Poincaré's Inequality on Sliding-Window Occupation Count Kernels and the Optimal Normalized Spectral-Gap Constant
Abstract
Let P be an irreducible reversible Markov kernel on an m-state space Ω, and denote its right spectral gap by Gap(P) =1−λ2(P). For a stationary trajectory, let Kt be the occupation-count vector of the length-n window beginning at time t. The stationary pair (K0,K1) defines a reversible projected count kernel P̃n. For each fixed m≥2, define cm⋆= infsubstack|Ω|=m, P irreducible and reversible n≥2 nGap(P̃n)/Gap(P). We prove 1/1080m≤ cm⋆≤ qm−2, q0=1/4, qr+1=qr(1−qr). We also show qm−2=(m+log m+O(1))−1, which implies that cm⋆=Θ(m−1). Thus, the reciprocal optimal comparison coefficient Cmopt=(cm⋆)−1 is of the order m. In the two-state case, we determine the exact values c2⋆=1/4 and C2opt=4. %the exact constants for m≥3 remain open. The lower bound also holds for n=1 and is uniform in P, including sparse and periodic kernels. The proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson--Hardy inequality. A nested rare-state construction produces finite m-state witnesses whose normalized Rayleigh quotients approach qm−2 through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, Gap(P̃n)=ΘP(n−1).
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BibTeX
@misc{xiang2026poincarés,
title = {{Poincaré's Inequality on Sliding-Window Occupation Count Kernels and the Optimal Normalized Spectral-Gap Constant}},
author = {Yanjin Xiang and Zhihua Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27836},
eprint = {2609.27836},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27836},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Poincaré's Inequality on Sliding-Window Occupation Count Kernels and the Optimal Normalized Spectral-Gap Constant
AU - Yanjin Xiang
AU - Zhihua Zhang
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2609.27836
UR - https://arxiv.org/abs/2609.27836
AB - Let P be an irreducible reversible Markov kernel on an m-state space Ω, and denote its right spectral gap by Gap(P) =1−λ_2(P). For a stationary trajectory, let K_t be the occupation-count vector of the length-n window beginning at time t. The stationary pair (K_0,K_1) defines a reversible projected count kernel P̃_n. For each fixed m≥2, define c_m^⋆= inf_substack|Ω|=m, P irreducible and reversible n≥2 nGap(P̃_n)/Gap(P). We prove 1/1080m≤ c_m^⋆≤ q_m−2, q_0=1/4, q_r+1=q_r(1−q_r). We also show q_m−2=(m+log m+O(1))^−1, which implies that c_m^⋆=Θ(m^−1). Thus, the reciprocal optimal comparison coefficient C_m^opt=(c_m^⋆)^−1 is of the order m. In the two-state case, we determine the exact values c_2^⋆=1/4 and C_2^opt=4. %the exact constants for m≥3 remain open. The lower bound also holds for n=1 and is uniform in P, including sparse and periodic kernels. The proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson--Hardy inequality. A nested rare-state construction produces finite m-state witnesses whose normalized Rayleigh quotients approach q_m−2 through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, Gap(P̃_n)=Θ_P(n^−1).
ER -