# expectations of limits of norms of convergent r.v.

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?

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

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 $$\mathsf{E} X^2=0$$.