Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

assume $\tau_{k}$'s are positive i.i.d. random variables. $\xi_{k}=\xi_{k-1}+\tau_{k}$, and $\xi_{0}$ is a given real number. $\chi_{A}$ is index function.$x_{0}$ is a given positive real number. I wonder if $E[\sum_{k=0}^{+\infty}(\prod_{i=1}^{k}(1+b_{i}(\tau_{i}))\chi_{[\xi_{k},\xi_{k+1})}(x_{0}))]=\sum_{k=0}^{+\infty}E(\prod_{i=1}^{k}(1+b_{i}(\tau_{i}))\chi_{[\xi_{k},\xi_{k+1})}(x_{0}))$ and why? ($b_{i}$'s are continuous functions)

share|improve this question

1 Answer 1

up vote 2 down vote accepted

The expectation of a sum of nonnegative random variables indexed by the integers is equal to the sum of the expectations, whether the resulting number is finite or infinite. This follows from Fubini's theorem for nonnegative functions (often called Tonelli's theorem) applied to the product measure which is the product of $P$ by the counting measure on the integers. If all your functions $1+b_i$ are nonnegative you are done and the continuity hypothesis is not useful. Otherwise you have to check the hypothesis of Fubini's theorem itself, that is, the convergence of the series of the expectations of the absolute values of the random variables involved.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.