# Strong law of large numbers for a sequence of random variables in different probability spaces

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 $$\lim_{N\to\infty}\frac{1}{N} \sum_{k=1}^N (X_k - \int X_k\,d\mu_k) = 0$$.

• 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. Sep 21 at 10:42
• Tried to clarify the question, removed mentions of "Bernoulli probability spaces" by which I meant a finite probability space Sep 21 at 11:41
• Unless I misunderstood your question, no: math.stackexchange.com/questions/3871332/… Sep 21 at 12:43
• 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) Sep 21 at 12:51

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.