Local Central Limit Theorem Research Articles

Multidimensional local central limit theorem of some non-uniformly hyperbolic systems

We consider Young’s nonuniformly hyperbolic system (X, T, ν) where ν is the SRB measure corresponding to the system (X, T), and show that if the components of a Holder observable f: X → ℝd are cohomologously independent, then f satisfies the multidimensional central limit theorem. Moreover if f is aperiodic, then f satisfies the local multidimensional central limit theorem.

A local central limit theorem on the Laguerre hypergroup

We consider here the Laguerre hypergroup (K,*α), where K=[0,+∞[×R and *α a convolution product on K coming from the product formula satisfied by the Laguerre functions Lm(α) (m∈N, α⩾0). We set on this hypergroup a local central limit theorem which consists to give a weakly estimate of the asymptotic behavior of the convolution powers μ*αk=μ*α⋯*αμ (k times), μ being a given probability measure satisfying some regularity conditions on this hypergroup. It is also given a central local limit theorem for some particular radial probability measures on the (2n+1)-dimensional Heisenberg group Hn.

Ordered Random Walks

We construct the conditional version of $k$ independent and identically distributed random walks on $R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson's Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob $h$-transform with some positive regular function $V$ that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson's Brownian motions, the continuous analogue.

On thermodynamics of rational maps on the Riemann sphere

We show that there is a spectral gap for the transfer operator associated with a rational map $f$ on the Riemann sphere. Using this and the method of perturbed operators, we establish the (local) central limit theorem for the measure of maximal entropy of $f$, with an estimate on the speed of convergence.

Tribute to Endre Csáki and Pál Révész

This article provides a glimpse of some of the highlights of the joint work of Endre Csaki and Pal Revesz since 1979. The topics of this short exploration of the rich stochastic milieu of this inspiring collaboration revolve around Brownian motion, random walks and their long excursions, local times and additive functionals, iterated processes, almost sure local and global central limit theorems, integral functionals of geometric stochastic processes, favourite sites--favourite values and jump sizes for random walk and Brownian motion, random walking in a random scenery, and large void zones and occupation times for coalescing random walks.

Local central limit theorems, the high-order correlations of rejective sampling and logistic likelihood asymptotics

Let I_1,...,I_n be independent but not necessarily identically distributed Bernoulli random variables, and let X_n=\sum_{j=1}^nI_j. For \nu in a bounded region, a local central limit theorem expansion of P(X_n=EX_n+\nu) is developed to any given degree. By conditioning, this expansion provides information on the high-order correlation structure of dependent, weighted sampling schemes of a population E (a special case of which is simple random sampling), where a set d\subset E is sampled with probability proportional to \prod_{A\in d}x_A, where x_A are positive weights associated with individuals A\in E. These results are used to determine the asymptotic information, and demonstrate the consistency and asymptotic normality of the conditional and unconditional logistic likelihood estimator for unmatched case-control study designs in which sets of controls of the same size are sampled with equal probability.

Partitions of n into [formula omitted] parts

Szekeres proved, using complex analysis, an asymptotic formula for the number of partitions of n into at most k parts. Canfield discovered a simplification of the formula, and proved it without complex analysis. We re-prove the formula, in the asymptotic regime when k is at least a constant times n , by showing that it is equivalent to a local central limit theorem in Fristedt’s model for random partitions. We then apply the formula to derive asymptotics for the number of minimal difference d partitions with a given number of parts. As a corollary, we find (explicitly computable) constants c d , β d , γ d , σ d such that the number of minimal difference d partitions of n is (1+o(1))c dn −3/4 exp(β d n ) (a result of Meinardus), almost all of them (fraction a(1+ o(1))) have approximately γ d n parts, and the distribution of the number of parts in a random such partition is asymptotically normal with standard deviation (1+ o(1)) σ d n 1/4. In particular, γ 2= 15 log[(1+ 5 )/2]/π .

