7 problems

Let $T^{(D)}_{n}(\alpha,\hat j_n)$ and $Q^{(D)}_{n}(\alpha)$ denote the $L_\infty$-type and $L_2$-type sample-splitting self-normalized (SS-SN) statistics, respectively. Suppose th…

Let $\Omega(n)$ denote the total number of prime factors of $n$, counted with multiplicity, let $[N]=\{1,\ldots,N\}$, and for a finite set $S$ write…

Let $\mathbb{P}_\ell$ denote the set of integers with the relevant almost-prime parameter $\ell$, let $\1_{\mathbb{P}_\ell}$ be its indicator, and let $\overline{\pi}_\ell(N)$ deno…

Let $\Omega(n)$ denote the total number of prime factors of $n$, counted with multiplicity, and let $[N]=\{1,\ldots,N\}$. For a function $f$ on a finite set, write…

Let $X$ be a positive random variable, and write … where $\lfloor\log_2 X\rfloor$ and $\{\log_2 X\}$ are respectively the whole and fractional parts of $\log_2 X$. Let $A$ and…

Bender–Helton–Richmond conjecture. Every composition has the asymptotic independence property: for a uniformly random permutation with the corresponding fixed descent set,…

Asymptotic independence conjecture. The centering factor $n m(\eta_{n,d})/A_n$ and the partial sum process are asymptotically independent under the conditions of Example.