# The minimum of the reciprocals of some Poisson random variables

Let $$X_1,\dots,X_k$$ denote a collection of independent samples of a Poisson random variable whose mean also happens to be equal to $$k$$. Does the quantity $$k\boldsymbol{E}\min\left\{ \frac{1}{1+X_{1}},\dots,\frac{1}{1+X_{k}}\right\}$$ have a strictly positive limit as $$k$$ becomes large?

• Yes and using a large deviation principle one can proves that this goes to 1. – RaphaelB4 Jan 18 at 14:57
• @RaphaelB4 : Can you expand your comment into a formal answer? From my answer, it seems clear that, to get the value of the limit, one needs to have the asymptotics of $P(X_1>x)$ uniformly in the zone $x=O(k)$ (which will be given by a certain analytic function of $\lfloor x\rfloor$, rather than by an analytic function of $x$). On the other hand, a large deviation principle only gives the asymptotics of $\ln P(X_1>x)$ -- which is much, much less informative than the needed asymptotics of $P(X_1>x)$. – Iosif Pinelis Jan 20 at 0:29
• Previous comment continued: Cf. e.g. the asymptotics $\int_0^1(1-ct)^k\,dt\sim1/(ck)$ for $c\in(0,1)$ as $k\to\infty$, which of course depends on $c$, whereas $\ln(ct)\sim\ln t$ as $t\downarrow0$. So, I don't see how a large deviation principle by itself could be enough here. – Iosif Pinelis Jan 20 at 0:32

Let $$$$Y_i:=\frac1{1+X_i}, \quad Y:=\min(Y_1,\dots,Y_k).$$$$ Then $$$$EY=\int_0^\infty P(Y>y)\,dy=\int_0^\infty P(Y_1>y)^k\,dy.$$$$ Next, for $$y\in(0,\frac1{1+2k})$$ and $$x:=\frac1y-1>2k$$, we have $$$$P(Y_1>y)=1-P(X_1>x),\quad P(X_1>x)\le P(X_1>2k)=P(X_1-k>k)\le1/k,$$$$ by Chebyshev's inequality, whence $$P(Y_1>y)\ge1-1/k$$ and $$$$EY\ge\int_0^{1/(1+2k)} P(Y_1>y)^k\,dy\ge\int_0^{1/(1+2k)} (1-1/k)^k\,dy\sim\frac1{2ek}$$$$ as $$k\to\infty$$. So, if the limit of $$k\,EY$$ exists, it must be $$\ge\frac1{2e}>0$$.
For $$X$$ a Poisson of mean $$k$$, $$\mathbb{P}(X\geq (1+\epsilon)k) =\mathbb{P}(e^{X\epsilon/4}\geq e^{(1+\epsilon)k\epsilon/4}) \\ \leq \frac{\mathbb{E}(e^{X\epsilon / 4})}{e^{(1+\epsilon)k\epsilon/4}} =\frac{\exp(k(e^{\epsilon /4}-1))}{e^{(1+\epsilon)k\epsilon/4}} \\ \approx\exp(-k \epsilon^2(\frac{1}{4}-\frac{1}{2*4^2})$$ for small $$\epsilon$$. One can then finish the proof of Pinelis and get that the limit is larger than $$1/(1+\epsilon)$$.