The problem is to prove that $E[\log|X_1-\bar{X}_n|]$ converges towards $E[\log|X_1|]$ for i.i.d. continuous random variables $X_1,\ldots,X_n$ with $E[X_i]=0$ and $Var[X_i]=1$, for example for Laplace variables with density $\exp(-|x|)/2$. Note that since $X_1-\bar{X}_n$ equals 0 with probability 0, we can try to prove the above result on the set $\{X_1\ne\bar{X}_n, n=2, 3,\ldots\}$ which has probability 0, so that the problem of $\log(0)$ above is discarded.
Thank you.