be inferred from the omission of critical information in a wiretap application. See Rivera v. United States, 928 F.2d 592, 604 (2d Cir.1991) (“Recklessness may be inferred where the omitted information was clearly critical to the probable cause determination.” (emphasis supplied) (internal quotation marks omitted)). Subjective intent, after all, is often demonstrated with objective evidence. But such an inference is not to be automatically drawn simply because a reasonable person would have included the omitted information, cf. Farmer, 511 U.S. at 842, 114 S.Ct. 1970, and the inference is particularly inappropriate where the government comes forward with evidence indicating that the omission resulted from nothing more than negligence, or that the omission was the result of a considered and reasonable judgment that the information was not necessary to the wiretap application.

In this case, Judge Holwell’s view that the SEC investigation was “clearly critical” is the only basis for his conclusion that the government omitted certain information about that investigation with “reckless disregard for the truth.” But as we now review all of the evidence presented at the Franks hearing, it points in the opposite direction. And, despite the inferences that Judge Holwell drew from the omitted “clearly critical” information, when discussing the subjective state of mind of each affiant, he too “comfortably con-elude[d] that no one acted with the deliberate intent to mislead Judge Lynch.” Ra-jaratnam, 2010 WL 4867402, at *19.

The evidence presented at the Franks hearing showed that no one in the USAO acted with “reckless disregard for the truth” by not detailing the SEC investigation of Rajaratnam. Former AUSA Goldberg testified that, when she “was drafting the affidavit, it never occurred to [her], never crossed [her] mind to put a section in [the wiretap application] that [discussed the] SEC investigation .... [because] [she] didn’t think about the SEC investigation as an alternative technique that was available to FBI agents, because [the USAO] can’t direct them what to do.” Franks Tr. 819. Similarly, FBI Special Agent Kang testified that he “didn’t think about including [the SEC investigation] in a criminal affidavit.... We just didn’t really think about it.” Although the District Court believed that this civil investigation by the SEC was relevant to the issue of necessity, the evidence presented at the Franks hearing in no way suggested that omitting certain information about SEC investigation was “designed to mislead” or was made with “reckless disregard of whether [it] would mislead.” Awa-dallah, 349 F.3d at 68 (internal quotation marks and emphasis omitted). Indeed, the evidence indicates that the wiretap application was reviewed by supervisors at the USAO, none of whom thought that additional information about the SEC’s civil investigation needed to be included.

On a more fundamental level, we cannot conclude that the government omitted certain information about the SEC investigation with “reckless disregard for the truth” when it is clear that fully disclosing the details of that investigation would only have strengthened the wiretap application’s “necessity” showing. The District Court tacitly recognized this fact, stating that “[m]any of the same documents that were used to compile the SEC chronologies strongly suggested that Rajaratnam had been careful to exchange nearly all of his inside information by telephone.” Rajaratnam, 2010 WL 4867402, at *21; cf. United States v. Young, 822 F.2d 1234, 1237 (2d Cir.1987) (“[W]iretapping is particularly appropriate when the telephone is routinely relied on to conduct the criminal enterprise under investigation.” (quotation marks omitted)). The District Court made the point explicitly in discussing whether the government should have pursued additional “normal investigative procedures” before seeking a Title III wiretap:

Could or should the government have done more with conventional techniques to test whether a wiretap was “necessary”? It is hard to make that argument with regard to document subpoenas, search warrants, and other forms of documentary investigation. Over four million documents from targets and third parties had already been gathered. Analysis of the documentary evidence was fairly sophisticated and while this revealed much circumstantial evidence of insider trading it also confirmed what one would expect: insider trading is typically conducted verbally. Thus it seems reasonably unlikely that additional documents would have produced qualitatively different evidence.

Rajaratnam, 2010 WL 4867402, at *22 (emphasis supplied).' In other words, the evidence does not support the inference that the government omitted information from a wiretap application with “reckless disregard for the truth,” and such an inference seems ever more inappropriate where the information omitted would only have further supported the government’s position.

After reviewing the evidence in the record-especially the Franks hearing testimony regarding the states of mind of the government agents-and applying the correct understanding of reckless disregard, we conclude that the record does not support the finding that the omission of the SEC investigation in the Title III wiretap application was made with “reckless disregard for the truth.”

In any event, even if we were to assume, arguendo, the opposite conclusion-that government officials omitted information about the SEC investigation with “reckless disregard for the truth”we are persuaded that this omission was not material, substantially for the reasons stated in the District Court’s analysis on that issue. Rajaratnam, 2010 WL 4867402, at *21-24 (holding that the wiretap application, as corrected, was sufficient to support a finding of “necessity”).

