4 problems

Self-normalized moderate deviation conjecture. The two displayed approximations hold in the stated ranges of $x$. These estimates would extend precise moderate-deviation results fo…

Let $X_1,X_2,\ldots$ be identically distributed random variables with $\mathsf{E}X_1=0$, set $S_k=\sum_{i=1}^k X_i$ and $V_n=(\sum_{i=1}^n X_i^2)^{1/2}$, and let $\Phi$ denote the…

Logan et al.'s conjecture. The sequence $\psi_n(\mathbf{1},\mathbf{Y})$ is asymptotically normal if and only if $\mathbf{Y}$ is in the domain of attraction of the normal law.

Logan–Mallows–Rice–Shepp conjecture. The ratio $S_n/V_n$ is asymptotically normal if, and perhaps only if, $X$ is in the domain of attraction of the normal law.