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Let $\{\mu_N\}$ be a sequence of random measures which converges almost surely in the weak sense to a deterministic measure $\mu$ with impact support.

The weak convergence does not necessarily imply the convergence of the moments. But, my question is that, if we add the assumption that the second moment of $\mu_N$ is almost surely bounded, can we deduce that the second moment of $\{\mu_N\}$ converges to the second moment of $\mu$?

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Suppose $\mu_N$ assigns mass $1/(2N^2)$ to each of $\pm N$ and mass $1-1/N^2$ to $0$. Then $\mu_N$ (which have second moment equal to $1$) converge weakly to the dirac measure at $0$.

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  • $\begingroup$ What if we add the assumption that the support is almost surely bounded? Do we have the result or not? $\endgroup$
    – Rostam22
    Jun 3 at 15:17
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    $\begingroup$ Yes, since in that case $x^2$ is a bounded continuous function on the support. $\endgroup$
    – Yuval Peres
    Jun 3 at 15:49

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