8 problems

Let $X^{(n)}_3$ denote the number of distinct adjacent pairs of distinct letters in a geometrically distributed word of length $n$. Gaussian-limit conjecture. The asymptotic distri…

Let $X^{(n)}_1$ be the number of distinct adjacent pairs of identical letters in a geometrically distributed word of length $n$, and let $\xi^{(n)}$ be the sum of independent rando…

Let $X^{(n)}_1$ be the number of distinct adjacent pairs of identical letters in a geometrically distributed word of length $n$, and let … where the independent random variables…

Let $M$ be an indecomposable, doubly stochastic matrix indexed by an alphabet of size $k$, satisfying … where $k+1\equiv1$. A word of length $N$ is generated by the Markov chain wi…

Let $T_r$ denote the correction to the $r$th cumulant of $X^{(n)}_3$ arising from dependence among the pair indicators. Cumulant-correction conjecture. … The surrounding argument e…

Let $Z$ be the number of occurrences of a pattern $w$ as a subsequence in a random text of length $n$, and let $\binom{n}{m}$ be the number of possible positions for a subsequence…

Let $V$ and $W$ be independent random words of length $2n$, written as concatenations $V=V_1V_2$ and $W=W_1W_2$, where each component has length $n$. Define … and write…

Fluctuation conjecture. If $0<\alpha<\frac12$, then