Discrete total variation under the multinomial, Part 3: expectation under a general distribution
Abstract
For multinomial counts X∼Mult(N;p) on n ordered bins and a reference law r we study the expectation of the reference-centered discrete total variation dr(X)=∑j=1n−1|(Xj+1−Nrj+1)−(Xj−Nrj)|, 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 (p=r) the statistic is the matched dp. 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≥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(pj+1−pj) (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 p≠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 ℓ1 distance between the adjacent-gradient profiles of p and r, on which threshold tests are consistent.
Related
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- Discrete total variation under the multinomial, Part 2: variance under the uniform null
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BibTeX
@misc{banic2026discrete,
title = {{Discrete total variation under the multinomial, Part 3: expectation under a general distribution}},
author = {Nikola Banić and Neven Elezović},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41255},
eprint = {2610.41255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41255},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discrete total variation under the multinomial, Part 3: expectation under a general distribution
AU - Nikola Banić
AU - Neven Elezović
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41255
UR - https://arxiv.org/abs/2610.41255
AB - For multinomial counts X∼Mult(N;p) on n ordered bins and a reference law r we study the expectation of the reference-centered discrete total variation d_r(X)=∑_j=1^n−1|(X_j+1−Nr_j+1)−(X_j−Nr_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 (p=r) the statistic is the matched d_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≥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 p≠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 ℓ_1 distance between the adjacent-gradient profiles of p and r, on which threshold tests are consistent.
ER -