I have a two-index succession of real-valued random variables $x_{t,n}$ such that $\lim_{n\to\infty} x_{t,n} = x_t$, for all $t$ and suitable limit r.v. $x_t$.

I would like to prove that $$\lim_{n\to\infty}\frac1n \sum_{t=1}^n x_{t,n} =\lim_{n\to\infty}\frac1n\sum_{t=1}^n x_t. \tag{*}$$

Of course this does not hold without additional hypotheses, even if the $x_{t,n}$ were numbers instead of random variables. In addition, my random variables are quite nasty: they are neither independent, nor identically distributed, nor bounded in norm (they have a Gaussian-like "tail" at infinity).

What additional hypotheses or criteria are there that could enable me to prove (*)?