Let $X_n \geq 0$ be an i.i.d. sequence with all moments finite. Let $E_n \geq 0$ be an increasing sequence of random variables with $\lim_{n \to \infty}E_n = \infty$. If $Y_n = \min\{X_n,E_n\}$, then is $Y_n \neq X_n$ only finitely many times.
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- $\begingroup$ Do you have a response to the answer below? $\endgroup$Iosif Pinelis– Iosif Pinelis2025-04-23 16:59:18 +00:00Commented Apr 23 at 16:59
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This is not true. For instance, let the $X_n$'s be i.i.d. standard exponential random variables, and let $E_n:=\ln n$. Then $$\sum _n P(Y_n\ne X_n)=\sum _n P(X_n>Y_n)=\sum _n P(X_n>E_n)=\sum _n \tfrac1n=\infty.$$ So, by the Borel--Cantelli lemma, with probability $1$ the events $\{Y_n\ne X_n\}$ occur infinitely often.
