BAUER, Circuit Judge.

Girvies L. Davis was convicted of murder and sentenced to death in the circuit court of St. Clair County, Illinois. After unsuccessfully appealing his conviction and sentence in the Illinois courts. Davis filed this petition for writ of habeas corpus pursuant to 28 U.S.C. § 2254. The district court denied Davis' petition and he appeals.

I.

On Friday, December 22, 1978, Charles Biebel was shot to death in his mobile home in St. Clair County, Illinois. Biebel, an 89 year-old man who was confined to a wheelchair, was shot twice while inside his trailer. Davis and Richard Holman were indicted and tried for the murder. The evidence at trial established that Davis and Holman burglarized Biebel's home in addition to killing him. Davis was convicted by a jury of one count of murder in violation of section 9-1(a)(1) of the Illinois Criminal Code of 1961. Ill.Rev.Stat. ch. 38, para. 9-1(a)(1) (1977) (now codified as 720 ILCS § 5/9-1(a)(1)). As required by Illinois law, the court conducted a sentencing hearing to determine whether Davis was eligible for the death penalty. At the sentencing hearing, the jury heard evidence that Davis had been convicted for the murders of two other people in addition to Biebel, and the attempted murder of a third. These prior murder convictions, along with the Biebel conviction, made Davis eligible for the death penalty under Illinois law. The jury determined that the necessary aggravating factors existed, and that no mitigating factors were sufficient to preclude the imposition of the death penalty. The jury returned a unanimous verdict that directed the court to sentence Davis to death. The court sentenced Davis to death and the Illinois Supreme Court affirmed his conviction and sentence. People v. Davis, 95 Ill.2d 1, 69 Ill. Dec. 136, 447 N.E.2d 353 (1983). The United States Supreme Court denied Davis' petition for writ of certiorari. Davis v. Illinois, 464 U.S. 1001, 104 S.Ct. 507, 78 L.Ed.2d 697 (Ill. 1983). Davis then filed a petition for post-conviction relief in the circuit court of St. Clair County, Illinois. That court dismissed Davis' petition and the Illinois Supreme Court affirmed. People v. Davis, 119 Ill.2d 61, 115 Ill.Dec. 553, 518 N.E.2d 78 (1987). The United States Supreme Court denied Davis' second petition for writ of certiorari. Davis v. Illinois, 489 U.S. 1059, 109 S.Ct. 1327, 103 L.Ed.2d 595 (1989). Davis then sought leave in the Illinois Supreme Court to file a second petition for post-conviction relief. The Illinois Supreme Court denied his request in an unpublished order. With all state remedies exhausted, Davis turned to federal court. He filed a petition for writ of habeas corpus pursuant to 28 U.S.C. § 2254. The district court denied his petition.

II.

On appeal, Davis alleges the following: (1) his sentencing hearing was fundamentally unfair; (2) the Illinois death penalty statute is unconstitutional on its face; and (3) the Illinois death penalty statute is unconstitutionally applied as a general matter and was unconstitutionally applied in this case.

A. Davis' Sentencing Hearing

Davis contends that his sentencing hearing violated constitutional standards in three ways. First, Davis claims that the use of a taped custodial interview of him by the prosecutor coupled with a remark by the prosecutor that the jury was only to "recommend" the death penalty unconstitutionally injected passion and emotion into the sentencing, violated his right to counsel, violated the Sixth Amendment's Confrontation Clause, and caused the jury to abdicate its responsibility in considering the death sentence. Second, Davis alleges that he was denied his right to the assistance of counsel at his sentencing hearing. Finally, Davis claims that the prosecutor at his trial systematically and purposefully excluded black members of the venire in violation of the Fourteenth Amendment Equal Protection Clause.

We begin with the videotape. At his sentencing hearing, the trial court allowed the State to play for the jury a videotape of a custodial interview of Davis. Davis said several things to the prosecutor at that interview, including: that he knew of his right to an attorney and did not want his attorney present; that he wanted to be executed quickly for his crimes because he was "tried of living"; that he was a murderer; that Ricky Holman gave him a gun; and that he committed a series of burglaries.

