Convergence of moments of binomial to Poisson - Cross Validated most recent 30 from stats.stackexchange.com 2022-01-20T17:12:06Z https://stats.stackexchange.com/feeds/question/167212 https://creativecommons.org/licenses/by-sa/4.0/rdf https://stats.stackexchange.com/q/167212 7 Convergence of moments of binomial to Poisson Luis Mendo https://stats.stackexchange.com/users/28285 2015-08-14T21:28:39Z 2016-12-30T18:32:31Z <p>As is well known, the $\mathsf{Binomial}(n,p)$ <em>distribution</em> converges to the $\mathsf{Poisson}(a)$ distribution as $n\rightarrow \infty$, $p\rightarrow 0$ with $np=a$.</p> <p>I'm pretty sure that the <em>moments</em> of $\mathsf{Binomial}(n,p)$ also converge to those of $\mathsf{Poisson}(a)$, but I don't know how to prove it. Convergence in distribution doesn't imply convergence of moments, in general. <strong>How can I prove that the moments converge?</strong></p> <p>I've found that the binomial <em>probability (mass) function</em> converges <em>uniformly</em> to the Poisson one. This is stronger than convergence in distribution, so perhaps it can be exploited (but if so I don't know how).</p> https://stats.stackexchange.com/questions/167212/-/167217#167217 4 Answer by Luis Mendo for Convergence of moments of binomial to Poisson Luis Mendo https://stats.stackexchange.com/users/28285 2015-08-14T22:04:32Z 2016-12-30T18:32:31Z <p>I think I found an answer using <a href="https://en.wikipedia.org/wiki/Factorial_moment" rel="nofollow noreferrer">factorial moments</a>. <em>Still, I will accept someone else's answer if they can shed some light into the more general case, such as giving sufficient conditions that assure convergence</em>.</p> <p>The $r$-th factorial moment of the binomial distribution is easily computed as $$\mathrm E[(X)_r] = (n)_r\, p^r,$$ where $(a)_r = a(a-1)\cdots(a-r+1)$ denotes the <a href="https://en.wikipedia.org/wiki/Pochhammer_symbol" rel="nofollow noreferrer">falling factorial</a>; and that of the Poisson distribution is $$\mathrm E[(X)_r] = a^r.$$ It is clear that $(n)_r\, p^r$ tends to $a^r$ as $n\rightarrow \infty$, $p \rightarrow 0$ with $np=a$. Thus the binomial factorial moments converge to the Poisson ones.</p> <p>The $r$-th moment is a linear combination of the $0$-th, ..., $r$-th factorial moments: $$\mathrm E[X^r] = \sum_{k=0}^r \left\lbrace\ r\atop k \right\rbrace \mathrm E[(X)_r],$$ where $\left\lbrace\ r\atop k \right\rbrace$ denotes the <a href="https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow noreferrer">Stirling numbers of the second kind</a>. This expression, together with the convergence of the factorial moments, implies the convergence of the moments.</p> https://stats.stackexchange.com/questions/167212/-/167219#167219 12 Answer by whuber for Convergence of moments of binomial to Poisson whuber https://stats.stackexchange.com/users/919 2015-08-14T22:19:22Z 2015-08-14T22:19:22Z <p>Because the characteristic function (cf) of a Bernoulli$(p)$ variate is</p> <p>$$\psi_p(t) = 1 + p(e^{it}-1),$$</p> <p>the <a href="https://en.wikipedia.org/wiki/Binomial_distribution">cf of a sum of $n$ independent such variates</a> (which is a Binomial$(p,n)$ variable) is</p> <p>$$\psi_p(t)^n = \left(1 + p(e^{it}-1)\right)^n = \left(1 + \frac{np(e^{it}-1)}{n}\right)^n.$$</p> <p>It is well known (and easy to show, even for Complex numbers) that</p> <p>$$\left(1 + \frac{x}{n}\right)^n$$</p> <p>converges to $\exp(x)$ as $n\to \infty$. Keeping $np=a$ constant as $n$ increases allows us to write</p> <p>$$x = np(e^{it}-1) = a(e^{it}-1).$$</p> <p>Therefore</p> <p>$$\psi_p(t)^n \to \exp(x) = \exp(a(e^{it}-1)).$$</p> <p>Because this is the <a href="https://en.wikipedia.org/wiki/Poisson_distribution">characteristic function of a Poisson$(a)$ distribution</a>, and all the characteristic functions we have considered are analytic in a neighborhood of $t=0$ with power series whose coefficients give the moments, the moments of the Binomial distributions must have converged to the moments of this Poisson distribution, <em>QED</em>.</p>