When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.:
$f_m(x)=\frac{n!}{(k!)^2}2^{-n}\frac{1}{b}e^{-(k+1)|x|/b}(2-e^{-|x|/b})^k$
where the p.d.f. of the underlying Laplace distribution is given as $f(y)=\frac{1}{2b}e^{-|y|/b}$.
The formula for p.d.f. of the median stems from the usual method of characterizing the distributions of order statistics and is found as equation (2.5.10) in Kotz's volume on Laplace distribution. There is another formula for the case when $n$ is even, but we shall not be concerned with it for now.
I am interested in the variance of the sample median. Since $f_m(x)$ is symmetric about $x=0$, I can get rid of the absolute value and write it as follows:
$\sigma^2_m=\frac{n!}{(k!)^2}2^{-n}\frac{1}{b}\int_{0}^{\infty}x^2e^{-(k+1)x/b}(2-e^{-x/b})^kdx$
What I need is to reduce the following definite integral to a more manageable form:
$\int_{0}^{\infty}x^2e^{-(k+1)x/b}(2-e^{-x/b})^kdx$
For fixed $k$ this is fairly easy integral to solve. However, I am interested in asymptotics of variance $\sigma_m^2$ as $n\rightarrow\infty$ and $b$ is a linear function of $\sqrt{n}$, and thus need a solution for an arbitrary $k$.
Any ideas?