Discrete total variation under the multinomial, Part 4: variance under a general distribution
Abstract
We study the variance of the reference-centered discrete total variation dp=∑j=1n−1|Xj+1−Xj−N(pj+1−pj)| of X∼Mult(N;p) for fixed n≥3 and fixed p with all pi>0, as N→∞. We prove Var(dp)=N v(p)+O(1), where v(p), the variance of the sum of absolute values of the limiting Gaussian edge vector, is expressed through the exact multinomial edge variances and correlations and Nabeya's covariance formula for folded bivariate normal variables. Unlike in the uniform case, non-adjacent edge pairs generally contribute to v(p). The O(1) remainder is controlled by a lattice Edgeworth expansion and Poisson summation; it oscillates with the lattice phases {N(pj+1−pj)}, and for n=3,4, where a pair of edges exhausts the bins, the summation runs over a sublattice coset. The formula reduces to the known uniform result (the classical Hoel--Kamat variance), gives a closed form when the reference law is a symmetric comb, and has a shape-independent large-bin limit for positive C1 profiles. For the statistic dr centered at a reference law r≠p the absolute values linearize: when every edge is mismatched the variance is that of a linear form in the counts up to an exponentially small error, and in mixed (triangular-array) regimes it is given to o(N) by bivariate folded-normal covariances. The two-bin formulas are recorded from the companion preprints, and exact finite-N reductions for n≤4 are used for validation.
Related
- Discrete total variation under the multinomial, Part 3: expectation under a general distribution
- Discrete total variation under the multinomial, Part 2: variance under the uniform null
- Variance Bounds for Minimum Perfect Matching Problem by Random Walk on Multinomial Distribution
- Discrete total variation under the multinomial, Part~1: expectation under the uniform null
- Closed-form evaluation of the cumulative distribution function and absolute moments of the variance-gamma distribution
BibTeX
@misc{banic2026discrete,
title = {{Discrete total variation under the multinomial, Part 4: variance under a general distribution}},
author = {Nikola Banić and Neven Elezović},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09592},
eprint = {2611.09592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09592},
}
RIS
TY - UNPB
TI - Discrete total variation under the multinomial, Part 4: variance under a general distribution
AU - Nikola Banić
AU - Neven Elezović
PY - 2026
DA - 2026/11/06
PB - arXiv
DO - 10.48550/arxiv.2611.09592
UR - https://arxiv.org/abs/2611.09592
AB - We study the variance of the reference-centered discrete total variation d_p=∑_j=1^n−1|X_j+1−X_j−N(p_j+1−p_j)| of X∼Mult(N;p) for fixed n≥3 and fixed p with all p_i>0, as N→∞. We prove Var(d_p)=N v(p)+O(1), where v(p), the variance of the sum of absolute values of the limiting Gaussian edge vector, is expressed through the exact multinomial edge variances and correlations and Nabeya's covariance formula for folded bivariate normal variables. Unlike in the uniform case, non-adjacent edge pairs generally contribute to v(p). The O(1) remainder is controlled by a lattice Edgeworth expansion and Poisson summation; it oscillates with the lattice phases {N(p_j+1−p_j)}, and for n=3,4, where a pair of edges exhausts the bins, the summation runs over a sublattice coset. The formula reduces to the known uniform result (the classical Hoel--Kamat variance), gives a closed form when the reference law is a symmetric comb, and has a shape-independent large-bin limit for positive C^1 profiles. For the statistic d_r centered at a reference law r≠p the absolute values linearize: when every edge is mismatched the variance is that of a linear form in the counts up to an exponentially small error, and in mixed (triangular-array) regimes it is given to o(N) by bivariate folded-normal covariances. The two-bin formulas are recorded from the companion preprints, and exact finite-N reductions for n≤4 are used for validation.
ER -