Partial sums of standard normals

Topic: Functions of random variables and limit theorems · more questions on this topic
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Let X_1,X_2,\ldots be independent and standard normal, and set Y_n=X_1+\cdots+X_n, with Y_1=X_1.

1. Describe a typical path of X_1,\ldots,X_n and of Y_1,\ldots,Y_n for large n.

2. Find \mathbb{E} Y_n and \operatorname{Var}(Y_n).

3. Find \operatorname{Cov}(Y_m,Y_n) for m\le n.

Worked answer (click to show)

1. The X_i fluctuate about 0 and do not wander. The partial sums Y_n form a random walk: the path is continuous only in the discrete-time sense of successive additions, and it drifts away from 0 on the scale \sqrt{n}.

2. \mathbb{E} Y_n=0 and \operatorname{Var}(Y_n)=n.

3. For m\le n, Y_n=Y_m+X_{m+1}+\cdots+X_n. The extra sum is independent of Y_m and has mean 0, so \operatorname{Cov}(Y_m,Y_n)=\operatorname{Var}(Y_m)=m.