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Let $X_{i,j}$ be $K \times N$ i.i.d. random variables such that $P(X_{i,j} = 1) = P(X_{i,j} = -1) = \frac{1}{2}$, and $S_p = \sum_{q=1}^{N} X_{p,q}$ be $K$ i.i.d. random variables, each of which is an $N$ step random walking.

How to prove $E[\max_{p \in [K]} S_p] \ge \Omega(\sqrt{N\log K})$?

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