Solved – Sum of iid random variables

Let $X_1, X_2,…,X_n$ be iid random variables. Let $Z_1, Z_2, Z_3$ be defined as $X_1, X_1+X_2, X_1+X_2+X_3$ respectively. Are $Z_1, Z_2$ and $Z_3$ also iid's?

The question is based on renewal processes. If the inter arrival durations are iid, are the arrival times also iid?

Assume that $E[X_i]=0$. If this is not true, then simply subtract the mean, it's a constant, so it will not change anything.

For independent random variables $Cov[Z_nZ_{n-1}]=E[Z_nZ_{n-1}]=0$.

Evaluate the left hand side $$E[Z_nZ_{n-1}] =E[(Z_{n-1}+X_n)Z_{n-1}] =E[X_nZ_{n-1}]+E[Z^2_{n-1}] =E[Z^2_{n-1}]>0 $$ So, $Z_n$ is not independent of $Z_{n-1}$.

Here, we used $E[X_nZ_{n-1}]=0$, because $X_n$ is independent of all $X_i$.

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