Problem 1 Let Yn be a symmetric random walk; that is Yn = X1 +_+Xn, where X1,X2, is a sequence of independent identically distributed random variables such that P(Xn = 1) = P(Xn = -1) = 0.5 Show that Y2 n is a martingale.
Added by Alba G.
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Since $Y_{2n} = X_1 + \cdots + X_{2n}$, we have: $$E[|Y_{2n}|] = E[|X_1 + \cdots + X_{2n}|] \leq E[|X_1|] + \cdots + E[|X_{2n}|]$$ Since $P(X_n = 1) = P(X_n = -1) = 0.5$, we have $E[|X_n|] = 0.5(1) + 0.5(1) = 1$. Show more…
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