Local Limit Theorem Research Articles

The Local Limit Theorem for Supercritical Branching Processes with Immigration

The Local Limit Theorem for Supercritical Branching Processes with Immigration

Local limit theorems in relatively hyperbolic groups II: the non-spectrally degenerate case

This is the second of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this second paper, we restrict our attention to non-spectrally degenerate random walks and we prove precise asymptotics of the probability$p_n(e,e)$of going back to the origin at time$n$. We combine techniques adapted from thermodynamic formalism with the rough estimates of the Green function given by part I to show that$p_n(e,e)\sim CR^{-n}n^{-3/2}$, where$R$is the inverse of the spectral radius of the random walk. This both generalizes results of Woess for free products and results of Gouëzel for hyperbolic groups.

Bell Numbers in Matsunaga's and Arima's Genjikō Combinatorics: Modern Perspectives and Local Limit Theorems

We examine and clarify in detail the contributions of Yoshisuke Matsunaga (1694?-1744) to the computation of Bell numbers in the eighteenth century (in the Edo period), providing modern perspectives to some unknown materials that are by far the earliest in the history of Bell numbers. Later clarification and developments by Yoriyuki Arima (1714-1783), and several new results such as the asymptotic distributions (notably the corresponding local limit theorems) of a few closely related sequences are also given.

An improved result in almost sure local central limit theorem for the partial sums

An improved result in almost sure local central limit theorem for the partial sums

Archimedes’ Principle for Ideal Gas

We prove Archimedes’ principle for a macroscopic ball in ideal gas consisting of point particles with non-zero mass. The main result is an asymptotic theorem, as the number of point particles goes to infinity and their total mass remains constant. We also show that, asymptotically, the gas has an exponential density as a function of height. We find the asymptotic inverse temperature of the gas. We derive an accurate estimate of the volume of the phase space using the local central limit theorem.

Local limit theorems for subgraph counts

We introduce a general framework for studying anticoncentration and local limit theorems for random variables, including graph statistics. Our methods involve an interplay between Fourier analysis, decoupling, hypercontractivity of Boolean functions, and transference between ``fixed-size'' and ``independent'' models. We also adapt a notion of ``graph factors'' due to Janson. As a consequence, we derive a local central limit theorem for connected subgraph counts in the Erd\H{o}s-Renyi random graph $G(n,p)$, building on work of Gilmer and Kopparty and of Berkowitz. These results improve an anticoncentration result of Fox, Kwan, and Sauermann and partially answers a question of Fox, Kwan, and Sauermann. We also derive a local limit central limit theorem for induced subgraph counts, as long as $p$ is bounded away from a set of ``problematic'' densities, partially answering a question of Fox, Kwan, and Sauermann. We then prove these restrictions are necessary by exhibiting a disconnected graph for which anticoncentration for subgraph counts at the optimal scale fails for all constant $p$, and finding a graph $H$ for which anticoncentration for induced subgraph counts fails in $G(n,1/2)$. These counterexamples resolve anticoncentration conjectures of Fox, Kwan, and Sauermann in the negative. Finally, we also examine the behavior of counts of $k$-term arithmetic progressions in subsets of $\mathbb{Z}/n\mathbb{Z}$ and deduce a local limit theorem wherein the behavior is Gaussian at a global scale but has nontrivial local oscillations (according to a Ramanujan theta function). These results improve on results of and answer questions of the authors and Berkowitz, and answer a question of Fox, Kwan, and Sauermann.

Local limit theorem in deterministic systems

On montre que dans tout système dynamique probabilisé ergodique et apériodique, il existe une fonction de carré intégrable à valeurs entières dont les sommes de Birkhoff satisfont le théorème limite local.

Optimal Local Law and Central Limit Theorem for $$\beta $$-Ensembles

In the setting of generic $\beta$-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of order $(\log N)/N$ in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale $\sqrt{\log N}/N$. (iii) The logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem. Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian $\beta$-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.

Independent sets of a given size and structure in the hypercube

AbstractWe determine the asymptotics of the number of independent sets of size $\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $\beta \in (0,1)$ as $d \to \infty$ , extending a result of Galvin for $\beta \in (1-1/\sqrt{2},1)$ . Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard-core model at any fixed fugacity $\lambda>0$ . In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.