Davis argues that the videotape, together with the prosecutor's remarks that the jury was only to "recommend" the death penalty, eroded the moral responsibility that the jury felt for Davis' sentence. Davis' suggestion that the videotape made the jury more likely to vote for the death penalty is pure speculation. In fact, Davis' trial counsel, Patrick Young, referred to Davis' stated desire to die as a mitigating circumstance because only someone who was "extremely emotionally disturbed" and not in his "right mind" would make statements like those Davis made during the taped interview. Tr. of Oct. 28, 1980 at 160. And the prosecutor's comment was an apparent off-hand statement made during the course of the closing argument to the jury. To determine that this lone assertion required reversal, we would have to find that the statement was so inflammatory and prejudicial as to deprive Davis of a fair trial. Jentges v. Milwaukee County Circuit Court, 733 F.2d 1238, 1242 (7th Cir. 1984). We examine the prosecutor's remark in the context of the proceeding as a whole. United States v. Easley, 994 F.2d 1241, 1245 (7th Cir. 1993). The jury in this case was repeatedly instructed that Davis would be sentenced to death if the jury unanimously decided that he deserved the death penalty. For example, the court instructed the jury in part: "If, at the conclusion of your deliberations in accordance with the court's instructions, you unanimously recommend that the death sentence be imposed, then the defendant will be sentenced to death by the court." Tr. of Oct. 28, 1980 at 180. The court also instructed the jury to consider only the testimony of the witnesses and the exhibits which the court received as evidence. So instructed, the jury returned a unanimous sentence of death. We presume that juries follow their instructions. United States v. Badger, 983 F.2d 1443, 1456 (7th Cir. 1993), cert. denied, ___ U.S. ___, 113 S.Ct. 2391, 124 L.Ed.2d 293 (1993). Nothing in this case even begins to rebut that presumption.

Davis also challenges the use of the videotape because, he says, the interview was conducted in violation of his right to have counsel present. The record reveals that Davis was made aware of his right to counsel and waived that right. As required by Miranda v. Arizona, 384 U.S. 436, 479, 86 S.Ct. 1602, 1630, 16 L.Ed.2d 694 (1966), Davis was warned prior to any questioning that he had a right to remain silent, that he had a right to the presence of an attorney, and that if he could not afford an attorney, one would be provided for him. At the beginning of the interview, Clyde Kuehn, the State's Attorney who prosecuted Davis, explained these rights to Davis:

MR. KUEHN: Okay. Now, listen to me. I want you to understand your constitutional rights. You understand?

MR. DAVIS: Yes, sir.

MR. KUEHN: Do you have a lawyer?

MR. DAVIS: Yes sir.

* * * * * *

MR. KUEHN: You don't mind talking to me without having him here?

MR. DAVIS: No, I don't mind talking to you without having him here.

* * * * * *

MR. KUEHN: Now, do you understand that you have a right and you can remain silent?

MR. DAVIS: Right.

MR. KUEHN: You can have a lawyer, you've got a lawyer. Maybe if you haven't paid him, you can have one appointed. Do you understand that?

MR. DAVIS: Yes, sir.

Tr. of Oct. 28, 1980 at 113-14.

This exchange demonstrates that Davis waived his right to counsel. A waiver of the right to have an attorney present at a custodial interview, like the one at issue in this case, must be knowingly and voluntarily made. United States v. Morrison, 946 F.2d 484, 502 n. 4 (7th Cir. 1991), cert. denied, ___ U.S. ___, 113 S.Ct. 826, 121 L.Ed.2d 696 (1992). Kuehn explained Davis' right to an attorney; Davis therefore knew about this right. He also voluntarily decided to abandon this known right. See Johnson v. Zerbst, 304 U.S. 458, 464, 58 S.Ct. 1019, 1023, 82 L.Ed. 1461 (1938) ("A waiver is ordinarily an intentional relinquishment or abandonment of a known right or privilege."). Davis" express statements of waiver amply demonstrate that he waived his right to have an attorney present. North Carolina v. Butler, 441 U.S. 369, 373, 99 S.Ct. 1755, 1757, 60 L.Ed.2d 286 (1979).

