Local Limit Theorem Research Articles

Nepriklausomų atsitiktinių dydžių k-tojo maksimumo tankio asimptotika

In this paper the local limit theorem for density of k-th maxima of independent identically distributed random variables is proved.

On Mukhin’s necessary and sufficient condition for the validity of the local limit theorem

Abstract Mukhin found in 1984 an important necessary and sufficient condition for the validity of the local limit theorem. Revisiting the succinct proof given in [A. B. Mukhin, Some necessary and sufficient conditions for the validity of local limit theorems, Dokl. Akad. Nauk USSR 1984, 8, 7–8], we could only prove rigorously a weaker necessary and sufficient condition, with a significantly different formulation. This is the object of this short note.

Local limit theorems for collective risk models

A probability model known as the collective risk model is used to describe the total claim from a portfolio of insurance contracts. It is essential to non-life insurance. Let N represent the number of claims and X1,X2,… represent the amount of loss in each claim where Xj′s are independent and identically distributed. For the collective risk model, the total claim is given by SN=X1+X2+⋯+XN. Local limit theorems estimate the probability at a particular point P(SN=k). In this paper, we provide local limit theorems for SN, where N is a random variable with a binomial, Poisson and negative binomial distribution. Our results give a better rate of convergence than Berry–Esseen’s Theorem. Explicit constants of the error bounds are also given.

Explicit conditions for the CLT and related results for non-uniformly partially expanding random dynamical systems via effective RPF rates

The purpose of this paper is to provide a first class of explicit sufficient conditions for the central limit theorem and related results in the setup of non-uniformly (partially) expanding non iid random transformations, considered as stochastic processes together with some random Gibbs measure. More precisely, we prove a central limit theorem (CLT), an almost sure invariance principle, a moderate deviations principle, Berry-Esseen type estimates and a moderate local central limit theorem for random Birkhoff sums generated by a non-uniformly partially expanding dynamical systems Tω and a random Gibbs measure μω, corresponding to a random potential ϕω with a sufficiently regular variation. In the partially expanding case the maps we consider are similar to the ones in [44], with the exception that the amount of expansion dominates the amount of contraction fiberwise and not only on the average, and with an additional regularity condition on a certain type of local variation of ϕω along inverse branches of Tω. A notable example when the maps are truly partially-expanding is the case when ϕω≡0 which corresponds to random measures of maximal entropy μω, but any potential with a sufficiently small (fiberwise) variation can be considered. Our results in the partially expanding case are new even in the uniformly random case, where all the random variables describing the maps are uniformly controlled. For properly expanding maps (as in [42,33]), the above local regularity condition allows applications also in the smooth case where the Gibbs measure is absolutely continuous with respect to the underlying volume measure and ϕω=−ln⁡JTω. For instance, we can consider certain fiberwise piecewise C2-perturbations of piecewise linear or affine maps. All of the above is achieved by first proving random, real and complex, Ruelle-Perron-Frobenius (RPF) theorems with rates that can be expressed analytically by means of certain random parameters that describe the maps (such rates will be referred to as “effective”). Using these effective rates, our conditions for the limit theorems involve some weak type of upper mixing conditions on the driving system (base map) and some integrability conditions on the norm of the random function generating the Birkhoff sums. A big part of the proof of the moderate deviations, the Berry-Esseen type estimates and the local CLT is to show how Rugh's theory [43] of complex cones contractions applies to the cones considered in [16] (and their random versions in [44]), which is new even for deterministic dynamical systems T, and in that case it yields explicit estimates on the spectral gap of appropriate deterministic complex perturbations of the transfer operator of T, as well as explicit constants in the corresponding Berry-Esseen theorem for deterministic partially expanding dynamical systems.

Asymptotic expansions in the local limit theorem for a branching Wiener process

Consider a supercritical branching Wiener process in Rd. Let Zn(A) be the number of the nth generation particles located in a given set A⊂Rd. Under the moment condition of EX(lnX)1+λ type, the complete asymptotic expansions of Zn(A) as n tends to infinity are obtained. This result gives an alternative version of the work by Révész, Rosen and Shi (2005), hence generalizing theirs by weakening the second moment condition therein.