Local limit theorem for the Lorentz process and its recurrence in the plane

For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Young's axioms (Lai-Sang Young, Ann. Math.147 (1998), 585–650), which imply exponential decay of correlations and the central limit theorem (CLT), a local CLT is proven. In fact, a unified version of the local CLT is found, covering, among others, the absolutely continuous and arithmetic cases. For planar Lorentz process with a finite horizon, this result implies (a) a local CLT and (b) recurrence. For the latter case (d = 2, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt (C. R. Acad. Sci. Paris Ser. 1 Math. 372(9) (1998), 837–842) and J.-P. Conze (Ergod. Th. & Dynam. Sys.19(5) (1999), 1233–1245) could already establish recurrence.

A generalised inductive approach to the lace expansion

The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions d > 4, and for sufficiently spread-out models of critical oriented percolation in dimensions d + 1 > 5, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models.

A Local Limit Theorem for Moderate Deviations

The purpose of this note is to establish a uniform estimate for the mass function P(Sm = y) of an integer-valued random walk when y → ∞ and ( y − m μ ) / m → ∞ where μ isthe mean of the step distribution. (The local central limit theorem provides such an estimate when ( ( y − m μ ) / m is bounded.) The assumptions are that the mass function p of the step distribution is regularly varying at ∞ with index −κ, where κ > 3, and that ∑ n = 0 ∞ n κ ′ p ( – n ) < ∞ for some κ′ > 2. From this result, a ratio limit theorem is derived, and this in turn is applied to yield some new information about the space–time Martin boundary of certain random walks.

Limit Theorems for General Markov Chains

Let S be a measurable space with a σ-algebra B(S) of measurable sets. Let Pn(x,B), x ∈ S, B ∈ B(S), be some transition probability on S; in this article, the parameter n, time, ranges in the set Z+ = {0, 1, 2, . . . } of nonnegative integers. The transition probability is not assumed homogeneous in time n. Consider the Markov chain X = {Xn, n ∈ Z+} with values in S and transition probability Pn(·, ·); i.e., P{Xn+1 ∈ B | Xn = x} = Pn(x,B). In § 2 we find out conditions under which f(Xn)/n converges almost surely to some limit, where f : S → Y is a function with values in a separable Banach space Y . We consider the so-called p-smooth Banach space. In § 3 we study the behavior in time of the characteristic functional of a Markov chain with values in an arbitrary separable Banach space. In § 4, for a Markov chain with values in a finite-dimensional Euclidean space, we state conditions under which Xn satisfies the central limit theorem. In § 5 we derive an upper estimate for the probability to belong to a compact set for an Rd-valued Markov chain asymptotically homogeneous in time and space (in some direction). In § 6 we prove a local central limit theorem for an asymptotically homogeneous Markov chain with values in the integral lattice Zd, and in § 7 we prove an analog of this theorem for a nonlattice Markov chain with values in Rd. Although in the conditions of the theorems we do not indicate explicitly whether the Markov chain {Xn} is positive recurrent, null recurrent, or transient, our results are most meaningful for nonrecurrent chains. Moreover, in some theorems we suppose explicitly that the value of the chain Xn “goes to infinity in some direction.” From the viewpoint of the strong law of large numbers and the central limit theorem, most publications are devoted to positive Harris recurrent (ergodic) time homogeneous Markov chains (see, for instance, [1, § 17]). Some results relating to the central limit theorem for time nonhomogeneous ergodic Markov chains can be found in [2, 3]. Observe that, unlike transient chains (that are the main subject of study in the present article), the most natural problem for ergodic chains is that of the asymptotic behavior of the distribution of sums of the values of a function f : S → R of a Markov chain, i.e., the distribution of f(X1) + · · ·+ f(Xn). In this case, employment of the cyclic structure of an ergodic chain (cycles of return to an atom) eventually reduces the problem to the familiar limit theorems for sums of independent random variables.

