Covariance of sum random process

Question

Sum random process is defined as: $$ X[n] = \sum_{i=0}^n U[i] \text{ for } n \ge 0 $$ $$ E[U[i]]=0 \text{, } \text{var}(U[i]) = \sigma^2_U \text{ for } i \ge 0 \text{ and } U[i] \text{ are IID} $$ This tells us that $E[X_n]=0$.
Show that $c_x[n_1, n_2] = \sigma^2\text{min}(n_1, n_2)$

Answer 1

The covariance is $E(X[n_1]E[n_2])=\sum_{i=0}^{n_1}\sum_{j=0}^{n_2}E(U(i)U(j))$.
For $j\ne i)$, $E(U(i)U(j))=0$.
The only terms remaining are $E(U(i)^2)=\sigma^2$ for $i\le min(n_1,n_2)$.
The answer is slightly incorrect- should be $\sigma^2 min(n_1+1,n_2+1)$.