Some questions on random sum of i.i.d random variables

Question

Recently, I have started studying stochastic processes based on Ross. However, at the start of the book, I have struggled with some difficulties. Actually, I can't understand the following example: Indeed, I can't understand why in the highlighted formula he argues that the sum in $\sum$ is independent of $N$? intuitively they are dependent.

The value of $N$ does not affect the individual $X_i$, due to independence. So while $\exp\left{t \sum\limits_1^N X_i\right}$ does depend on $N$, for any given $n$ you have $\exp\left{t \sum\limits_1^n X_i\right}$ is independent of $N$ — Henry, Mar 11 '22 at 08:59
Thanks a lot, why $E[exp(t\sum_{i}^{N}X_i)|N=n]=E[exp(t\sum_{i}^{n}X_i)|N=n]$? — Optimized Life, Mar 11 '22 at 09:07

score 2 · Accepted Answer · answered Mar 11 '22 at 08:41

2

$N$ is independent of the entire sequence $(X_1,X_2,...)$ so it is independent of $(X_1,X_2,...,X_n)$. This implies that it is independent of $ f(X_1,X_2,...,X_n)$ for any measurable function $f$ on $\mathbb R^{n}$. Take $f(x_1,x_2,...,x_n)=e^{ t\sum\limits_{i=1}^{n} x_i}$. This function is continuous, hence measurable.

answered Mar 11 '22 at 08:41

Kavi Rama Murthy

311,013

thanks a lot, why $E[exp(t\sum_{i}^{N}X_i)|N=n]=E[exp(t\sum_{i}^{n}X_i)|N=n]$? – Optimized Life Mar 11 '22 at 09:07
@OptimizeLife $E(Y|A)=\frac 1 {P(A)} EYI_A$. Take $A=N, Y=e^{t\sum\limits_{i=1}^{N}X_i} $ and note that $e^{t\sum\limits_{i=1}^{N}X_i} I_A$ is same as $e^{t\sum\limits_{i=1}^{n}X_i} I_A$. – Kavi Rama Murthy Mar 11 '22 at 09:18
Can you suggest to me a good book in order to learn these kinds of perquisites? – Optimized Life Mar 11 '22 at 09:29
K L Chung's A Course in Probability Theory is a good book. @OptimizeLife – Kavi Rama Murthy Mar 11 '22 at 09:33
Thanks a lot for your response – Optimized Life Mar 11 '22 at 09:34

Some questions on random sum of i.i.d random variables

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