When do supremum and expectation commute?

Question

This is an alternative form of the question in When do maximum and expectation commute?

When we looking for conditions on $G(t,x(t))$ such that $$ \sup\limits_{t\in [0,N]}E[G(t,X(t))]=E[\sup\limits_{t\in [0,N]}G(t,X(t))] $$ where $X(t)$ is a sequence of random variables indexed by $t$ (it's easy to see that $\sup\limits_{t\in [0,N]}E[G(t,X(t))]\leq E[\sup\limits_{t\in [0,N]}G(t,X(t))]$).

Notice that, we relax the space for $t$ from $[0, 1]$ to $[0, N]$, and also relax $X$ to a sequence of random variables indexed by $t$.

Here, $t$ can be viewed as the time instant especially for discrete-time case. A background of this question can be given, for example, for a discret dynamic input-output mapping: $$z=(A_{0}(t)+A(X(t))s$$ where $A_{0}(t)$ is a deterministic matrix with its entries to be time-variant, and $A(X(t))$ is a random matrix with its entries to be given from $X(t)$ in $t\in[0, N]$. In order to measure the size of the input-output of the matrix or operator $A_{0}(t)+A(X(t))$, we can choose $$G(t,X(t))=\frac{s'(A_{0}(t)+A(X(t))'(A_{0}(t)+A(X(t))s}{s's}$$

Any suggestion or reference is greatly appreciated!

Answer 1

Let $Y_t:=G(t,X(t))$. Let $M:=\max(Y_0,\dots,Y_N)$. Let $s$ be an integer in $[0,N]$ such that $EY_s\ge EY_t$ for all integers $t$ in $[0,N]$, so that $EY_s=\max_t EY_t$. Then $M\ge Y_s$ and (by your first display) $EM=EY_s$, whence $M=Y_s$ almost surely (a.s.), that is, $Y_s\ge Y_t$ a.s. for all integers $t$ in $[0,N]$. Vice versa, if $Y_s\ge Y_t$ a.s. for some integer $s$ in $[0,N]$ and all integers $t$ in $[0,N]$, then $\max_t EY_t=EY_s=E\max_t Y_t$.

Thus, the condition in your first display holds if and only if $G(s,X(s))\ge G(t,X(t))$ a.s. for some integer $s$ in $[0,N]$ and all integers $t$ in $[0,N]$.