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?

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

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