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Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions $$ \lim_{N\rightarrow\infty}\prod_{n=1}^{N}1\{X_n\in A\} $$ and am interested in its limit behavior. Specifically, are there conditions on the dependence between the $X_n$, that are weaker than iid, and ensure convergence to zero in $L_1$ norm? That is $$ \lim_{N\rightarrow\infty}E\left(\prod_{n=1}^{N}1\{X_n\in A\}\right) = \lim_{N\rightarrow\infty}P\left(\bigcap_{n=1}^{N}X_n\in A\right)=0. $$

I have already asked a question regarding this issue here. I know that ergodicity implies that the product converges to zero almost surely, but in the link Anthony Quas argues that this does not hold in $L_1$ norm.

Any hint or reference is appreciated, thank you in advance!

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  • $\begingroup$ I think perhaps you misunderstood Anthony's comment. When the product converges to zero almost surely, so that $P(\bigcap_{n=1}^\infty \{X_n \in A\}) = 0$, then it certainly converges in $L^1$ as well. This is just the "continuity from above" property of probability measures and follows immediately from countable additivity. (Or you could use the dominated convergence theorem.) I understood Anthony's comment to be saying that you can't always get exponentially fast convergence to zero. $\endgroup$
    – Nate Eldredge
    Commented Oct 19, 2018 at 15:27
  • $\begingroup$ You are right. I guess I stared at this problem for too long, I'll delete this question. Thank you for the reply! $\endgroup$
    – Marc
    Commented Oct 21, 2018 at 11:57

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