The minimum of the reciprocals of some Poisson random variables

Let \begin{equation} Y_i:=\frac1{1+X_i}, \quad Y:=\min(Y_1,\dots,Y_k). \end{equation} Then \begin{equation} EY=\int_0^\infty P(Y>y)\,dy=\int_0^\infty P(Y_1>y)^k\,dy. \end{equation} Next, for $y\in(0,\frac1{1+2k})$ and $x:=\frac1y-1>2k$, we have \begin{equation} 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, \end{equation} by Chebyshev's inequality, whence $P(Y_1>y)\ge1-1/k$ and \begin{equation} 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} \end{equation} as $k\to\infty$. So, if the limit of $k\,EY$ exists, it must be $\ge\frac1{2e}>0$.