Martingale-like properties of extreme value process?

Question

Imagine I have a sequence of i.i.d. random variables $x_1,x_2,...,x_n,...$ and let $M_n = \max(x_1,...,x_n)$. Is there something I can say about the expectation $\mathbb{E}[M_n | M_{n-1}]$? What if $x_1,...,x_n$ have a known distribution, such as the exponential distribution or an extreme value distribution?

Answer 1

Since the cumulative maximum will only increase if the next $X_n$ is greater than the current value of the maximum, we have $$E(M_n\mid M_{n-1}) = P(X\le M_{n-1})M_{n-1} + P(X> M_{n-1})E(X\mid X> M_{n-1}).$$

In particular, for an exponential with mean $\theta$, this works out to $$ (1-e^{-M_{n-1}/\theta})M_{n-1} + e^{-M_{n-1}/\theta}(M_{n-1}+\theta)=M_{n-1} +\theta e^{-M_{n-1}/\theta}$$ where we used the memorylessness property to get the conditional expectation.

Answer 2

We also know $M_n$ is a sub-martingale since $M_{n+1} \geq M_n$ almost surely, so $$\mathbb{E}[M_{n+1}|M_1,\ldots,M_n] = \mathbb{E}[M_{n+1} | M_n] \geq M_n.$$