Davis says that he lacked the requisite mental capacity to waive his right to counsel. Since Davis was competent to stand trial, (and does not claim otherwise) he was competent enough — and had the capacity — to waive his right to counsel. Godinez v. Moran, ___ U.S. ___, ___-___, ___, 113 S.Ct. 2680, 2685-86, 2688, 125 L.Ed.2d 321 (1993). By expressly stating that he did not want an attorney present, Davis waived his right to counsel.

Next, Davis argues that Kuehn violated his Fifth Amendment right to remain silent. Toward the end of the interview, Davis told Kuehn that "I don't want to talk no more" and then, as the interview continued, said, "I don't want to talk about it any more." Tr. of Oct. 28, 1980 at 128, 131. The interview continued despite Davis' expressed desire not to talk. By continuing the interview, Davis contends, Kuehn deprived him of his right to remain silent.

A suspect who is subject to a custodial interview enjoys the right to terminate questioning at any time. Miranda, 384 U.S. at 473-74, 86 S.Ct. at 1627-28; Bobo v. Kolb, 969 F.2d 391, 395 (7th Cir. 1992). In this case, Davis refused to answer any questions about other crimes and other murders in which he was involved. No new evidence was elicited after Davis invoked his right to remain silent. In fact, the rest of the conversation was simply follow-up by the prosecutor. The prosecutor asked Davis such questions as: why did you make these statements? did anyone in jail threaten you? and, were you promised anything by anybody? If Kuehn violated Davis' right to remain silent, we would reverse the district court and grant habeas relief if the error had "`substantial and injurious effect or influence in determining the jury's verdict.'" Brecht v. Abrahamson, ___ U.S. ___, ___, 113 S.Ct. 1710, 1714, 123 L.Ed.2d 353 (1993) (quoting Kotteakos v. United States, 328 U.S. 750, 776, 66 S.Ct. 1239, 1253, 90 L.Ed. 1557 (1946)). Any error here in failing to respect fully Davis' right to remain silent could not have had any substantial or injurious effect or influence on the jury's verdict. Having said all that, we would like to restate a caution to prosecutors and other law enforcement agents involved in custodial interrogations: when the arrestee says he wants to quit talking, and says he doesn't want to answer any more questions, stop the questioning. Here, the answers were inconsequential and had no effect on the proceeding. The next time may be different. The subject of the inquiry has a constitutional right not to talk; the government and its agents have a constitutional duty to respect that right.

Davis also complains that the videotaped material presented to the jury contained hearsay statements that violated the Sixth Amendment's Confrontation Clause. That provision guarantees that "[i]n all criminal prosecutions, the accused shall enjoy the right . . . to be confronted with the witnesses against him." U.S. Const. amend. VI. But the Confrontation Clause does not apply at sentencing. United States v. Hammer, 3 F.3d 266, 272 (8th Cir. 1993). See also United States v. Corbin, 998 F.2d 1377, 1385 n. 15 (7th Cir. 1993) (noting that at least eight circuits have held that the Confrontation Clause does not apply in sentencing proceedings). The defendant at sentencing is no longer the "accused" referred to in the Sixth Amendment; instead, the defendant stands before the jury as a convicted criminal. Any violation of the hearsay rule that may have occurred at Davis' state trial is a matter of state law and outside the scope of our review in this habeas case. Estelle v. McGuire, ___ U.S. ___, ___, 112 S.Ct. 475, 480, 116 L.Ed.2d 385 (1991); Reed v. Clark, 984 F.2d 209, 210 (7th Cir. 1993).

