Discrete total variation under the multinomial, Part 2: variance under the uniform null

Abstract

For the counts $(X_1,\dots,X_n)\sim\mathrm{Mult}(N;1/n,\dots,1/n)$ of $N$ observations in $n$ ordered bins we study the variance of the discrete total variation $d=\sum_{i=1}^{n-1}|X_{i+1}-X_i|$. An exact decomposition splits $\mathrm{Var}(d)$ into per-edge, adjacent-pair and disjoint-pair terms and yields, for fixed $n$ as $N\to\infty$, $\mathrm{Var}(d)/N=α(n)-β(n)/N+γ(n)/N^2+o(N^{-2})$. For every $n\ge 2$, $Nα(n)$ is the classical normal-theory variance (Hoel, Kamat) of the sum of absolute successive differences of $n$ independent $N(0,N/n)$ variables; the correction $β(n)$ is obtained in closed form for $n\ge 3$ (the case $n=2$ is treated separately). Two disjoint edge differences $D_1=X_2-X_1$ and $D_3=X_4-X_3$ are exactly uncorrelated and asymptotically independent, yet $\mathrm{Cov}(|D_1|,|D_3|)=-1/(πn)+O(1/N)$: a correlation of relative order $1/N$, invisible in the Gaussian limit and absent under Poissonized sampling, whose constant is proportional to the exact joint fourth cumulant $κ_{2,2}(D_1,D_3)=-4N/n^2$ and is proved here on the lattice. The adjacent pair contributes the lattice-Edgeworth constant $a_*=-2\sqrt{3}/(9π)$, of which one quarter is a smooth Edgeworth term and three quarters a lattice-corner term; the disjoint skeleton $h(m)$ is closed through a Bessel generating function; and $γ(n)$ ($n\ge 5$) is expressed through a single triangular-lattice constant whose closed form we conjecture. Together with the mean of Part 1, the variance supplies the two moments used by moment-matched approximations to the null law of $d$. This is Part 2 of a series; the cyclic statistic $d^{c}$ will be treated in Part 5.