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Let $\widehat{\sum}=\left(a_{i,j}^{N}\right)_{1\leq i\leq j\leq N}$ and $\sum=\left(a_{i,j}\right)_{1\leq i\leq j\leq N}$.

If $a_{i,j}^{N}\overset{P}{\to} a_{i,j}$ for any $1\leq i\leq j\leq N$, then $\widehat{\sum}\overset{P}{\to} \sum$? Is this true or are there additional conditions required?

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This is true! No further conditions required here. The given condition is enough to prove that the maximum difference of the entries with the limiting matrix converge in probability to zero, and every norm on matrices is equivalent (in terms of convergence) with the sup norm.

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  • $\begingroup$ Thanks for your kindly help. I am a rookie in random matrix. $\endgroup$
    – user381975
    Sep 15 at 0:49
  • $\begingroup$ @user381975 no problem! If you are satisfied with the answer, please click the green check mark. That way no-one else will bother looking at it, and I get meaningless internet points. It is a win-win! $\endgroup$ Sep 15 at 2:48
  • $\begingroup$ Many thanks for your comments. $\endgroup$
    – user381975
    Sep 15 at 11:47

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