Davis argues next that his trial attorney, Patrick Young, performed so ineffectively that he was deprived of his right "to have the Assistance of Counsel for his defense." U.S. Const. amend. VI. To succeed on an ineffective assistance of counsel claim, a convicted defendant must show (1) that the defendant's trial counsel performed deficiently and (2) that the deficient performance prejudiced the defense so as to deprive the defendant of a fair trial. Strickland v. Washington, 466 U.S. 668, 687, 104 S.Ct. 2052, 2064, 80 L.Ed.2d 674 (1984); United States v. Levine, 5 F.3d 1100, 1108 (7th Cir. 1993). For these purposes, a fair trial is a trial whose result is reliable. Id.

Davis claims that Young was ineffective because he did not present evidence of Davis' alleged mental illness, retardation, and troubled family background. For this omission, however, Davis has one person to blame — himself. Davis prevented Young from presenting this evidence to the jury during the mitigation phase of the sentencing hearing. Young informed the court outside the presence of the jury that discussions he had with the St. Clair County (Illinois) Mental Health Department led him to believe that Davis had suffered for years from a mental condition and that Davis had received mental treatment for a number of years, going back to early childhood. Young also told the court that those he talked with at the St. Clair County Mental Health Department would testify that Davis received a severe blow to his head when he was a child, that the blow to his head damaged his brain, and that Davis' E.E.G. readings were very abnormal. Young offered to put on witnesses and enter records into evidence that would confirm what he said about Davis' mental condition. Young said this would show that "the murder was committed while the defendant was under the influence of extreme mental or emotional disturbance, although not such as to constitute a defense to the prosecution." Tr. of Oct. 28, 1980 at 18. Young explained that Davis did not want any of that evidence to go before the jury. The court asked Davis if Young was right, that is, if Davis did not want his mental health history presented to the jury. Davis responded with a terse "[t]hat's correct." As a result of this, the jury never heard this proposed mitigating evidence. Id. at 19.

Davis was fully informed of the option before him — whether to introduce evidence of his background and mental health history — and informed his attorney and the court that he did not want this evidence to go to the jury. He will not be heard to complain that the trial court's decision to grant his request deprived him of his right to the assistance of counsel. See United States v. Weaver, 882 F.2d 1128, 1140 (7th Cir.) ("Where a defendant fully informed of the reasonable options before him, agrees to follow a particular strategy at trial, that strategy cannot later form the basis of a claim of ineffective assistance of counsel."), cert. denied, 493 U.S. 968, 110 S.Ct. 415, 107 L.Ed.2d 380 (1989). We note as an additional matter that Young did present mitigating evidence in the form of testimony by Davis' wife, Cindy Davis. She testified that her husband once took her to the hospital, was never violent towards her, and, if the jury did not give her husband the death penalty, she would visit him in jail. The jury was not moved, or at least not moved enough, and voted unanimously to impose the death penalty.

Davis relies on our decision in Brewer v. Aiken, 935 F.2d 850 (7th Cir. 1991), as support for his claim that Young's performance was constitutionally defective. His reliance is misplaced. In Brewer, we held that the defendant's attorney was ineffective during the death penalty phase of the trial because he failed to investigate the mental history of a defendant who had a low intelligence quotient and a deprived background. See Brewer, 935 F.2d at 857-59. In this case, unlike Brewer, Young did investigate Davis' mental health history and background. Young's performance easily satisfied the requirements of the Sixth Amendment.