The local limit theorem for general weighted sums of Bernoulli random variables

The local limit theorem (LLT) is one of the well-known limit theorems which can be used to estimate the probability at a particular point of a random variable. In this paper, we generalize weighted sums which are introduced by Giuliano and Weber and give the LLT with explicit error bounds.

Richter’s local limit theorem, its refinement, and related results*

We give a detailed exposition of the proof of Richter’s local limit theorem in a refined form and establish the stability of the remainder term in this theorem under small perturbations of the underlying distribution (including smoothing).We also discuss related quantitative bounds for characteristic functions and Laplace transforms.

Effective de Sitter space, quantum behaviour and large-scale spectral dimension (3+1)

De Sitter space-time, essentially our own universe, is plagued by problems at the quantum level. Here we propose that Lorentzian de Sitter space-time is not fundamental but constitutes only an effective description of a more fundamental quantum gravity ground state. This cosmological ground state is a graph, appearing on large scales as a Riemannian manifold of constant negative curvature. We model the behaviour of matter near this equilibrium state as Brownian motion in the effective thermal environment of graph fluctuations, driven by a universal time parameter. We show how negative curvature dynamically induces the asymptotic emergence of relativistic coordinate time and of leading ballistic motion governed by the isometry group of an “effective Lorentzian manifold” of opposite, positive curvature, i.e. de Sitter space-time: free fall in positive curvature is asymptotically equivalent to the leading behaviour of Brownian motion in negative curvature. The local limit theorem for negative curvature implies that the large-scale spectral dimension of this “effective de Sitter space-time” is (3+1) independently of its microscopic topological dimension. In the effective description, the sub-leading component of asymptotic Brownian motion becomes Schrödinger quantum behavior on a 3D Euclidean manifold.

Periodic Lorentz gas with small scatterers

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size rho may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive sqrt{nlog n} scaling (i) for fixed infinite horizon configurations—letting first nrightarrow infty and then rho rightarrow 0—studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first rho rightarrow 0 and then nrightarrow infty —studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .

Maximum entropy and integer partitions

We derive asymptotic formulas for the number of integer partitions with given sums of \(j\)th powers of the parts for \(j\) belonging to a finite, non-empty set \(J \subset \mathbb N\). The method we use is based on the `principle of maximum entropy' of Jaynes. This principle leads to an intuitive variational formula for the asymptotics of the logarithm of the number of constrained partitions as the solution to a convex optimization problem over real-valued functions. Finding the polynomial corrections and leading constant involves two steps: quantifying the error in approximating a discrete optimization problem by a continuous one and proving a multivariate local central limit theorem.Mathematics Subject Classifications: 05A17, 05A16, 60F05Keywords: Integer partitions, maximum entropy, asymptotic enumeration, local central limit theorem, limit shape

Local limit theorem for the number of empty cells in a scheme of random equiprobable allocations

Abstract A classical scheme of random equiprobable allocations of n particles into N cells is considered. We find an asymptotic formula for the probability that the number of empty cells is equal to k under the condition that n, N → ∞ in such a way that n/(N − k) is bounded and separated from 1 from below.

Local limit theorems without assuming finite third moment

One of the most fundamental probabilities is the probability at a particular point. The local limit theorem is the well-known theorem that estimates this probability. In this paper, we estimate this probability by the density function of normal distribution in the case of lattice integer-valued random variables. Our technique is the characteristic function method. We complete to relax the third moment condition of Siripraparat and Neammanee (J. Inequal. Appl. 2021:57, 2021) and the references therein and also obtain explicit constants of the error bound.

A refined continuity correction for the negative binomial distribution and asymptotics of the median

In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a $\mathrm{Negative\hspace{0.5mm}Binomial}\hspace{0.2mm}(r,p)$ random variable jittered by a $\mathrm{Uniform}\hspace{0.2mm}(0,1)$, which answers a problem left open in Coeurjolly & Tr\'epanier (2020). This is used to construct a simple, robust and consistent estimator of the parameter $p$, when $r > 0$ is known. The case where $r$ is unknown is also briefly covered. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments.

