Let $X_1, X_2, X_3, ...$ be i.i.d variables, and for every $i$ $X_i$ has variance.
Define $S_k=\sum_{i=1}^{k}X_i$.
Calculate $\rho(S_m,S_n)$ for $m\leq n$.
Well, I know it should be $\sqrt{ m/n }$, but I don't know how to show that. Can someone please give me a hint?
Hint: The covariance is bi-linear, that is, for any two sequences $(Y_i)_{1\leq i\leq n}$ and $(Z_j)_{1\leq j\leq m}$ ones has $$ \mathrm{Cov}\Big(\sum_{i=1}^n Y_i,\sum_{j=1}^m Z_j\Big)=\sum_{i=1}^n\sum_{j=1}^m \mathrm{Cov}(Y_i,Z_j). $$
Apply this to $S_m$ and $S_n$ and note that many of the covariances are zero (why?). Next, find the variance of $S_m$ and $S_n$ and then apply the definition of correlation: $$ \rho(Y,Z):=\frac{\mathrm{Cov}(Y,Z)}{\sqrt{\mathrm{Var}(Y)\mathrm{Var}(Z)}}. $$