5 problems

Simultaneous upper LIL conjecture. For every $\epsilon>0$,

Lower-capacity replacement conjecture. The assertion of Proposition 1.1 remains true when $\mathcal V^{\mathscr{P}}$ is replaced by $\widehat{\mathcal V}^{\ast}$. This conjecture c…

Let $Y_1,Y_2,dots$ be the identically distributed random variables and let $U_n^2=sum_{j=1}^n Y_j^2$. The self-normalized cluster set is … Under the general moment conditions of th…

Lower-deviation scale conjecture. The scale $(\log\log n)^{1/3}$ is the correct scale of fluctuation for the lower deviations of $Z_n$. This concerns the lower-tail analogue of the…

Regular-variation-only conjecture. The law of the iterated logarithm