Albanese Maps and Off Diagonal Long Time Asymptotics for the Heat Kernel

We discuss long time asymptotic behaviors of the heat kernel on a non-compact Riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient. A local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted Laplacians. Our ideas and techniques are motivated partly by analogy with Floque–Bloch theory on periodic Schrodinger operators. For the asymptotic expansion, we make careful use of the classical Laplace method. In the course of a discussion, we observe that the notion of Albanese maps associated with the abelian group action is closely related to the asymptotics. A similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph. The results obtained in the present paper refine our previous ones [4]. In the asymptotics, the Euclidean distance associated with the standard realization of the lattice graph, which we call the Albanese distance, plays a crucial role.

On the Accuracy of Normal Approximation for the Densities of Sums of Independent Identically Distributed Random Variables

The structure of the nonuniform estimate of convergence rate in the local central limit theorem for the densities of sums of independent identically distributed random variables is made more accurate. The absolute constants are written out explicitly.

A new inductive approach to the lace expansion for self-avoiding walks

We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ℤ d where loops of length m are penalised by a factor e −β/m p (0 4, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0.

Hardy–Ramanujan's Asymptotic Formula for Partitions and the Central Limit Theorem

Letf(z) be the generating function of the sequence {p(n)} of unrestricted partitions ofn, and letXtbe an integral random variable taking the valuenwith probability (f(t))−1p(n)tn. It is shown here that, ast→1, the normalizedXtare asymptotically Gaussian. The mode of convergence is sufficiently strong for the conclusion of a local central limit theorem to hold, leading to the classical formula of Hardy–Ramanujan,[formula].

The hydrodynamic limit for the reaction diffusion equation — An approach in terms of the GPV method

We study the hydrodynamic limit of the reaction diffusion process by means of the GPV technique (Guoet al.(4)). To this end, we first derivea priori bounds on the moments of the occupation numbers using the local central limit theorem and results of stochastic analysis. The result of De Masi and Presutti(2) for the hydrodynamic limit of the reaction diffusion process is generalized here.

Weight functions and pathwise local central limit theorems

This paper is concerned with weak convergence together with convergence rates in weighted almost sure local central limit theorems for random walks. The main tools are stochastic calculus and strong approximations.

An elementary proof of the local central limit theorem

We give an elementary proof of the local central limit theorem for independent, non-identically distributed, integer valued and vector valued random variables.

On almost sure local and global central limit theorems

LetS 1,S 2,... be a sequence of sums of i.i.d. random variables. The properties of the logarithmic average $$\frac{1}{{\log n}}\sum\limits_{k = 1}^n {\frac{{I\{ a_k \leqq S_k< b_k \} }}{{kP(a_k \leqq S_k< b_k )}}}$$ will be studied under some conditions.

On a Paditz’s Problem Concerning the Restricted Convergence on a Half-Axis In the Local Central Limit Theorem

Previous article Next article On a Paditz’s Problem Concerning the Restricted Convergence on a Half-Axis In the Local Central Limit TheoremH. HerrmannH. Herrmannhttps://doi.org/10.1137/1135044PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, the Netherlands, 1971, 443– 48:1287 0219.60027 Google Scholar[2] R. Ranga Rao and , V. S. Varadarajan, A limit theorem for densities, Sankhyā, 22 (1960), 261–266 24:A1734 0101.11603 Google Scholar[3] Hans-Joachim Rossberg, , Bernd Jesiak and , Gerhard Siegel, Analytic methods of probability theory, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], Vol. 67, Akademie-Verlag, Berlin, 1985, 311– 87h:60060 0583.60013 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Limit Theorems Involving Restricted ConvergenceH.-J. RossbergTheory of Probability & Its Applications, Vol. 39, No. 2 | 28 July 2006AbstractPDF (2035 KB) Volume 35, Issue 2| 1991Theory of Probability & Its Applications205-409 History Submitted:30 September 1987Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135044Article page range:pp. 347-352ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics