We modify our original opinion, 743 F.2d 1091, by withdrawing Sections VI and VII, 743 F.2d at 1099-1102, and substituting the following. In all other respects the Petitions for Rehearing and Rehearing En Banc are denied.
VI. PROPORTIONALITY REVIEW
The proportionality review accorded Pre-jean’s death sentence — comparison to capital murder cases in two districts — did not violate any federal constitutional right because the Louisiana Supreme Court, in some other cases, has engaged in a broader search for similar cases.
*483A
The Louisiana Supreme Court is required to review every death sentence to determine if it is excessive. La.Code Crim.Proe. Ann. art. 905.9 (1984). State law further authorizes the Louisiana Supreme Court to enact such rules and procedures “as are necessary to satisfy constitutional criteria for review.” Id. Supreme Court Rule 28 establishes the guidelines for effecting the court’s review of capital sentences for ex-cessiveness. According to these guidelines, the Louisiana Supreme Court must determine, among other things, “whether the sentence is disproportionate to the penalty imposed in similar cases, considering both the crime and the defendant.” La. Sup.Ct.R. 28, § 1(c), La.Code Crim.Proe. Ann. art. 905.9 (1984). To aid the court in its review, the district attorney is to file a memorandum listing each first degree murder case after January 1, 1976 in the district in which the sentence was imposed. Id. § 4(b)(i).
Prejean contends these guidelines establish an obligation to review similar cases, rather than an obligation to review cases in the district where the sentence is imposed. Prejean argues that the district attorney’s memorandum, compiling first degree murder cases from the district where sentence is imposed, is merely a convenience to the court and not a geographic limitation on the scope of comparative review the court conducts. According to Prejean, the court must review some similar cases, wherever they arise. Thus, whenever the district attorney cannot uncover similar cases in his district, the Louisiana Supreme Court must look elsewhere within the State to find similar cases with which it can make its mandatory comparison for excessiveness. To support his interpretation of the guidelines, Prejean cites cases in which the Louisiana Supreme Court looked outside the district after failing to find sufficiently similar cases within the district. See, e.g., State v. Moore, 432 So.2d 209, 227 (La.1983) (“[W]e have found cases from other jurisdictions which allow for a more fair comparison____”); State v. Narcisse, 426 So.2d 118, 139 (La.1983) (“A review of cases from other parishes ... will justify the death sentence.”); State v. Williams, 383 So.2d 369, 374-75 (La.1980) (court looked outside district to compare similar mitigating factors).
Prejean urges that to determine his sentence is not excessive merely because no similar cases exist in the district1 is a denial of proportionality review as actually practiced by the Louisiana Court. Prejean asserts that considering its willingness to compare murder cases from other districts, the Louisiana Supreme Court violated Louisiana law and its own rules by failing to look for and consider cases similar to his own.
Prejean is not arguing that he has a constitutional right to proportionality review, nor that the Louisiana Supreme Court must consider all similar cases arising in the state. His claim is that because the Louisiana Supreme Court failed to consider any similar, cases, as required by statute and by that court’s own practices, his due process rights were violated. We find the procedures followed in Prejean’s case comported with due process.
B
Louisiana’s single district proportionality review is constitutionally sufficient. See Williams v. Maggio, 679 F.2d 381, 394-95 (5th Cir.1982) (en banc), cert. denied, 463 U.S. 1214, 103 S.Ct. 3553, 77 L.Ed.2d 1399, stay vacated, 464 U.S. 46, 104 S.Ct. 311, 314, 78 L.Ed.2d 43 (1983); Martin v. Maggio, 711 F.2d 1273, 1286 (1983), petition for rehearing, 739 F.2d 184, 185, 186 (5th Cir. 1984) . In Williams v. Maggio, this court, en banc, rejected an eighth amendment challenge to Louisiana’s single district corn-*484parative review. 679 F.2d at 394-95. In upholding the Louisiana statute, the en banc court recognized that, in upholding the death penalty in Georgia, Florida, and Texas, the United States Supreme Court focused on the fact that these states had implemented provisions to adequately safeguard against freakish imposition of capital punishment. Id. at 395. The Georgia, Florida, and Texas statutes were found to comply with the Constitution by controlling against the arbitrary and capricious exercise of jury discretion. See Gregg v. Georgia, 428 U.S. 153, 96 S.Ct. 2909, 49 L.Ed.2d 859 (1976); Proffitt v. Florida, 428 U.S. 242, 96 S.Ct. 2960, 49 L.Ed.2d 913 (1976); Jurek v. Texas, 428 U.S. 262, 96 S.Ct. 2950, 49 L.Ed.2d 929 (1976). Thus, a constitutionally sanctioned death penalty must provide adequate review to protect against the arbitrary imposition of capital punishment. Proportionality review, even of cases within a single district, is one means of providing such protection.2
Just as a venire chosen from a cross section of the community in which the crime is committed is an adequate constitutional safeguard against arbitrary imposition of verdicts and sentences, so a review of the murder convictions imposed within that venire community is sufficient to ensure against arbitrary imposition of the death penalty.
