We study sums over boundary layers on the infinite d-regular tree that arise from finite-radius factors of i.i.d. We give an exact spectral identity for the variance and identify the limiting variance constant through an explicit function acting on the spectral measure. In addition, we prove a quantitative normal approximation based on local dependence. For radius one, we compute closed-form constants for two simple local rules.
It is numerically well known that Gaussian approximations underlying moment-based tests and estimators may deteriorate at higher moment orders. In this work, we quantify this phenomenon for Hermite-based moment statistics under i.i.d. standard normal data. More precisely, for Sn=n−1∕2∑k=1nΦ(Gk), where Φ:R→Rd collects the first d normalised Hermite polynomials, we derive explicit upper and lower bounds for the Gaussian approximation of Sn to its limit Z. We show that there exist explicit constants c,C>0 such that dC(Sn,Z)≤Cd3∕4ed⋅3log(2)∕2n−1∕2, while, for every d≥2, there exists Nd≥1 such that dK(Sn,Z)≥cd−3∕4ed⋅3log(2)∕2n−1∕2, for all n≥Nd. Thus, the upper and lower bounds are exponentially sharp in the maximal Hermite order d. The lower bound is obtained through a one-dimensional projection onto the highest Hermite coordinate, showing that the exponential dependence is driven by the maximal Hermite order rather than by high-dimensional geometry alone. We also illustrate the resulting finite-sample bounds for classical procedures such as the Shenton–Bowman test and higher-order moment tests.
We study closed absorbing sets and invariant ergodic measures for Markov processes on Polish spaces. In the Feller setting, the support of any invariant ergodic measure is a closed absorbing set, and distinct ergodic measures with different supports can only intersect along a set with empty interior. We show that minimality among closed absorbing sets does not imply ergodic minimality, so that the closed absorbing structure may fail to reflect the full ergodic decomposition. At the opposite end, the strong Feller property ensures separation by disjoint open sets of full measure. Between these regimes, we identify quasi strong Feller regularity as an intermediate condition under which ergodic measures remain separable modulo a negligible exceptional set.
Due to the regularization effect of the stochastic noise, the quantitative entropy-cost type propagation of chaos for mean field interacting particle system is established. The result shows that the relative entropy between particles and the limit McKean-Vlasov SDEs at any positive time can be expressed in terms of the initial L2-Wasserstein distance. Moreover, under dissipative assumption, the long time entropy-cost type propagation of chaos can also be captured.
We prove a Hoeffding-type moment generating function bound for weighted sums of bounded exchangeable random variables centered by their finite-population average. The bound improves the finite-population inflation factor in a recent weighted exchangeable Hoeffding inequality from logarithmic order to the rate-optimal inverse-population-size order, with an explicit constant. The proof reduces the problem to Hamming slices, identifies two-level extremizers for the relevant symmetric variational problem, and applies a hypergeometric martingale bound. We also give a lower bound showing that an inverse-population-size inflation is unavoidable.
