Timeline for The minimum of the reciprocals of some Poisson random variables

7 events

when toggle format	what		by	license	comment
Jan 20, 2019 at 8:43	answer	added	RaphaelB4		timeline score: 1
Jan 20, 2019 at 7:40	vote	accept	Chuck Newton
Jan 20, 2019 at 0:32	comment	added	Iosif Pinelis		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.
Jan 20, 2019 at 0:29	comment	added	Iosif Pinelis		@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 $P(X_1>x)$.
Jan 18, 2019 at 14:57	comment	added	RaphaelB4		Yes and using a large deviation principle one can proves that this goes to 1.
Jan 18, 2019 at 14:35	answer	added	Iosif Pinelis		timeline score: 3
Jan 17, 2019 at 3:18	history	asked	Chuck Newton	CC BY-SA 4.0