C. “Probable Cause”: Did the District Court Correctly Determine that the Wiretap Application’s Misstatements About Khan and “Paraphras-ings” Did Not Require Suppression?

Rajaratnam argues that the District Court also erred by concluding that the alleged deficiencies in the wiretap application regarding probable cause were not “material,” and therefore that suppression was not required. Specifically, he argues that the “probable-cause determination comes up short when the materially false and misleading allegations of probable' cause are eliminated, because the government’s recklessly false and misleading claims about Roomy Khan and her conversations with [Rajaratnam] were the heart of its probable cause allegations.” Rajaratnam’s Br. 50.

We disagree because, even assuming, arguendo, that these alleged misstatements and omissions regarding Roomy-Khan and the two paraphrased conversations between Khan and Rajaratnam were indeed made with “reckless disregard for the truth,” we agree with the District Court that they were not “material,” substantially for the reasons stated in the District Court’s analysis on that issue. See Rajaratnam, 2010 WL 4867402, at *11-13 (“Adding it all up, and correcting the affidavit to account for the government’s misstatements and omissions, the Court believes that there were enough facts for Judge Lynch to have found probable cause.”).

III. Were the Jury Instructions on the Use of Inside Information Erroneous?

Finally, Rajaratnam argues that his convictions on the substantive securities fraud counts (Counts 6 through 14) should be vacated because the District Court instructed the jury that it could convict Rajaratnam if the “material nonpublic information given to the defendant was a factor, however small, in the defendant’s decision to purchase or sell stock.” Joint App’x 433 (emphasis supplied). In particular, he asserts that the emphasized fragment of the jury instructions allowed the jury to convict him without finding the necessary causal connection between the inside information he possessed and the trades he executed. On appeal, our review of jury instructions for legal error is de novo. See United States v. Robinson, 702 F.3d 22, 30 (2d Cir.2012).

“Insider trading — unlawful trading in securities based on material non-public information — is well established as a violation of section 10(b) of the Securities Exchange Act of 1934 and Rule 10b-5.” SEC v. Obus, 693 F.3d 276, 284 (2d Cir.2012). There are two theories of insider trading: (1) a “classical theory” involving corporate insiders, and (2) a “misappropriation theory” involving “persons who are not corporate insiders but to whom material non-public information has been entrusted in confidence and who breach a fiduciary duty to the source of the information to gain personal profit in the securities market.” Id. The second of these theories is at issue in this case. As relevant here, it “holds that a person commits fraud ‘in connection with’ a securities transaction, and thereby violates § 10(b) and Rule 10b-5, when he misappropriates confidential information for securities trading purposes, in breach of a duty owed to the source of the information.” United States v. O’Hagan, 521 U.S. 642, 652, 117 S.Ct. 2199, 138 L.Ed.2d 724 (1997).

In United States v. Teicher, 987 F.2d 112 (2d Cir.1993), we stated, in dicta, that a “knowing possession” standard satisfied the “in connection with” requirement of § 10(b) and Rule 10b-5, see note 22, ante, despite the defendant’s argument that a “causal connection” was required between the inside information and the executed transaction. Id. at 120-21. In discussing the appropriate standard, we noted that “[a] number of factors weigh in favor of a ‘knowing possession’ standard,” including that: (1) “§ 10(b) and Rule 10b-5 require only that a deceptive practice be conducted in connection with the purchase or sale of a security”; (2) “a knowing possession standard comports with the oft-quoted maxim that one with a fiduciary or similar duty to hold material nonpublic information in confidence must either disclose or abstain with regard to trading”; and (3) “a knowing possession standard has the attribute of simplicity.” Id. at 120 (internal quotation marks omitted).

Fifteen years later, in United States v. Royer, 549 F.3d 886 (2d Cir.2008), we elevated the dicta of Teicher to the law of the Circuit, when we “adhere[d] to the knowing possession standard articulated in Teicher,” which “was the product of sustained and detailed consideration.Id. at 899. In doing so, we noted that no developments since Teicher persuaded us to resolve the issue differently and, “[o]n the contrary, the SEC[’s] subsequent! ] enactment] [of] Rule 10b5-1,” counseled in favor of applying the knowing possession standard. Id.