Asymptotic behavior for supercritical branching processes

Let {Z(t);t⩾0} be a Markov branching process (MBP). There exists a well-known sequence {C(t);t⩾0} such that W(t)≔Z(t)/C(t) a.s. converges to a non-degenerate random variable W as t→∞. This paper attempts to study the asymptotic behavior of P(Z(t)=kt) and P(0⩽Z(t)⩽kt) with kte−λt→0 as t→∞ for MBPs, which helps to study large deviations of Z(t+s)/Z(t). Moreover, we obtain the local limit theorem of this process as an additional finding. During the argumentation, the Cramér method is applied to analyze the large deviation of the sum of random variables.

Berry–Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices

Let $$\mu $$ be a probability measure on $$\textrm{GL}_d(\mathbb {R})$$ and denote by $$S_n:= g_n \cdots g_1$$ the associated random matrix product, where $$g_j$$ ’s are i.i.d.’s with law $$\mu $$ . We study statistical properties of random variables of the form $$\begin{aligned} \sigma (S_n,x) + u(S_n x), \end{aligned}$$ where $$x \in \mathbb {P}^{d-1}$$ , $$\sigma $$ is the norm cocycle and u belongs to a class of admissible functions on $$\mathbb {P}^{d-1}$$ with values in $$\mathbb {R}\cup \{\pm \infty \}$$ . Assuming that $$\mu $$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry–Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $$\mathbb {R}$$ and Hölder continuous target functions on $$\mathbb {P}^{d-1}$$ . As particular cases, we obtain new limit theorems for $$\sigma (S_n,x)$$ and for the coefficients of $$S_n$$ .

Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster

We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions d+1 with d≥3 being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous with respect to the annealed measure and satisfies certain concentration properties.

Quenched limit theorems for random U(1) extensions of expanding maps

In this paper we provide quenched central limit theorems, large deviation principles and local central limit theorems for random U(1) extensions of expanding maps on the torus. The results are obtained as special cases of corresponding theorems that we establish for abstract random dynamical systems. We do so by extending a recent spectral approach developed for quenched limit theorems for expanding and hyperbolic maps to be applicable also to partially hyperbolic dynamics.

Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions

The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transformation h=logu, where u is the solution of the multiplicative stochastic heat equation(SHE). In [CD20, CSZ20, G20], they consider the solution of two dimensional KPZ equation via the solution uε of SHE with the flat initial condition and with noise which is mollified in space on scale in ε and its strength is weakened as βε=βˆ 2π −logε, and they prove that when βˆ∈(0,1), 1 βε(loguε−E[loguε]) converges in distribution as a random field to a solution of Edwards-Wilkinson equation. In this paper, we consider a stochastic heat equation uε with a general initial condition u0 and its transformation F(uε) for F in a class of functions F, which contains F(x)=xp (0<p≤1) and F(x)=logx. Then, we prove that 1 βε(F(uε(t,⋅))−E[F(uε(t,⋅))]) converges in distribution as a random field to a centered Gaussian field jointly in finitely many F∈F, t, and u0. In particular, we show the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depend on u0. Our main tools are Itô’s formula, the martingale central limit theorem, and the homogenization argument as in [CNN22]. To this end, we also prove a local limit theorem for the partition function of intermediate disorder 2d directed polymers.

Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group

Consider the random walk Gn:=gn…g1, n⩾1, where (gn)n⩾1 is a sequence of independent and identically distributed random elements with law μ on the general linear group GL(V) with V=Rd. Under suitable conditions on μ, we establish Cramér type moderate deviation expansions and local limit theorems with moderate deviations for the coefficients 〈f,Gnv〉, where v∈V and f∈V∗. Our approach is based on the Hölder regularity of the invariant measure of the Markov chain Gn⋅x=RGnv on the projective space of V with the starting point x=Rv, under the changed measure.

Elliptic Harnack inequality for ℤd

We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of EHI in ${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the Local Central Limit Theorem for simple symmetric random walks on ${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability function. The uniform Green inequality and the classical Balayage formula then imply the EHI.