Let $X_1, X_2, ..., X_n$ be independent, identically and exponentially distributed random variables, $P(x) = k \exp(-k x)$. Define $Y$ as the sum of the sequence of cumulative maxima:
$Y = X_1 + \max(X_1, X_2) + \max(X_1, X_2, X_3) + \dots + \max(X_1, X_2, \dots, X_{n})$
What distribution is followed by $Y$?
I know the formula for the distribution of $\max(X_1, X_2, \dots, X_{n})$ is:
$P(z) = nk(1-\exp(-kz))^{n-1}\exp(-kz)$
But I can't see how to find the distribution of $Y$, because none of the terms in the summation are independent of each other. I suspect some trick exploiting $\max(X_1,X_2,X_3) = \max(\max(X_1,X_2),X_3)$ will be useful here, but I'm stuck...