Question

IfXi, X2, are independent and identically distributed random variables having finite expectations, and if N is a stopping time for X,, X2, such that E[N1<, then Proof Letting ifN n lo if N < n, we have that n-t Hence,
Why the equation 3.3.2 is true？Why can the limitation symbol and the expectation symbol be interchanged?

Answer 1

Given $X_1,X_2,..,X_n$ are IID random variables with $E\left (X_i \right )=E\left (X \right );i=1,2,..,n$

Define the index function as

$I_n=\left\{\begin{matrix} 1; & N\geqslant n\\ 0; & N<n \end{matrix}\right.$

Then $\sum_{n=1}^{N}X_n=\sum_{n=1}^{\infty }X_nI_n$

Now using the linearity of expectation, that is $E\left ( U&plus;V \right )=E\left ( U \right )&plus;E\left ( V \right )$

$E\left (\sum_{n=1}^{N}X_n \right )=E\left [ \sum_{n=1}^{\infty }X_nI_n \right ]\\ E\left (\sum_{n=1}^{N}X_n \right )= \sum_{n=1}^{\infty }E\left [X_nI_n \right ]\\ E\left (\sum_{n=1}^{N}X_n \right )= \sum_{n=1}^{\infty }E\left [X_nI_n \right ]\\$

$E\left (\sum_{n=1}^{N}X_n \right )= \sum_{n=1}^{N }E\left [X_nI_n \right ]\\E\left (\sum_{n=1}^{N}X_n \right )= \sum_{n=1}^{N }E\left [X_nI_n \right ]\\ E\left (\sum_{n=1}^{N}X_n \right )= \sum_{n=1}^{N }E\left [NX\right ]\\$

$E\left (\sum_{n=1}^{N}X_n \right )=E\left ( X \right )E\left ( N \right )=RHS$

The proof is complete.  The limit symbol and the expectation symbols are interchanged.