Williams v. Maggio, 679 F.2d at 395.
Prejean’s assertion here is not grounded in the eighth amendment, but rather in the fourteenth amendment’s due process clause. However, the reasoning is the same under either amendment. The critical determination is whether Prejean received adequate protection against arbitrary imposition of the death penalty. We find that he did. His sentence was duly reviewed in accordance with a valid capital punishment statute which has been found to be constitutionally correct. The fact that the Louisiana Supreme Court chose to go outside the district in other cases cannot convert its action, proper under the eighth amendment, into a fourteenth amendment due process violation. Substantive due process requires that a state follow procedures that insure against arbitrary or capricious imposition of the death penalty. Williams v. Maggio adjudges Louisiana’s same-district review as comporting with this standard. 679 F.2d at 394-95.
C
The constitutional sufficiency of the proportionality review of Prejean’s sentence is also supported by Pulley v. Harris, 465 U.S. 37, 104 S.Ct. 871, 79 L.Ed.2d 29 (1984). Pulley held that the Constitution does not require proportionality review of capital sentences. 104 S.Ct. at 876, 879; see also Williams v. Maggio, 679 F.2d 381, 395 (5th Cir.1982) (en banc), cert. denied, 463 U.S. 1214, 103 S.Ct. 3553, 77 L.Ed.2d 1399 (1983). Pulley, too, is an eighth amendment case. Again, it is the reasoning behind the decision, rather than the number of the applicable amendment, that controls Pulley’s precedential value to Prejean’s case.
In Pulley, Harris, the habeas corpus petitioner, complained that he was denied the comparative proportionality review practiced by California courts. Although California’s death penalty statute did not provide for proportionality review, Harris argued it had become the practice of the state supreme court to protect against ex-cessiveness by exercising a comparative review of similar cases. Because this practice had evolved as part of the review process, Harris claimed he was entitled to such review. Harris claimed the failure to provide such review deprived him of protection against a capital sentence “wantonly and freakishly” imposed in violation of the eighth amendment's protection against cruel and unusual punishment.
In essence, Prejean’s claim, like Harris’, protests a perceived variation from stan*485dard state procedure. As in Pulley, Preje-an’s claim is that because of an evolution of Louisiana law as practiced by the Louisiana courts, he is entitled to proportionality review broader in scope than he received. However, Pulley tells us that "[e]ven if an error of state law could be sufficiently egregious to amount to a denial of ... due process of law guaranteed by the Fourteenth Amendment, [petitioner’s] submission is not persuasive.” 104 S.Ct. at 875.
The United States Supreme Court noted in Pulley that the California Supreme Court had twice rejected Harris’ demand for proportionality review without suggesting that it was departing from precedent. Id. Similarly, the Louisiana Supreme Court has repeatedly rejected Prejean’s demand for a broader proportionality review without suggesting it was departing from precedent.
When Prejean’s Application for Rehearing to the Louisiana Supreme Court was denied, 379 So.2d at 249 (La.1979), three justices voted to grant rehearing on the penalty phase only. Id. In a dissent from denial of rehearing, these justices set forth their belief that failure to select cases from throughout the state for comparison in a proportionality review was a serious constitutional flaw in the scheme for imposing the death penalty. Id. at 250. The dissent further indicates that
a majority of this Court has today adamantly refused to expand the basis of its proportionality review by either granting a rehearing in this case or by amending its rule to require annual reporting of capital sentence proceedings results throughout the state. Apparently, a majority of this Court is of the opinion that state-wide proportionality review is not required as part of a constitutional capital punishment scheme.
