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The jury did, in fact, rely on the hearsay evidence. According to the link provided by OP:
"It was the hearsay evidence against Peterson, the juror said, that convinced him and other jurors to convict Peterson of murder on Thursday in the 2004 death of his third wife, Kathleen Savio."
There are certain exceptions to the hearsay evidence rule however, none of them apply in the Drew Peterson case. The conviction will certainly be overturned on appeal.
What Stacy Peterson said about Drew Peterson's conversations with her would have been considered hearsay even if she had taken the witness stand to offer this evidence. However, When Savio’s divorce attorney, Harry Smith, testified what Stacy Peterson had told him of Drew Peterson's conversation, his testimony was hearsay on hearsay – double hearsay! Smith's account of what Stacy Peterson allegedly said she saw was simple hearsay, but hearsay nonetheless.
The United States Constitution (Amendment VI) gives each defendant the right to confront his accusers. By using hearsay evidence against him Drew Peterson was denied that right and the appellate court will not allow his conviction to stand.
Look, I am convinced that Drew Peterson is guilty, but the prosecution messed up big time. Even a first year law student knows the fundamentals of the hearsay evidence rule and a defendant's constitutional right to confront his accusers. The prosecution should never have called the witnesses they did, because even if the circumstantial evidence alone may have been sufficient to convict Peterson (and there is no way of knowing this), the fact that the jury ADMITTED to relying on the hearsay evidence will result in the verdict being overturned - count on it. The prosecution blew it … big time.