Like the jury instructions in Royer; the phrase deployed by Judge Holwell (“was a factor, however small”) was “if anything more favorable,” id. at 899 n. 12 (emphasis supplied), to Rajaratnam than the “knowing possession” standard that is the law of this Circuit. Instead of instructing the jury that “[i]t is sufficient if the government proves that the defendant!] purchased or sold securities while knowingly in possession of the material nonpublic information,” Teicher, 987 F.2d at 119, the instructions given by Judge Holwell, if anything, went beyond the “knowing possession” standard because they required that the inside information be “a factor, however small, in the defendant’s decision to purchase or sell stock,” Joint App’x 433.

Undeterred by these precedents, Rajar-atnam argues that the Supreme Court’s recent decision in CSX Transportation, Inc. v. McBride, — U.S.-, 131 S.Ct. 2630, 180 L.Ed.2d 637 (2011), casts doubt on the law of our Circuit. We are not persuaded. In CSX Transportation, the Supreme Court held that, under the Federal Employers’ Liability Act (“FELA”), a railroad worker need only demonstrate that the railroad’s negligence “played a part — -no matter how small — in bringing about the injury.” Id. at 2644. Rajarat-nam relies on CSX Transportation because the Court noted that the statutory causation requirement in the FELA — that the injury “result!] in whole or in part from [the defendant’s] negligence” — was “as broad as could be framed,” id. at 2636, and it contrasted the FELA causation requirement with “traditional notions of proximate causation under the RICO, antitrust, and securities fraud statutes,” id. at 2644 n. 14 (emphasis supplied). In substance, Rajaratnam contends that the Supreme Court’s reasoning in CSX Transportation implies that securities fraud cases require some causation element greater than the formulation that the inside information “played a part — no matter how small.”

To the contrary, the Supreme Court’s statements about the FELA causation requirement and the causation requirement in “securities fraud statutes,” do not call into question our decisions in Royer and Teicher. Indeed, the Supreme Court’s reference to “securities fraud statutes” in CSX Transportation was accompanied by a citation to Dura Pharmaceuticals, Inc. v. Broudo, 544 U.S. 336, 125 S.Ct. 1627, 161 L.Ed.2d 577 (2005), which addressed suits for civil fraud — not criminal fraud prosecutions. Compare id. at 343, 125 S.Ct. 1627 (“Judicially implied private securities fraud actions resemble in many (but not all) respects common-law deceit and misrepresentation actions.”), with Neder v. United States, 527 U.S. 1, 25, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999) (“By prohibiting the ‘scheme to defraud,’ rather than the completed fraud, the elements of reliance and damage would clearly be inconsistent with the statutes Congress enacted.”). Moreover, as noted above, the District Court’s jury instructions are consistent with SEC Rule 10b5-l, which clarifies that “a purchase or sale of a security of an issuer is ‘on the basis of material nonpublic information ... if the person making the purchase or sale was aware of the material nonpublic information when the person made the purchase or sale.” 17 C.F.R. § 240.10b5-l(b) (emphasis supplied).

CONCLUSION

To summarize, we hold that:

(1) The District Court properly analyzed the misstatements and omissions in the government’s Title III wiretap application under the analytical framework prescribed by the Supreme Court in Franks v. Delaware, 438 U.S. 154, 98 S.Ct. 2674, 57 L.Ed.2d 667 (1978);
(2) The alleged misstatements and omissions in the wiretap application did not require suppression, because (a) contrary to the District Court’s conclusion, the government did not omit information about the SEC investigation of Rajarat-nam with “reckless disregard for the truth,” and (b) as the District Court correctly concluded, all of the alleged misstatements and omissions in the wiretap application were not “material”;
(3)The District Court’s jury instructions on the use of inside information— which instructed the jury that it could convict Rajaratnam if the “material nonpublic information given to the defendant was a factor, however small, in the defendant’s decision to purchase or sell stock” — satisfied the “knowing possession” standard that is the law of this Circuit.

For the reasons stated, we AFFIRM the District Court’s October 25, 2011 judgment of conviction. 
      
      . In his capacity then as a United States District Judge, Judge Gerard E. Lynch approved the first wiretap application on March 7, 2008. As the wiretap authorization expired after a 30-day period, the government filed subsequent wiretap applications. Ultimately, eight wiretap applications were approved by six judges of the United States District Court for the Southern District of New York, including: Judge Lynch, Judge Denise Cote, Judge Deborah A. Batts, Judge Laura Taylor Swain, Judge Richard J. Sullivan, and Judge Denny Chin. Like the parties and the District Court below, because "[t]he first 30 days of wiretapping Rajaratnam yielded enough evidence of criminal conduct to justify renewals of the wiretap,” we focus only on the initial wiretap application approved by Judge Lynch. See United States v. Rajaratnam, No. 09 Cr. 1184(RJH), 2010 WL 4867402, at *7 n. 11 (S.D.N.Y. Nov. 24, 2010).
     