Id. Following denial of his Application for Rehearing, Prejean petitioned the United States Supreme Court for a Writ of Certio-rari, which was denied. 449 U.S. 891, 101 S.Ct. 253, 66 L.Ed.2d 119 (1980). Prejean then sought and was denied habeas relief from the state court. The three justices who voted for rehearing when the case was before them on appeal voted to deny the writ. These justices specially stated “[although each of us voted for a rehearing ... on the question of appellate review of the proportionality of sentence, that issue was presented to the United States Supreme Court in a writ application ... and the application was denied.” 397 So.2d 517 (La.1981). Thus, the Louisiana Supreme Court conclusively has laid to rest any question concerning the procedure for proportionality review of Prejean’s death sentence.
We can neither ignore nor contravene this clear statement of Louisiana law by that state’s highest court. State courts are the ultimate expositors of their own state’s laws. This court is bound by the construction placed on a state’s criminal statutes by the courts of that state. See Mendiola v. Estelle, 635 F.2d 487, 489 (5th Cir.1981).
Prejean’s complaint of state error does not raise a constitutional violation. This court may not issue a writ of habeas corpus on the basis of a perceived error of state law. Pulley v. Harris, 465 U.S. 37, 104 S.Ct. 871, 875, 79 L.Ed.2d 29 (1984). Under 28 U.S.C. § 2241, a writ of habeas corpus disturbing a state court judgment may issue only if it is found that a prisoner is in custody “in violation of the Constitution or laws or treaties of the United States.” 28 U.S.C. § 2241(c)(3) (1976). Because the refusal of the Louisiana Supreme Court to look outside the district for similar cases does not constitute a due process violation, Prejean’s petition for a writ of habeas corpus was properly denied. See Pulley, 104 S.Ct. at 875.
Louisiana’s Supreme Court conducted the proportionality review it deemed appropriate under Louisiana law and its own rules. It looked at murder cases from two districts, the one in which the crime occurred and the one in which Prejean’s sentence was imposed. It has considered and rejected Prejean’s claim that this procedure was insufficient and an improper departure from state practice. The first degree murder cases in those districts revealed no *486similar crimes and therefore no sentences less severe than Prejean’s. Certainly the first person convicted of murdering a peace officer in these districts cannot negate an otherwise valid capital sentence because his crime is unique. Prejean was not denied fourteenth amendment due process by this proportionality review.
VII. DISCRIMINATORY APPLICATION
Prejean contends that the district court erred in refusing to hold an evidentiary hearing on his claim of discriminatory application of the death penalty based on race. Prejean argues that he tendered statistical proof which would establish that Louisiana juries act on a racial basis in imposing the death sentence against blacks who have killed white victims. To create a fact issue warranting an evidentiary hearing, a statistical proffer must be “so strong that the results would permit no other inference but that they are the product of racially discriminatory intent or purpose.” Smith v. Balkcom, 671 F.2d 858, 859, modifying 660 F.2d 573 (5th Cir.), cert. denied, 459 U.S. 882, 103 S.Ct. 181, 74 L.Ed.2d 148 (1982). Prejean’s tender does not meet this standard.
Smith v. Balkcom, id., and Spinkellink v. Wainwright, 578 F.2d 582 (5th Cir.1978), remain the touchstone of our analysis. In both Smith and Spinkellink, we said the proof must establish specific acts evidencing intentional or purposeful discrimination “against the petitioner” on the basis of race. Id. at 614 n. 40, quoted in Smith, 660 F.2d at 585 (emphasis supplied in Smith). We found the tenders in both Spinkellink and Smith to be based on conclusory assumptions of discrimination against those particular petitioners. In both cases, the sentencing discrepancies were explainable on nonracial grounds. However, in Smith we left open the possibility that statistical evidence might be adduced which would compel the conclusion that a particular defendant’s conviction was the product of intentional racial discrimination. 671 F.2d at 859. Prejean contends he has created a statistical proffer that fits this hole. He errs.
