This follows up on my earlier posts about time and eternal recurrence: If time is 2-way infinite and there is eternal recurrence, then I think persons are reincarnated. Each person has lived before and will live again, infinitely many times. Here is a paper I wrote explaining this: https://philpapers.org/rec/HUEEIE
Tl;dr:
Premise: There is a nonzero initial probability that persons are repeatable (can have multiple lives).
Also, the probability that you would be alive now given that persons are repeatable is nonzero.
Evidence: You are alive now.
Claim: The probability that you would be alive now, given that persons are unrepeatable, and that there is an infinite past, is zero. Rough explanation: there were infinite opportunities for you to exist in earlier centuries, which, if persons are unrepeatable, would have prevented you from existing now.
But you do exist now, so either the past is finite, or persons are repeatable. Bayesian calculation: Let H=[persons are repeatable], E=[You exist now]:
P(H|E) = P(H)*P(E|H) / [P(H)*P(E|H) + P(~H)*P(E|~H)]
= P(H)*P(E|H) / [P(H)*P(E|H) + P(~H)*0]
= P(H)*P(E|H) / P(H)*P(E|H)
= 1 (provided P(H), P(E|H) are nonzero)
If persons are repeatable, then they will repeat, given sufficient time. Conclusion: if the past is infinite, then persons are reincarnated infinitely many times.