      
      . For ease of reference, we refer to the government’s wiretap application and its accompanying affidavit as “the wiretap application.” See note 1, ante.
      
     
      
      . In full, 18 U.S.C. § 2518(l)(b)-(c) provides: (1) Each application for an order authorizing or approving the interception of a wire, oral, or electronic communication under this chapter shall be made in writing upon oath or affirmation to a judge of competent jurisdiction and shall state the applicant's authority to make such application. Each application shall include the following information:
      
        (b) a full and complete statement of the facts and circumstances relied upon by the applicant, to justify his belief that an order should be issued, including (i) details as to the particular offense that has been, is being, or is about to be committed, (ii) except as provided in subsection (11), a particular description of the nature and location of the facilities from which or the place where the communication is to be intercepted, (iii) a particular description of the type of communications sought to be intercepted, (iv) the identity of the person, if known, committing the offense and whose communications are to be intercepted;
      (c) a full and complete statement as to whether or not other investigative procedures have been tried and failed or why they reasonably appear to be unlikely to succeed if tried or to be too dangerous ....
     
      
      . Even though Judge Holwell denied Rajarat-nam’s request for a Franks hearing regarding probable cause, Rajaratnam asked the District Court to reconsider its prior probable cause determination. In its November 24, 2010 Memorandum Opinion and Order, the District Court explained why the alleged misstatements and omissions regarding probable cause were not material. See Rajaratnam, 2010 WL 4867402, at *11-13. In particular, Judge Holwell found that "[ajdding ... all [the evidence] up, and correcting the affidavit to account for the government's misstatements and omissions, the Court believes that there were enough facts for Judge Lynch to have found probable cause.” Id. at *13.
     
      
      . Even though the District Court concluded that Rajaratnam had failed to demonstrate that a Franks hearing was needed on the issue of probable cause, we describe below all the alleged misstatements, including those relating to the "probable cause” determination.
     
      
      . The paraphrased portion of the conversation also omitted "a piece of the conversation in which Rajaratnam said that he thought margins the next quarter 'will be below,’ and explained that he took this view '[b]ecause of [sic] the volumes are down, right?' ” Rajarat-nam, 2010 WL 4867402, at *11 (quoting transcripts of the recorded conversations).
     
      
      . The government contends that the wiretap application "disclosed, among other things, information provided to the [USAO and FBI] by the SEC and by Khan” and notes that the wiretap application “reli[ed] on documents collected or information provided by the SEC in seven different places.” Gov't’s Br. 23. Although the wiretap application refers to the fact that the SEC had passed along some information pertinent to the criminal investigation of Rajaratnam, see, e.g., Joint App’x 99 ("Based on conversations with the [SEC], I have learned that Santhanam is a risk manager and portfolio manager at Galleon.”), it does not suggest the extent of the SEC investigation, the fact that interviews and depositions were taken of Galleon employees, including Rajaratnam, or the fact that the SEC gained access to millions of documents.
     
      
      . The wiretap application did state that "certain" trading records had been reviewed. See Joint App'x 108.
     
      
      . The District Court's analysis necessarily assumes that notice of an investigation by civil authorities would be viewed by Rajaratnam, or the general public, as notice of an investigation by criminal authorities. We doubt that this is the case.
     
      
      .The District Court concluded that a Franlcs hearing was not warranted on the "probable cause” issue because the alleged misstatements and omissions were not "material.” Rajaratnam, 2010 WL 3219333, at *1 n. 2 (rejecting Rajaratnam’s argument that the misstatements or omissions were material to the existence of probable cause). It is unclear, however, whether the District Court made a finding regarding whether the misstatements and omissions on the issue of probable cause were made with "reckless disregard for the truth.” Although Judge Hol-well noted that these deficiencies in the wiretap application "[do not] win high marks for candor” and "evince a lack of [the] frankness that should be found in all ex parte applications,” he made no explicit finding that these deficiencies were made with reckless disregard for the truth. Rajaratnam, 2010 WL 4867402, at *10-11.
     
      
      . These "indicia” of reliability included the fact that: (1) Khan was a “known informant”; (2) she made statements against "her own penal interest”; and (3) the government was able to corroborate some of her statements. Rajaratnam, 2010 WL 4867402, at *12.
     
