# tightness and convergence for Hilbert spaces

Let $$(X_n,n\geqslant 1)$$ and $$(Y_n,n\geqslant 1)$$ be sequences of stochastic processes taking values in some Hilbert space $$H$$ and defined on the same probability space. Assume that $$(X_n,n\geqslant 1)$$ converges weakly to some limit process.

Next, suppose that $$\lVert Y_n\rVert_{H} \leq \lVert X_n\rVert_{H}$$ for all $$n$$.

Is $$(Y_n,n\geqslant 1)$$ tight?

In general, I am confused with the following: Assume me have a majorating weakly convergent process. Then, another process, majorated by convergent one, is not necessarily convergent, but it it not (is it?) tight either.

In general, your conjecture is not true. You can see it by the obversation that: One direct implication from your conjecture is that
"Every sequence of random variables bounded in $$H$$ is tight"
Which is clearly wrong for any infinite-dimensional normed vector space (not hibert space alone).
For example,taking $$H= l_2$$ ; $$Y_n= e_n$$ . The sequence $$(e_n)$$ is bounded, while it is clearly not a relatively compact subset of $$H$$.
(Though it converges in weak topology, but this topology is not the canonical topology when we regard $$H$$ as a normed vector space)

• More precisely, it is :" If $x_n$ converges to something in H and $y_n$ satisfies that $|x_n| \ge |y_n|$ , then $\{y_n\}$ is a relative compact subset of $H$" Nov 17 '20 at 16:23