# Rate of convergence of the maxium of a random sequence

I came across a problem which requires the rate of convergence of $\sup_{1\le j \le N} |\sum_{i=1}^{N}X_{i,j}/N|$. If sup over a finite number of objects, this is a simple application of LLN. However, what if the number of objects is growing?

• What assumptions do you have on your grid $(X_{ij})_{1\le i,j\le N}$?. – Marc May 15 '17 at 14:41
• $X_{i,j}$ is independent across j but weakly dependent across i. Any kind of relaxation of identically distributed assumption can be imposed. – Yanhg May 15 '17 at 14:58