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In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the mean ergodic theorem preceding the point-wise ergodic theorem and Shannon-McMillan theorem preceding Shannon-McMillan-Breiman theorem.

Commenters are invited to mention results proven for convergence in probability but that are still open problems regarding to convergence almost-surely.

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    $\begingroup$ Let $X=(X_n)_{n\geq 0}$ be a simple random walk on $\mathbb{Z}$ with $p\in(1/2,1)$, where $p$ is the probability of an upward step. Let $Y_n=I(\text{both $X_n$ and $X_n+2$ are prime})$. It's known that $Y_n\to 0$ in probability but it's not known if the a.s. convergence holds. (In comparison, for $Z_n=I(\text{$X_n$ is prime})$ then again the convergence in probability holds, but the a.s convergence is known to fail.) $\endgroup$
    – James Martin
    Commented Nov 29 at 10:52
  • $\begingroup$ Thanks. Though I am more interested in convergence in probability results that are expected to also be true for convergence almost-surely. $\endgroup$
    – Matan Tal
    Commented Nov 29 at 14:57
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    $\begingroup$ I was being slightly flippant (apologies). If you want an example in a similar spirit where convergence almost surely is expected, you can take $W_n$ to be the indicator function of the event that $X_n >\max(X_1, \dots, X_{n-1})$ and $X_n$ is not the sum of three primes. $\endgroup$
    – James Martin
    Commented Nov 29 at 20:14
  • $\begingroup$ But this proven, no? $\endgroup$
    – Matan Tal
    Commented Nov 29 at 21:35
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    $\begingroup$ It would imply Goldbach's conjecture, wouldn't it? If 100 is the sum of three primes then 98 is the sum of two primes etc. $\endgroup$
    – James Martin
    Commented Nov 30 at 1:11

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