Convergence of maximum of dependent RVs

Question

I've been stumped on the following for a while and I am.hoping you could be of some help. We have some (not necessarily independent) real valued random variables $Y_i$ such that $\frac{Y_n}{\sqrt{n}}\to 0$ and $\frac{Y_0}{\sqrt{n}}\to 0$. I was wondering if we could conclude that $\frac{\max\limits_{k=0,...,n} |Y_k|}{\sqrt{n}}\to 0$. I suspect it does hold (it is not too difficult to show if we have sequences of real numbers), but I am having trouble to show it in this case. Could anyone help? Thanks!

Answer 1

If you deal with almost sure convergence, then it follows from the case of real numbers.

If you deal with convergence in probability, then this may not be true. Define $Y_n=\sqrt n\mathbf 1_{A_n}$, where $p_n:=\mathbb P\left(A_n\right)\to 0$ as $n$ goes to infinity. Then $Y_n/\sqrt n\to 0$ in probability but if $\left(\mathbb P\left(\bigcup_{j=n+1}^{2n}A_j\right)\right)_{n\geqslant 1}$ does not converge to $0$, then the sequence $\left(\max_{1\leqslant j\leqslant 2n} \left|Y_j\right| /(2\sqrt n) \right)_{n\geqslant 1}$ does not converge to $0$ in probability. To get an explicit example, consider independent sets $(A_n)_{n\geqslant 1}$ such that $p_j=1/2$ and use Bonferroni's inequality to get $$ \mathbb P\left(\bigcup_{j=n+1}^{2n}A_j\right ) \geqslant \sum_{j=n+1}^{2n}p_j-\sum_{n+1\leqslant i\lt j\leqslant 2n}p_ip_j \geqslant \sum_{j=n+1}^{2n}p_j-\frac 12\left(\sum_{j=n+1}^{2n}p_j\right)^2.$$