Davis, a black man, next contends that at his trial the prosecutor systematically and purposefully excluded black members of the venire. This, he says, violated the constitutional guarantee that no state shall "deny to any person within its jurisdiction the equal protection of the laws." U.S. Const. amend. XIV, § 1. Davis asks us to apply the rule of Batson v. Kentucky, 476 U.S. 79, 106 S.Ct. 1712, 90 L.Ed.2d 69 (1986), to his case. In Batson, the Supreme Court held that a prosecutor violates a defendant's equal protection rights if the prosecutor purposefully uses peremptory challenges to exclude members of the defendant's race from the jury. Id. 476 U.S. at 86, 106 S.Ct. at 1717. The principle announced in Batson was not new, but changed the quantum of proof necessary to substantiate a particular claim. Ford v. Georgia, 498 U.S. 411, 420, 111 S.Ct. 850, 856, 112 L.Ed.2d 935 (1991). In one of the forerunners to Batson, Swain v. Alabama, 380 U.S. 202, 227, 85 S.Ct. 824, 839, 13 L.Ed.2d 759 (1965), the Court held that a defendant could establish an equal protection violation if the defendant showed that the prosecutor systematically used peremptory challenges against blacks over a period of time. Batson did away with Swain's requirement of proof of prior discrimination and held it possible for a defendant to make out a prima facie equal protection argument merely by reference to the prosecution's use of peremptory challenges in the defendant's own case. Ford, 498 U.S. at 418, 111 S.Ct. at 855; Williams v. Chrans, 945 F.2d 926, 941 (7th Cir. 1991), cert. denied, ___ U.S. ___, 112 S.Ct. 3002, 120 L.Ed.2d 877 (1992). Under the Batson standard, a defendant need only demonstrate that the use of peremptory challenges to eliminate members of a particular race raises an inference that the prosecutor used the peremptories to deny a defendant an impartial jury. When a prima facie case of this improper use of peremptory challenges has been made, the burden shifts to the prosecution to explain a constitutionally permissible reason for the challenges.

Davis urges us to apply Batson retroactively. In Williams, we considered and rejected the retroactive application of Batson. Id. at 946. Batson would, however, apply if Davis' conviction was not yet final when Batson was decided on April 30, 1986. Richardson v. Gramley, 998 F.2d 463, 464 (7th Cir. 1993). A conviction is final for these purposes when the defendant has exhausted his state appellate remedies and either the United States Supreme Court has denied the defendant's petition for certiorari or the time for filing the petition has expired. Id. In this case, the Court denied Davis' petition for certiorari on November 28, 1983, long before Batson was decided. Davis v. Illinois, 464 U.S. 1001, 104 S.Ct. 507, 78 L.Ed.2d 697 (1983). Batson does not apply and we therefore examine Davis' claim under the Swain standard.

As we have explained, Swain requires Davis to show that the prosecutor systematically used peremptory challenges against blacks over a period of time. Davis has not satisfied this burden of proof. Although Davis alleges that in his case the prosecutor purposefully excluded black venire-persons, he fails to show a history of purposeful discrimination. We cannot find from this record that the prosecutor systematically and purposefully excluded black members of the jury in violation of the Fourteenth Amendment.

Given all this, we conclude that there was nothing about the videotaped custodial interview, the prosecutor's remark to the jury, the prosecutor's use of peremptory challenges during jury selection, or anything else about Davis' sentencing hearing that violated the Constitution.

B. Facial Validity of the Illinois Death Penalty Statute

Davis claims that the Illinois death penalty statute is facially unconstitutional because of defects in the statute and, in particular, the statute's accompanying jury instructions. He charges that the jury instructions do not inform the jury that they may consider nonstatutory mitigating factors and do not instruct jurors on the standard of proof on aggravation and mitigation. Davis also contends that the jury instructions are defective because the instructions cause jurors to believe that capital defendants, like Davis, bear the burden of persuasion on the issue of life or death. All of this, Davis says, violates the Eighth Amendment requirement that "cruel and unusual punishments [shall not be] inflicted." U.S. Const. amend. VIII. We have described the intricacies of the Illinois death penalty statute before and need not do so here to address Davis' complaint. See, e.g., Gacy v. Wellborn, 994 F.2d 305, 306-07 (7th Cir. 1993), cert. denied, ___ U.S. ___, 114 S.Ct. 269, 126 L.Ed.2d 220 (1993).

