Strong law of large numbers for triangular arrays

Question

Consider a triangular array $X_{n,1},\ldots,X_{n,n}$ of rowwise i.i.d. real random variables with $ \sup_{n \in \mathbb{N}} \mathbb{E}\vert X_{n,1} \vert < \infty$ and $ \lim_{n \rightarrow \infty} \mathbb{E} X_{n,1} := \mu < \infty $ exists. Does under the standing assumptions the strong law of large numbers hold, i.e. \begin{align} \frac{1}{n} \sum_{i=1}^n X_{n,i} \stackrel{a.s.}{\longrightarrow} \mu \qquad \text{as } n \longrightarrow \infty? \end{align} If yes and if it is not a trivial conclusion from a well known theorem, do you know some references where the statement is written down?

Answer 1

The answer is negative, moreover, even the convergence in probability may fail.

For example, let $X_{n,i} = n$ with probability $1/n$ and $0$ with probability $1-1/n$. Then, $\mathrm{E}[X_{n,i}] = 1$ for all $n,i$, but $$ \mathrm P\Bigl(\frac 1n \sum_{i=1}^n X_{n,i} =0 \Bigr) = \Bigl(1-\frac1n\Bigr)^n \to \frac1e,\quad n\to\infty. $$ (Actually, by the Poisson limit theorem, $\frac 1n \sum_{i=1}^n X_{n,i}$ converges in law to the Poisson distribution with parameter $1$.)

Concerning some positive answers under additional assumptions, see e.g. this topic.