Correlation between two points on a random walk

Question

A random walk is Xt = X{t-1} + \varepsilont with X0 = 0 and shock variance \sigma^2 = 1. Find the correlation between X2 and X5.

Accepted Answer

For a random walk starting at 0, the covariance between two points equals the smaller of the two time indices times the shock variance: \operatorname{Cov}(Xs, Xt) = \min(s, t)\,\sigma^2. This holds because Xs and Xt share all the shocks up to time \min(s,t). With s = 2 and t = 5: \operatorname{Cov}(X2, X5) = 2 	imes 1 = 2, \quad \operatorname{Var}(X2) = 2, \quad \operatorname{Var}(X5) = 5. The correlation is \operatorname{Corr}(X2, X5) = \frac{2}{\sqrt{2 	imes 5}} = \frac{2}{\sqrt{10}} \approx 0.632. Because this correlation depends on the actual times 2 and 5, not just their gap of 3, the …

Correlation between two points on a random walk

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