4 problems

For a real number $\alpha>1$, write $\{\alpha^n\}$ for the fractional part of $\alpha^n$. The sequence $(\{\alpha^n\})_{n\ge1}$ has Poissonian $k$-point correlation when its $k$-po…

Let $f(\eta)$ be a continuous, symmetric jump distribution with finite second moment, and let $d_{k,n}=M_{k,n}-M_{k+1,n}$ be the gap between successive maxima of a random walk. In…

Let $f(\eta)$ be a symmetric jump distribution with finite variance $\sigma^2<\infty$. For the gap $d_{k,n}$ between successive maxima near the edge, suppose that its typical fluct…

Let $M_{k,n}$ denote the $k$-th maximum among the positions of a random walk of length $n$, and let $d_{k,n}=M_{k,n}-M_{k+1,n}$ be the gap between successive maxima. For long rando…