I'm working on this question and it's stumping me.
Let Sn = X1 + ... + Xn (with n>=1) be a random walk with X1,...,Xn be iid RV's. E(Xk)=mu Var(Xk)=sigma^2.
Find the covariance of Sn and Sm
Can anyone help out? I am trying to use the equation
Cov[Sn, Sm] = E[SnSm] - E[Sn]E[Sm]
Just use properties of covariance (including its value under independence assumption).
Say $n<m$.
$${\rm Cov} \left(\sum_{i=1}^n X_i, \sum_{j=1}^n X_j\right) = \sum_{i=1}^n {\rm Var}(X_i),$$ $${\rm Cov} \left(\sum_{i=1}^n X_i, X_j\right) = {\rm Var}(X_j),$$ if $j\in \{1,\ldots,n\}$, and $0$, if $j\in \{n+1,\ldots,m\}$.
$${\rm Cov} \left(\sum_{i=1}^n X_i, \sum_{j=1}^m X_j\right) = {\rm Cov} \left(\sum_{i=1}^n X_i, \sum_{j=1}^n X_i\right) + {\rm Cov} \left(\sum_{i=1}^n X_i, \sum_{j=n+1}^m X_j\right) $$ $$ = \sum_{i=1}^n {\rm Var}(X_i) + \sum_{j=n+1}^m{\rm Cov} \left(\sum_{i=1}^n X_i, X_j\right) $$ $$= \sum_{i=1}^n {\rm Var}(X_i) + 0 = \sum_{i=1}^n {\rm Var}(X_i).$$
So the answer is: $${\rm Cov}(S_n, S_m)= \min(n,m)\cdot\sigma^2. $$