Davis first contends that the jurors in his case were not instructed that they could consider nonstatutory mitigating factors. The record reveals otherwise. The trial judge at Davis' sentencing hearing instructed the jury that mitigating factors may include the five explicitly set out in the statute, but also added a sixth: "Any other facts or circumstances that provide reasons for imposing less than the death penalty." Tr. of Oct. 28, 1980 at 185. As this instruction indicates, the jury could consider literally anything that in its judgment provided a reason for imposing a sentence other than death.

We considered a similar challenge to similar jury instructions in Silagy v. Peters, 905 F.2d 986 (7th Cir. 1990), cert. denied, 498 U.S. 1110, 111 S.Ct. 1024, 112 L.Ed.2d 1106 (1991). In point of fact, the instructions in this case would seem more favorable to Davis than those we upheld in Silagy. The instructions at issue in Silagy listed mitigating factors as: "One, the murder was committed while the Defendant was under the influence of extreme mental or emotional disturbance although not such as to constitute a defense to prosecution; Two, any other facts or circumstances that provide reasons for imposing less than the most severe sentence." 905 F.2d at 1006. In this case, the jurors were given a list of the five statutory mitigating factors in addition to the all-encompassing "any other facts or circumstances" instruction. As in Silagy, the jury instructions in this case informed the jury that they could consider nonstatutory mitigating factors and therefore did not violate the Constitution.

Davis makes other challenges to the instructions. He contends that the instructions did not instruct jurors on the standard of proof on aggravation and mitigation and that jurors incorrectly understood that he bore the burden of persuasion on the issue of life or death. The portion of the instructions challenged by Davis reads:

A person found guilty of murder may be sentenced to death only if the jury unanimously recommends that the death sentence be imposed.

If, at the conclusion of your deliberations in accordance with the court's instructions, you unanimously recommend that the death sentence be imposed, then the defendant will be sentenced to death by the court.

On the other hand, if, at the conclusion of your deliberations, you do not unanimously recommend that the death sentence be imposed, then the defendant will be sentenced to imprisonment for a number of years, not less than 20, to be determined by this court.

Tr. of Oct. 28, 1980 at 180. The court also instructed the jury that:

If, after your deliberations, in this second stage of the sentencing hearing, you unanimously determine that there are no mitigating factors sufficient to preclude the imposition of the death sentence on the defendant, you should sign the verdict form which so indicates. In such a situation, you will be returning one verdict form indicating your verdict on the absence of any sufficiently mitigating factors. If you sign this form, the Court must sentence the defendant to death.

If, after your deliberations, in this second stage of the sentencing hearing, you are not unanimous in concluding that there is no mitigating factors sufficient to preclude the imposition of the death sentence, you must sign the "No Unanimous Verdict" Form. In this situation you will sign one verdict form indicating that you did not unanimously agree on the absence of any sufficiently mitigating factors sufficient to preclude the death penalty. If you sign this form, the Court will sentence the defendant to imprisonment.

Id. at 185-86. In addition to these boilerplate instructions (which track the Illinois death penalty statute), the court explained to the jury that Davis would be sentenced to death only if the jury's verdict was unanimous and that if the jury was not unanimous he would be sentenced to imprisonment. With all this, and additional clarifying instructions, the jury returned a unanimous verdict directing the court to sentence Davis to death. After the court read the verdict, it asked each member of the jury if each person voted to sentence Davis to death. All twelve jurors answered individually that they voted for the death penalty.

Davis points to a study conducted by the late Professor Hans Zeisel that purports to demonstrate that the Illinois Pattern Instructions in death penalty cases leave jurors hopelessly confused as to the standard of proof, the burden of persuasion, and a whole host of other factors. We considered Professor Zeisel's study in both Gacy and Free v. Peters, 12 F.3d 700, (7th Cir. 1993), along with jury instructions that were materially identical to those challenged by Davis, and concluded that there was no constitutional violation. See Free, 12 F.3d at 705-706; Gacy, 994 F.2d at 307-10. As in Gacy, we invoke the presumption that the jurors understood and followed their instructions. Id. at 313. The instructions pass constitutional muster.

