If I let $S = \min\{k \geq 1: X_k - Y_k > \sqrt{\log k}\}$ and have that $X_1, \ldots X_S$ are iid samples from $N(0,1)$ and that $Y_1, \ldots, Y_S$ are iid from $N(0,1)$ and also independent of the $X$ sample (Here the number of samples of X and Y depend stochastically on the set $S$), will $S$ be finite almost surely? In other words, I am trying to determine if $P(S < \infty) = 1$. Does anyone have an idea how I can attack this problem? Thanks!
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I don't see how $_i$ and $Y_i$ depend on $S$? – gt6989b Dec 14 '15 at 18:49
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$X_k-Y_k\sim N(0,2)$. Use asymptotics of $P(N(0,2)>\sqrt{\log k})$ to see if it's summable and the Borel Cantelli lemmas will decide whether $S$ happens a.s. or not. Overall, Law of Iterated Logarithm would imply that $P(S<\infty)<1$. – A.S. Dec 14 '15 at 18:56
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I was wrong re LIL. See http://math.stackexchange.com/a/1483735/274197 – A.S. Dec 14 '15 at 20:44
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Is that a stopping time? I think we can use http://math.stackexchange.com/questions/1467193/showing-stopping-is-finite-almost-surely/1468800#1468800 or https://books.google.com.hk/books?id=e9saZ0YSi-AC&pg=PA233&lpg=PA233&dq=%22what%20always%20stands%22%20%22probability%22&source=bl&ots=AWEio5s6Xq&sig=6kAyfQZorHRSFhYEVjQ83V7XGU4&hl=en&sa=X&ved=0ahUKEwje6t-I0MTJAhUleaYKHUa7AfIQ6AEIGjAA#v=onepage&q=%22what%20always%20stands%22%20%22probability%22&f=false – BCLC Dec 14 '15 at 22:48