# 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.

## 1 Answer

Answer
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)

Comments
My two-cent comment: If I'm not wrong, what you want to search is essentially the relation between the tightness and the norm of value space (the codomain). As I showed in the previous example, the first step you need to overcome is to form the bridge between the compacity and norm. However, while that relationship is clear in finite dimension, it is extremely weak when you deal with infinite-dimensional space. Much work needs to be done, some change in approach might be nessessary.

• Dear @Paresseux Nguyen , I am confused about the following: "norm reduction does not guarantee tightness"...
– LrM
Nov 17 '20 at 15:41
• I don't know if I ever gave that line. Perhaps my explanation is not clear. Would you mind elaborating more on the part that you are confused about in my comment? Nov 17 '20 at 15:44
• no... this is my question, it is not about you comment
– LrM
Nov 17 '20 at 16:15
• Hm, so let's put it this way. If every random variable you have is deterministic ( no randomness in it) , what will your statement be? So, it will be a statement telling a relation between the compacity and norm. Nov 17 '20 at 16:20
• 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