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$X_i$ be a simple symmetric random walk. $Y_i= \operatorname{sign}(X_i)$. Define $Z_n= \sum_{i=1}^{n} Yi$. What is an upper bound on $Var(Z_n)$?

I feel the main issue is that $Y_i$ are not independent. Maybe there is some martingale hidden around which I’m not sure.

PinkyWay
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pulpfictional
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  • What is $sign(0)$? Is it 0 or 1 or -1? – Michael Jul 25 '19 at 17:59
  • It is $0$. But should it actually affect the answer? – pulpfictional Jul 25 '19 at 18:43
  • With $sign(0)=0$ and assuming $X_1=0$ then we get $E[Z_n]=0$ for all $n$ and so $$Var(Z_n)=E[Z_n^2]=\sum_{i=2}^nP[X_n\neq 0] + \sum_{i,j \in {2, ..., n}, i\neq j} E[Y_iY_j]$$ and the right-hand-side is at most $(n-1) + (n-1)(n-2) =(n-1)^2$. – Michael Jul 25 '19 at 20:50

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