      
      . Although the District Court held that information about the SEC investigation had been omitted with "reckless disregard for the truth,” it "comfortably conclude[d] that no one acted with the deliberate intent to mislead Judge Lynch.” Rajaratnam, 2010 WL 4867402, at *19.
     
      
      . Rajaratnam does not challenge any aspect of his sentence on appeal.
     
      
      . A "roving bug” is "interception by electronic means of oral communications without specifying in advance exactly where or when the interception would occur.” Bianco, 998 F.2d at 1117-18.
     
      
      . In particular, 18 U.S.C. § 2518(l)(b)(ii) requires:
      (1) Each application for an order authorizing or approving the interception of a wire, oral, or electronic communication under this chapter shall be made in writing upon oath or affirmation to a judge of competent jurisdiction and shall state the applicant's authority to make such application. Each application shall include the following information ... (ii) except as provided in subsection (11), a particular description of the nature and location of the facilities from which or the place where the communication is to be intercepted....
     
      
      . In addition to these two precedents of our Court, every Court of Appeals to have considered this question has relied on Franlcs to analyze whether alleged misstatements and omissions in Title III wiretap applications warrant suppression. See, e.g., United States v. Poulsen, 655 F.3d 492, 505 (6th Cir.2011) (affirming a district court’s denial of a Franlcs hearing where defendant had failed to make a preliminary showing that a Title III wiretap application affiant had “made any of the purportedly false statements intentionally or with reckless disregard for their truth” (quotation marks omitted)); United States v. Becton, 601 F.3d 588, 597-98 (D.C.Cir.2010) (applying Franlcs to a challenge to a Title III wiretap application); United States v. Small, 423 F.3d 1164, 1172 (10th Cir.2005) (“This court reviews alleged misrepresentations and omissions in a wiretap application under ... Franks v. Delaware." (citation omitted)); United States v. Gonzalez, Inc., 412 F.3d 1102, 1110 (9th Cir.2005) (“To obtain a Franlcs hearing, defendants must make a preliminary showing that the wiretap applications contained material misrepresentations or omissions.”); United States v. Guerra-Marez, 928 F.2d 665, 670 (5th Cir.1991) (“Where, as here, the affidavit falls ... within the dictates of section 2518, application of the Franlcs standard is ... appropriate.”); United States v. Leisure, 844 F.2d 1347, 1354-57 (8th Cir.1988) ("noting that affidavits in support of electronic surveillance orders are to be judged by the same standards as conventional search warrants” and applying Franlcs to a Title III wiretap application); United States v. Cole, 807 F.2d 262, 268 (1st Cir.1986) (“We have read carefully the wiretap application and the testimony adduced at the Franlcs hearing. We find that the district court properly applied the test required under Franks v. Delaware .... ”); United States v. Williams, 737 F.2d 594, 602 (7th Cir.1984) ("In challenging [the Title III wiretap application’s] affidavit in the district court, the defendants' task was defined by [Franlcs]: they had to prove that the ... allegations were intentional lies or made with reckless disregard for the truth.”).
     
      
      . See note 16, ante.
      
     
      
      . In addition to its relevance to the "materiality” inquiry, whether an omission would have strengthened or weakened the wiretap application is also probative of whether the omission occurred with "reckless disregard for the truth.” Indeed, it is difficult to imagine a situation where the government would intentionally or "with reckless disregard” omit information that would strengthen its "probable cause” or “necessity” showing.
     
      
      . We note that Rajaratnam challenges the District Court’s conclusion as to materiality generally and does not specifically challenge the District Court’s determination that he did not make a preliminary showing that the wiretap application included material misstatements or omissions sufficient to justify a Franks hearing as to probable cause. We also recognize uncertainty both in our own Circuit and in our sister Circuits as to whether to review the denial of a Franks hearing for clear error or de novo. See Falso, 544 F.3d at 126 n. 21. Moreover, it is unclear whether either of these standards is appropriate, inasmuch as that we generally review discretionary decisions on whether a district court ought to conduct a hearing for "abuse of discretion." See United States v. Pena, 961 F.2d 333, 339 (2d Cir.1992). Nonetheless, we need not decide the appropriate standard of review here, because we conclude, for the reasons stated above, that the District Court did not err in denying a Franks hearing as to probable cause under any of these standards.
     
