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Assume we have two sequences of random vectors $(X_{n})_{n > 1}$ and $(Y_{n})_{n > 1}$ taking values in $\mathbb{R}^{k}$ and defined on the same probability space. Also, we know that $(X_{n})_{n > 1}$ converges weakly to some random $X$ and $E[||X_{n}||_{2}^{2}] < \infty$ for all $n$.

Next, for some fixed continuous function $g: \mathbb{R}^{k}\times \mathbb{R}^{k} \to \mathbb{R}_{+}$ let $Z_{n}$ be the following random variable

$$ Z_{n} = g(X_{n}, Y_{n}) $$ and assume there exists $n_{0}$ such that for all $n>n_{0}$ the following holds almost surely $$ Z_{n} \leq C ||X_{n}||_{2}^{2}, $$ for some constant $C$.

Is the following correct?

$$ E[\limsup_{n}g(X_{n}, Y_{n})] \leq C \, E[||X||_{2}^{2}]. $$

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No, this is not correct. Draw $X_n=n$ with probability $1/n$ and $0$ with probability $1-1/n$, independently for all $n$. Pick $Z_n=X_n$. Then:

  • $(X_n)$ converges in distribution to $X\equiv 0$.
  • $(X_n\ge n)$ is a sequence of independent events whose sum of probabilities diverges, so by the second Borel-Cantelli lemma $(X_n\ge n)$ occurs infinitely often almost surely. In particular $\lim\sup X_n=+\infty$ almost surely.

So $X_n\le X_n^2$, $\mathsf{E} \lim\sup X_n=+\infty$ and $\mathsf{E} X^2=0$.

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