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.

Let $\ (\Omega,F,P)$ a probability space $X, X_n, n=1,2,\ldots$ are real valued random variables on $ (\Omega,F,P)$. Assume that $\ E[e^{c|X|}]< \infty$ for some $c>0$. Define $\ X_n = n(e^{X/n}-1), n\geq 1 $. By MVT for every $\ n\geq 1$ and every $\omega \in \Omega $ there exists $\ t_n(\omega) \in (0,1/n)$ s.t $\ X_n(\omega)$= $\ X(\omega)$ $\ e^{t_n(\omega)} $. $\ X(\omega)$ choose $\ n_0$ s.t $c>2/n_0$.

Find an integrable random variable $Y$ on $\ (\Omega,F,P)$ s.t $|X_n| \leq Y $ for all $\ n\geq n_0$

share|improve this question
add comment

1 Answer

You proved that $|X_n|\leqslant|X|\mathrm e^{|X|/n}$. Note that, for every $x\geqslant0$, $x\leqslant (2/c)\mathrm e^{cx/2}$ hence $|X_n|\leqslant (2/c)\mathrm e^{c|X|/2}\mathrm e^{|X|/n}$. If $n\geqslant 2/c$, $|X|/n\leqslant c|X|/2$ hence $|X_n|\leqslant Y$ with $Y=(2/c)\mathrm e^{c|X|}$ which is integrable.

share|improve this answer
add comment

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.