      
      . As noted above, see note 10, ante, the District Court did not conduct a Franks hearing on the "probable cause” issue because it determined that Rajaratnam had not made a sufficient preliminary showing that the alleged misstatements and omissions were "material.” Rajaratnam, 2010 WL 3219333, at *1 n. 2. Because it is unclear whether the District Court determined that the misstatements and omissions regarding probable cause were made with "reckless disregard for the truth,” we only consider whether those misstatements and omissions were "material.”
     
      
      . Indeed, although the District Court found "[plarticularly disturbing ... the omission of highly-relevant information regarding Khan’s prior criminal record,” Rajaratnam, 2010 WL 4867402, at *11, it held that other indicia of Khan’s reliability along with the accurate statements in the wiretap application "suffice[d] for probable cause” purposes, id. at *13.
     
      
      . In relevant part, § 10(b) of the Securities Exchange Act of 1934 provides:
      It shall be unlawful for any person, directly or indirectly, by the use of any means or instrumentality of interstate commerce or of the mails, or of any facility of any national securities exchange-—
      (b) To use or employ, in connection with the purchase or sale of any security registered on a national securities exchange or any security not so registered, ... any manipulative or deceptive device or contrivance in contravention of such rules and regulations as the [Securities and Exchange] Commission may prescribe as 'necessary or appropriate in the public interest or for the protection of investors.
      15 U.S.C. § 78j (emphasis supplied). Pursuant to its § 10(b) rulemaking authority, the SEC adopted Rule 10b-5, which, as relevant here, provides:
      It shall be unlawful for any person, directly or indirectly, by the use of any means or instrumentality of interstate commerce, or of the mails or of any facility of any national securities exchange,
      (a) To employ any device, scheme, or artifice to defraud,
      ... or
      (c) To engage in any act, practice, or course of business which operates or would operate as a fraud or deceit upon any person,
      
        in connection with the purchase or sale of any security.
      
      
        17 C.F.R. § 240.1 Ob-5 (emphasis supplied).
     
      
      . The Supreme Court further explained the misappropriation theory as follows: “In lieu of premising liability on a fiduciary relationship between company insider and purchaser or seller of the company’s stock, the misappropriation theory premises liability on a fiduciary-turned-trader’s deception of those who entrusted him with access to confidential information.” O'Hagan, 521 U.S. at 652, 117 S.Ct. 2199; see also 17 C.F.R. § 240.10b5-2(b) (defining "duties of trust or confidence” under Section 10(b) and Rule 10b-5 as circumstances where "a person agrees to maintain information in confidence” or where "the person communicating the material nonpublic information and the person to whom it is communicated have a history, pattern, or practice of sharing confidences, such that the recipient of the information knows or reasonably should know that the person communicating the material nonpublic information expects that the recipient will maintain its confidentiality”). In other words, the misappropriation theory focuses on the relationship between the trader and the insider and “is thus designed to protect the integrity of the securities markets against abuses by 'outsiders’ to a corporation who have access to confidential information that will affect the corporation's security price when revealed, but who owe no fiduciary or other duty to that corporation's shareholders.” Id. at 653, 117 S.Ct. 2199 (internal quotation marks and brackets omitted).
      As relevant here, we have also clarified that "Section 10(b) and Rule 10b-5 also reach situations where the insider or misappropriator tips another who trades on the information.” Obus, 693 F.3d at 285. Accordingly, "[w]hen an unlawful tip occurs, the tippee is ... liable if he knows or should know that the information was received from one who breached a fiduciary duty (such as an insider or a misappropriator) and the tippee trades or tips for personal benefit with the requisite scienter.” Id. (relying on Dirks v. SEC, 463 U.S. 646, 660, 103 S.Ct. 3255, 77 L.Ed.2d 911 (1983)).
     
      
      . In relevant part, Rule 10b5 — 1(b) provides: [A] purchase or sale of a security of an issuer is "on the basis of” material nonpublic information about that security or issuer if the person making the purchase or sale was aware of the material nonpublic information when the person made the purchase or sale.
      17 C.F.R. § 240.10b5 — 1 (b).
     
      
      . Specifically, the jury instructions contested in Royer stated: “A purchase or sale of a security is 'on the basis of material nonpublic information about that security, if the person making the purchase or sale was aware of the material non-public information when the person made the purchase or sale, and the information in some way informed the investment decision." 549 F.3d at 899 n. 12 (emphasis in original).
     
      
      .Indeed, any alleged error in the jury instructions in this case operated to the benefit of Rajaratnam and was therefore harmless. See Neder v. United States, 527 U.S. 1, 9-10, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999) (collecting cases applying harmless-error review to jury instructions in criminal cases).
     