Discrete total variation under the multinomial, Part 3: expectation under a general distribution

Abstract

For multinomial counts $X\sim \mathrm{Mult}(N;\mathbf{p})$ on $n$ ordered bins and a reference law $\mathbf{r}$ we study the expectation of the reference-centered discrete total variation $d_{\mathbf{r}}(X)={\sum }_{j=1}^{n-1}|(X_{j+1}-N{r}_{j+1})-(X_j-N{r}_j)|$, the natural extension of the comb-test statistic to a non-uniform null. For a non-uniform null the raw total variation of the counts carries a deterministic staircase of order $N$ that no centering of the statistic removes; the counts are therefore centered at the reference law, and under the null ($\mathbf{p}=\mathbf{r}$) the statistic is the matched $d_{\mathbf{p}}$. Exact finite-sample expectations are obtained by conditioning on the total mass in each adjacent pair, which reduces every edge to a translated binomial absolute-deviation problem. For fixed $n\ge 3$ and a sampling law bounded away from the boundary of the simplex, the matched statistic admits a three-term asymptotic expansion in powers of $1/N$ whose coefficients are explicit continuous periodic functions of $N(p_{j+1}-p_j)$ (a generalized, non-Poincaré asymptotic expansion), obtained by a weighted local Edgeworth estimate and a corner Euler--Maclaurin summation; all half-integer orders vanish by parity. Each correction separates into a smooth Edgeworth part and an oscillatory lattice part, the latter an Edgeworth perturbation of one formal Euler--Maclaurin series of a Gaussian lattice sum. Under a sampling law $\mathbf{p}\ne \mathbf{r}$ the leading expectation is the folded-normal profile implied by the central limit theorem; under local alternatives we identify the $O(1)$ correction (a skewness term, plus a variance-gradient term for Pitman alternatives), and fixed alternatives give a law-of-large-numbers limit, the ${\ell }_1$ distance between the adjacent-gradient profiles of $\mathbf{p}$ and $\mathbf{r}$, on which threshold tests are consistent.

arXiv comment43 pages, 2, tables