Explain it to me each equality at a time? lim_(n -> oo) E[e^(-X ** t/n)]^n ~~ lim_(n -> oo) E(1−(E[X]t)/(n))^n=e^(-E[X] ** t)

Aubrie Aguilar

Aubrie Aguilar

Answered question

2022-09-21

Explain it to me each equality at a time?
lim n E [ e X t n ] n lim n E ( 1 E [ X ] t n ) n = e E [ X ] t

Answer & Explanation

rae2721

rae2721

Beginner2022-09-22Added 8 answers

You need some assumption on X. I will assume that it is a positive random variable and that t 0. It is not clear what means so I will give a proof in two steps. The inequality e s 1 s is true for all s 0 so [ E e t X / n ] n [ 1 t E X / n ) ] n and the limit of this is e t E X . On the other hand ( E Y ) n E Y n for any positive random variable Y so [ E e t X / n ] n E [ e t X ]. You can combine these two fact to get the required result.

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