Why does law of large numbers require the independence in IID?

Question

Why does law of large numbers require independence? What happens if the variables aren't independently distributed?

Answer 1

In [Jan20], I present results for random variables that are not identically distributed and not necessarily independent based on the approach by Etemadi. For instance, the following is true:

Theorem. Let $(X_n)_{n\in\mathbb N}$ be a sequence of non-negative real random variables such that for $S_n=X_1+\dots+X_n$ we have

• $\mathsf V S_n \le c\sum_{k=1}^n \mathsf V X_k$ for some $c\in\mathbb R$ and all $n\in\mathbb N$,
• each $X_n$ has finite variance and $\sum_{n\in\mathbb N} \frac{\mathsf V X_n}{n^2} < \infty$,
• $\sup_{n\in\mathbb N} \frac{\mathsf E{S_n}}n < \infty$.

Then the $X_n$ satisfy the Strong Law of Large Numbers, i.e. \begin{equation} \lim_{n\to\infty} \frac{S_n-\mathsf E S_n}n = 0 \quad \text{almost surely.} \end{equation}

Additionally, it should be noted that the previous Theorem is wrong if the condition of non-negativity is removed.

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[Jan20]: Maximilian Janisch, Kolmogoroff's Strong Law of Large numbers holds for pairwise uncorrelated random variables. In: Theory of Probability and its Applications, 2021, Volume 66, Issue 2, Pages 263-275. Available online at https://arxiv.org/abs/2005.03967.