Limit of a rescaled random sum of i.i.d. random variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:07:22Z http://mathoverflow.net/feeds/question/94516 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94516/limit-of-a-rescaled-random-sum-of-i-i-d-random-variables Limit of a rescaled random sum of i.i.d. random variables Ale Zok 2012-04-19T11:40:57Z 2012-04-19T19:53:06Z <p>Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$</p> <p>For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, independent of $X_i$ for every $i \in \mathbb N$. Suppose that, as $\alpha \to +\infty$, $$\frac{N(\alpha)}{E(N(\alpha))} \stackrel{d}{\longrightarrow} Y,$$ where $Y$ is a non-degenerate continuous probability distribution on $[0,+\infty[$.</p> <p>Is possible to say something about the limit in distribution of $\displaystyle \frac{S_{N(\alpha)}}{E(S_{N(\alpha)})}$, as $\alpha \to +\infty$?</p> http://mathoverflow.net/questions/94516/limit-of-a-rescaled-random-sum-of-i-i-d-random-variables/94538#94538 Answer by mike for Limit of a rescaled random sum of i.i.d. random variables mike 2012-04-19T15:01:01Z 2012-04-19T15:01:01Z <p>Do you want $E(N(\alpha))$ in the denominator ?</p> http://mathoverflow.net/questions/94516/limit-of-a-rescaled-random-sum-of-i-i-d-random-variables/94563#94563 Answer by Hans Engler for Limit of a rescaled random sum of i.i.d. random variables Hans Engler 2012-04-19T19:53:06Z 2012-04-19T19:53:06Z <p>A closely related problem was treated by H. Robbins, The asymptotic distribution of the sum of a random number of random variables, Bull. AMS 54(1948), 1151--1161, Math Reviews MR0027974. </p> <p>In essence, under suitable nondegeneracy assumptions and assuming the existence of finite second moments, Robbins proves that the asymptotic (as $\alpha \to \infty$) distribution of $\frac{S_N - E(S_N)}{\sqrt{var(S_N)}}$ is related to the asymptotic distribution of a linear combination of $\frac{N - E(N)}{\sqrt{var(N)}}$ and another normal r.v. $Z$. </p> <p>For the case where $var(N) = o(E(N)^2)$, the implication seems to be that $\frac{S_N}{E(S_N)} \stackrel{d}{\longrightarrow} 1$, a constant.</p>