Correlation of a random walk at two times

Question

A symmetric random walk starts at 0 and takes independent \pm 1 steps. Let S{20} be the position after 20 steps and S{45} the position after 45 steps. What is the correlation between S{20} and S{45}?

Accepted Answer

Each step has variance 1, so 	ext{Var}(S{20}) = 20 and 	ext{Var}(S{45}) = 45. Write S{45} = S{20} + D, where D = S{45} - S{20} is the sum of steps 21 through 45, independent of S{20}. Covariance. 	ext{Cov}(S{20}, S{45}) = 	ext{Var}(S{20}) + 	ext{Cov}(S{20}, D) = 20 + 0 = 20. Correlation. ho = \frac{20}{\sqrt{20 \cdot 45}} = \frac{20}{\sqrt{900}} = \frac{20}{30} = \frac{2}{3} \approx 0.667. Equivalently, ho(Ss, St) = \sqrt{s/t} = \sqrt{20/45} = \sqrt{4/9} = 2/3. Takeaway. A random walk's positions at times s < t satisfy ho(Ss, St) = \sqrt{s/t}, the discrete twin of Brownian motion's…

Correlation of a random walk at two times

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