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Is it known whether the following version of the strong law of large numbers holds?

For each $k\in\mathbb{N}$, let $\Omega_k$ be a finite set and $\mu_k$ be a probability measure on $\Omega_k$. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of random variables, $X_k: \Omega_k \to [0,+\infty)$. Suppose in addition that $\sup_k \int X_k\,d\mu_k < \infty$.

Then with $\prod_{k=1}^\infty \mu_k$-probability one, we have $\lim_{N\to\infty}\frac{1}{N} \sum_{k=1}^N (X_k - \int X_k\,d\mu_k) = 0$.

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  • $\begingroup$ What is a "Bernoulli probability space" and what do you mean by "Bernoulli random variables" in the title? Normally, "Bernoulli random variables" are ones taking only values $0$ and $1$ - but then your additional condition of uniformly bounded expectations would be trivial, so I imagine you mean something else. $\endgroup$
    – James Martin
    Sep 21 at 10:42
  • $\begingroup$ Tried to clarify the question, removed mentions of "Bernoulli probability spaces" by which I meant a finite probability space $\endgroup$
    – Aleksi
    Sep 21 at 11:41
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    $\begingroup$ Unless I misunderstood your question, no: math.stackexchange.com/questions/3871332/… $\endgroup$
    – unwissen
    Sep 21 at 12:43
  • $\begingroup$ But maybe you have use for a version of the strong law which holds when the variances of $(X_k)_k$ do not grow too fast (as mentioned in en.wikipedia.org/wiki/Law_of_large_numbers#Strong_law) $\endgroup$
    – unwissen
    Sep 21 at 12:51

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Boundedness of the expectations is not enough.

You could consider the case where $X_k$ takes value $0$ with probability $1-1/(k+1)$ and value $k+1$ with probability $1/(k+1)$, so that $\mathbb{E} X_k=1$ for all $k$.

Then by Borel-Cantelli, with probability $1$ there are infinitely many $k$ with $X_k-\mathbb{E} X_k=k$, which means that $\frac{1}{N}\sum_{k=1}^N \left(X_k-\mathbb{E} X_k\right)$ does not converge to $0$ as $N\to\infty$.

By the way, giving all these details about the structure of the probability space is unusual and seems irrelevant. All that matters is the joint distribution of the random variables $X_k$. You could just as well say: let $X_k$ be a sequence of independent non-negative random variables, each one taking finitely many values.

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