Discrete total variation under the multinomial, Part 2: variance under the uniform null
Abstract
For the counts (X1,…,Xn)∼Mult(N;1/n,…,1/n) of N observations in n ordered bins we study the variance of the discrete total variation d=∑i=1n−1|Xi+1−Xi|. An exact decomposition splits Var(d) into per-edge, adjacent-pair and disjoint-pair terms and yields, for fixed n as N→∞, Var(d)/N=α(n)−β(n)/N+γ(n)/N2+o(N−2). For every n≥ 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≥ 3 (the case n=2 is treated separately). Two disjoint edge differences D1=X2−X1 and D3=X4−X3 are exactly uncorrelated and asymptotically independent, yet Cov(|D1|,|D3|)=−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 κ2,2(D1,D3)=−4N/n2 and is proved here on the lattice. The adjacent pair contributes the lattice-Edgeworth constant a*=−2√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≥ 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 dc will be treated in Part 5.
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BibTeX
@misc{banic2026discrete,
title = {{Discrete total variation under the multinomial, Part 2: variance under the uniform null}},
author = {Nikola Banić and Neven Elezović},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17046},
eprint = {2610.17046},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17046},
}
RIS
TY - UNPB
TI - Discrete total variation under the multinomial, Part 2: variance under the uniform null
AU - Nikola Banić
AU - Neven Elezović
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17046
UR - https://arxiv.org/abs/2610.17046
AB - For the counts (X_1,…,X_n)∼Mult(N;1/n,…,1/n) of N observations in n ordered bins we study the variance of the discrete total variation d=∑_i=1^n−1|X_i+1−X_i|. An exact decomposition splits Var(d) into per-edge, adjacent-pair and disjoint-pair terms and yields, for fixed n as N→∞, Var(d)/N=α(n)−β(n)/N+γ(n)/N^2+o(N^−2). For every n≥ 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≥ 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 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√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≥ 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.
ER -