As to the burden of persuasion issue raised by Davis, we considered and rejected a similar claim in Silagy. We noted that the imposition of the burden of persuasion on a defendant is constitutional because at this point in the hearing, the defendant stands guilty of murder and the prosecution has proven that the requisite statutory aggravating factor exists, thereby making the defendant eligible for the death penalty. Silagy, 905 F.2d at 998. The prosecution established that Davis had two prior murder convictions. By doing so, it satisfied its statutory obligation of proving the existence of aggravating factors. Even if the jury in Davis' case believed that Davis bore the burden of persuasion on the issue of life or death, or on the issue of mitigation, his sentence would not be unconstitutional. Id. Also, the instructions did not discuss which party bore the burden of persuasion because at that stage there is no burden.

The instructions given to the jury at Davis' sentencing hearing did not violate the Constitution. The Illinois death penalty statute is not unconstitutional on its face.

C. As Applied Challenge to the Illinois Death Penalty

Davis contends that the Illinois death penalty statute was and is applied in an unconstitutional manner. He charges that in Illinois the death penalty is meted out in an arbitrary and capricious manner so as to discriminate on the grounds of race. Davis also alleges that prosecutors did not request the death penalty for other criminal defendants whose crimes, like his, made them statutorily eligible for the death penalty. Davis points to a study by Samuel Gross and Robert Mauro as support for his claim that in Illinois a person accused of homicide against a white victim is approximately six times more likely to receive the death penalty than a person accused of the same type of homicide against a black victim. The Gross and Mauro study also alleges that a black person who kills a white person is 3.75 times more likely to receive the death penalty than is a white person who kills a white person. Davis argues that the Gross and Mauro study along with the prosecutor's decision to exclude black members of the venire from the jury demonstrates that the decisionmaking process in his case was infected with racial bias.

We start with the Gross and Mauro study and, for purposes of this case, assume as true the statistical conclusions that Davis describes. Our analysis begins and ends with McCleskey v. Kemp, 481 U.S. 279, 107 S.Ct. 1756, 95 L.Ed.2d 262 (1987). Like Davis, the defendant in that case offered a study that purported to show a disparity in the imposition of the death sentence in Georgia based on the race of the victim and, to a lesser extent, the race of the defendant. Id. at 286, 107 S.Ct. at 1764. The study at issue in McCleskey suggested that defendants charged with killing white persons received the death penalty more often than those who killed black persons. Id. Also, black defendants received the death penalty more often than white defendants. Id. The study found that prosecutors sought the death penalty in 70% of the cases involving black defendants and white victims, 32% of the cases involving white defendants and white victims, 15% of the cases involving black defendants and black victims, and 19% of the cases involving white defendants and black victims. Id. at 287, 107 S.Ct. at 1764. Another part of the study at issue in McCleskey concluded that defendants charged with killing white victims were 4.3 times as likely to receive a death sentence as defendants charged with killing blacks and that black defendants were 1.1 times as likely to receive a death sentence as other defendants. Id.

The Court held that the study was not enough to support an inference that any of the decisionmakers in the defendant's case acted with discriminatory purpose. Id. at 297, 107 S.Ct. at 1769. Only "exceptionally clear proof" of a discriminatory purpose by the decisionmakers in the defendant's case, the Court stated, would be sufficient for a constitutional violation. Id. As the defendant in McCleskey relied solely on the study, the Court held that there was nothing to suggest that his case was constitutionally unsound.

McCleskey disposes of Davis' claim with respect to the Gross and Mauro study. As described by Davis, the Gross and Mauro study purports to show the same things as the study in McCleskey: black defendants are more likely to get the death penalty than white defendants, blacks who murder whites are more likely to receive the death penalty than whites who murder blacks, defendants who murder whites are more likely to receive a death sentence than defendants who murder blacks, and so on. Like the Court in McCleskey, we do not accept Davis' proposition that his sentence was imposed along racial lines.

