Please give a Name to this lemma

Asked Jul 02 '18 at 17:10

Active Jul 02 '18 at 17:20

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I'm reading a .pdf file about Probability. Here's the link

The author introduces the lemma 4.1, suggesting to consult the proof in a specific book. I tried to search the lemma in the book but i didn't find it. Please help me to find a name for this lemma so as i can consult its proof...

Lemma 4.1. Let $Z_1, Z_2, \dots$ be a sequence of random variables having distribution functions $F_{Z_n}$ and moment generating functions $M_{Z_n}$ s.t. $n \geq 1$. Furthermore, let $Z$ be a random variable having distribution $F_Z$ and moment generating function $M_Z$. If $M_{Z_n}(t) \to M_Z$ for all $t$, then $F_{Z_n}(t) \to F_{Z}(t)$ for all $t$ at which $F_Z(t)$ is continuous.

edited Jul 02 '18 at 17:20

Clarinetist

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asked Jul 02 '18 at 17:10

Qwerto

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If you won't take the trouble to retype the lemma, perhaps you could say what page it's on. – saulspatz Jul 02 '18 at 17:13
Could you give us directly the lemma (copy-paste it, for instance) instead of linking to a whole pdf? – paf Jul 02 '18 at 17:14
picture of the lemma added in the main post – Qwerto Jul 02 '18 at 17:15
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Most treatments of that proof are not appropriate for the level of probability expected that you've linked. You will need measure theory. See Arnaud's link. – Clarinetist Jul 02 '18 at 17:17
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This looks like the MGF version of Levy's continuity theorem. In any canse I'm not sure that this can be easily proven without using measure theory. – gcc-6.0 Jul 02 '18 at 17:32

Please give a Name to this lemma

0 Answers0