Statistics of multipliers for hyperbolic rational maps

<p style='text-indent:20px;'>In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.</p>

On the local limit theorems for lower psi-mixing Markov chains

On the local limit theorems for lower psi-mixing Markov chains

Refined normal approximations for the Student distribution

In this paper, we develop a local limit theorem for the Student distribution. We use it to improve the normal approximation of the Student survival function given in Shafiei & Saberali (2015) and to derive asymptotic bounds for the corresponding maximal errors at four levels of approximation. As a corollary, approximations for the percentage points (or quantiles) of the Student distribution are obtained in terms of the percentage points of the standard normal distribution.

A quenched local limit theorem for stochastic flows

We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by a Gaussian random velocity field that is white in time and smooth in space. We show that conditional on the velocity field, the quenched density of the particle after a long time can be approximated pointwise by the product of a deterministic Gaussian density and a spacetime-stationary random field U. If the velocity field is additionally assumed to be incompressible, then U≡1 almost surely and we obtain a local central limit theorem.

Quenched local central limit theorem for random walks in a time-dependent balanced random environment

We prove a quenched local central limit theorem for continuous-time random walks in {mathbb {Z}}^d, dge 2, in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. We also obtain Gaussian upper and lower bounds for quenched and (positive and negative) moment estimates of the transition probabilities and asymptotics of the discrete Green’s function.

Invertibility of adjacency matrices for random d-regular graphs

Let d≥3 be a fixed integer, and let A be the adjacency matrix of a random d-regular directed or undirected graph on n vertices. We show that there exists a constant d>0 such that P(A is singular in R)≤n−d, for n sufficiently large. This answers an open problem by Frieze and Vu. The key idea is to study the singularity probability over a finite field Fp. The proof combines a local central limit theorem and a large deviation estimate.

Expansions in the Local and the Central Limit Theorems for Dynamical Systems

We study higher order expansions both in the Berry–Esséen estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical assumptions, discuss the verification of these assumptions and illustrate our results by different examples (subshifts of finite type, Young towers, Sinai billiards, random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting.

Berry–Esseen bound and precise moderate deviations for products of random matrices

Let {(g_{n})_{n\geq 1}} be a sequence of independent and identically distributed (i.i.d.) {d\times d} real random matrices. For {n\geq 1} set {G_n = g_n \ldots g_1} . Given any starting point {x=\mathbb R v\in\mathbb{P}^{d-1}} , consider the Markov chain {X_n^x = \mathbb R G_n v } on the projective space {\mathbb P^{d-1}} and define the norm cocycle by {\sigma(G_n, x)= \log (|G_n v|/|v|)} , for an arbitrary norm {|\cdot|} on \smash{\mathbb R^{d}} . Under suitable conditions we prove a Berry–Esseen-type theorem and an Edgeworth expansion for the couple {(X_n^x, \sigma(G_n, x))} . These results are established using a brand new smoothing inequality on complex plane, the saddle point method and additional spectral gap properties of the transfer operator related to the Markov chain {X_n^x} . Cramér-type moderate deviation expansions as well as a local limit theorem with moderate deviations are proved for the couple {(X_n^x, \sigma(G_n, x))} with a target function {\varphi} on the Markov chain {X_n^x} .

A. de Moivre theorem revisited

Cramér’s large deviations of order o(n) in the operator form are investigated. Consequently a generalization of de Moivre’s local limit theorem for binary Markov chains is obtained in the form with tied up (quenched) ends.

Interval type local limit theorems for lattice type random variables and distributions

In this paper, we propose a new interpretation of local limit theorems for univariate and multivariate distributions on lattices. Using elementary techniques, we show that – given a local limit theorem in the standard sense – the distributions are approximated well by the limit distribution, uniformly on intervals of possibly decaying length. We identify the maximally allowable decay speed of the interval lengths. Further, we show that for continuous distributions, the interval type local law holds without any decay speed restrictions on the interval lengths. We show that various examples fit within this framework, such as standardized sums of i.i.d. random vectors or correlated random vectors induced by multidimensional spin models from statistical mechanics.

Local limit theorems for generalized scheme of allocation of particles into ordered cells

Abstract A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.