Davis was convicted of murdering Biebel. The jury heard additional evidence of Davis' many other crimes, including two other murder convictions. The jury also heard the testimony of a survivor of one of Davis' attacks. Davis attempted to kill that person, but failed.

We decline to hold, as Davis urges, that an all-white jury was incapable of rendering a fair decision in accord with Illinois law. As to the apparent discrepancies along racial lines that the Gross and Mauro study announced, we note that such social science studies can cut a variety of ways. For example, the Department of Justice studied murder in large urban counties in 1988. That study concluded that whites and blacks were equally likely to receive the death penalty and that there were no statistically measurable differences in sentencing outcomes between white and black murder defendants. See Bureau of Just. Stats., U.S. Dep't of Just., Special Report, Murder in Large Urban Counties, 1988 (1993). We need not decide the relative merits of the Gross and Mauro study or the Department of Justice study. We note only that nothing in them, one way or the other, establishes any constitutional violation and certainly does not provide the exceptionally clear proof of discrimination required by McCleskey. That is all we need decide.

Finally, we consider Davis' assertion that prosecutors in other murder cases did not seek the death penalty and that doing so in his case (and not others) is arbitrary and capricious. Again, McCleskey is dispositive. The Court considered the same argument there and concluded that the defendant could not prove a constitutional violation by demonstrating that other defendants who might have been similarly situated did not receive the death penalty. McCleskey, 481 U.S. at 306-07, 107 S.Ct. at 1774-75.

One simple fact exists in this case: Davis was convicted of crimes for which the Constitution and Illinois law permit the imposition of the death penalty. The jury heard the evidence, including aggravating and mitigating evidence, and concluded that Davis' actions warranted the death penalty. That decision was within their discretion, was neither arbitrary nor capricious, and did not violate the Constitution.

III.

The State of Illinois complied with the Constitution when it sentenced Girvies Davis to death. The district court's decision to deny Davis' petition for writ of habeas corpus is therefore

AFFIRMED.

* * * * * *

3. the defendant has been convicted of murdering two or more individuals under subsection (a) of this Section. . . .

* * * * * *

(c) Consideration of factors in Aggravation and Mitigation. The court shall consider, or shall instruct the jury to consider any aggravating and any mitigating factors which are relevant to the imposition of the death penalty. Aggravating factors may include but need not be limited to those factors set forth in subsection (b). Mitigating factors may include but need not be limited to the following:

1. the defendant has no significant history of prior criminal activity;

2. the murder was committed while the defendant was under the influence of extreme mental or emotional disturbance, although not such as to constitute a defense to prosecution;

3. the murdered individual was a participant in the defendant's homicidal conduct or consented to the homicidal act;

4. the defendant acted under the compulsion of threat or menace of the imminent infliction of death or great bodily harm;

5. the defendant was not personally present during commission of the act or acts causing death.

(d) Separate sentencing hearing. Where requested by the State, the court shall conduct a separate sentencing proceeding to determine the existence of factors set forth in Subsection (b) and to consider any aggravating or mitigating factors as indicated in Subsection (c). The proceeding shall be conducted:

1. before the jury that determined the defendant's guilt[.]

* * * * * *

(g) Procedure — Jury. . . . If there is a unanimous finding by the jury that one or more of the factors set forth in Subsection (b) exist, the jury shall consider aggravating and mitigating factors as instructed by the court and shall determine whether the sentence of death shall be imposed. If the jury determines unanimously that there are no mitigating factors sufficient to preclude the imposition of the death sentence, the court shall sentence the defendant to death.

Unless the jury unanimously finds that there are no mitigating factors sufficient to preclude the imposition of the death sentence the court shall sentence the defendant to a term of imprisonment under Chapter V of the Unified Code of Corrections.