Finite Expected Value of Martingale

Question

I don't know how demonstrate that an expected value of a sum of iid r.v. is finite. I know that an expected value is finite in a continuos random variable when its integral is convergent.

But how to demonstrate for example:

$E[|V_n|]< ∞$ where $V_n=\displaystyle\sum_{i=1}^{n}exp(X_i)$ and $X_i$~$N(μ,σ^2)$

or another one

$E[|exp(S_n)|]< ∞$ where $S_n=\displaystyle\sum_{i=1}^{n}Xi$ and $X_i$~$Bernoulli(p)$

i know it's really easy question but i never did this before. thank you in advance

Answer 1

You don't really need absolute value bars here since $e^x>0.$

We have $$ E(V_n) = E(e^{X_1}+\ldots+e^{X_n}) \le E(e^{X_1} + \ldots +e^{X_n}) = nE(e^{X}).$$

For the second one, $$ E(e^{S_n}) = E(e^{X_1}e^{X_2}\ldots e^{X_n})=E(e^X)^n.$$

Answer 2

You have $e^{X_k}>0$ so $V_n = \sum_k e^{X_k} > 0$ and hence $$ \mathbb{E}[|V_n|] = \mathbb{E}[V_n] = \sum_{k=1}^n \mathbb{E}\left[e^{X_k}\right] $$ by independence and the only thing left is to show that $\mathbb{E}\left[e^{X_k}\right] < \infty$.