Essentially, Prejean’s proffer3 consists of two parts. The first part is comprised of a detailed description of the preliminary findings of a study tabulating statistics drawn from data reported by Louisiana law enforcement agencies to the FBI from 1976 to 1980. It also includes nationwide statistics presented to place the Louisiana data in national perspective. These statistics show no more than the rough correlation that, among all of those sentenced to die for every type of murder in Louisiana, youthful blacks and blacks who killed white victims made up a disproportionate number. This finding, by itself, does not entitle Prejean to an evidentiary hearing. As in Smith, the statistics proffered do not account, as they must, for the numerous racially neutral variables that this court has recognized are present in the sentencing of capital crimes. Smith, 671 F.2d at 859; see also Moore v. Maggio, 740 F.2d 308, 321-22 (5th Cir.1984); McCorquodale v. Balkcom, 705 F.2d 1553, 1556 (11th Cir.1983), cert. denied, — U.S. -, 104 S.Ct. 2161, 80 L.Ed.2d 546 (1984).
In the second part of his proffer, Prejean promises to submit to the district court at the evidentiary hearing the final results of the statistical study that would, by the use of certain unspecified statistical techniques, account for nonracial variables in the sentencing decision. These findings, Prejean alleges, would sufficiently compel the conclusion that his conviction was the product of intentional racial discrimination. While this tender does not suffer from the exact deficiencies we noted in Smith and Spinkellink, it is too speculative and con-clusory to entitle Prejean to an evidentiary hearing. In general, a district court is not required to conduct a hearing in the absence of specific, non-conclusory allega*487tions justifying issuance of the writ. Celestine v. Blackburn, 750 F.2d 353, 358 (5th Cir.1984); Baldwin v. Blackburn, 653 F.2d 942, 947 (5th Cir.1981), cert. denied, 456 U.S. 950, 102 S.Ct. 2021, 72 L.Ed.2d 475 (1982). The requirement of a detailed tender is particularly appropriate here where the burden on the petitioner is high indeed. Cf. McCleskey v. Kemp, 753 F.2d 877 (11th Cir.1985) (en banc) (concluding that a 230-variable regression statistical study, revealing a 6% disparity in capital sentencing, does not compel the conclusion of racial discrimination). Because the proffer was not detailed and did not set forth the specific methodology used and its bottom-line results, we cannot conclude that the district court erred in refusing to hold an evidentiary hearing on this issue. Cf. Maggio v. Williams, 464 U.S. 46, 104 S.Ct. 311, 78 L.Ed.2d 43 (1983) (per curiam) (finding no abuse of discretion in refusal to grant evidentiary hearing on guilt proneness of Witherspoon jury when evidence proffered was “tentative and fragmentary”).4
The Petition for Rehearing is DENIED and no member of this panel nor judge in regular active service on the court has requested that the court be polled on rehearing en banc (Federal Rules of Appellate Procedure and Local Rule 35), the Suggestion for Rehearing En Banc is DENIED.
. The Louisiana Supreme Court determined that there were a total of three first degree murder cases in both the district where the crime was committed and the district where the sentence was imposed. All three involved killings of family members; none involved a teenage defendant; and none resulted in a death sentence. State v. Prejean, 379 So.2d 240, 249 n. 3 (La.1979).
. In vacating a stay of execution in Maggio v. Williams, the Supreme Court expressly stated that in regards to the Louisiana statute "[o]ur prior actions are ample evidence that we do not believe that the challenge to district-wide, rather than state-wide, proportionality review is an issue warranting a grant of certiorari." 464 U.S. 46, 104 S.Ct. 311, 314, 78 L.Ed.2d 43 (1983).
. Prejean’s proffer is fully described in the prior opinion. 743 F.2d at 1100-01. It will not be repeated here.
. We are not called on to express, and thus do not express, any view on the precise statistical tender required to raise a fact issue that would' require a hearing. Moreover, we need not and do not reach the broader issue of how any such statistical tender would impact the verdict reached by a properly chosen and instructed jury operating under a constitutionally valid capital punishment statute.