Proof Of Central Limit Theorem Research Articles

On the Joint Distribution of Descents and Signs of Permutations

We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers.

Law of large numbers and central limit theorem under nonlinear expectations

The main achievement of this paper is the finding and proof of Central Limit Theorem (CLT, see Theorem 12) under the framework of sublinear expectation. Roughly speaking under some reasonable assumption, the random sequence {1/sqrt {n}(X_{1}+cdots +X_{n})}_{i=1}^{infty } converges in law to a nonlinear normal distribution, called G-normal distribution, where {X_{i}}_{i=1}^{infty } is an i.i.d. sequence under the sublinear expectation. It’s known that the framework of sublinear expectation provides a important role in situations that the probability measure itself has non-negligible uncertainties. Under such situation, this new CLT plays a similar role as the one of classical CLT. The classical CLT can be also directly obtained from this new CLT, since a linear expectation is a special case of sublinear expectations. A deep regularity estimate of 2nd order fully nonlinear parabolic PDE is applied to the proof of the CLT. This paper is originally exhibited in arXiv.(math.PR/0702358v1).

Random polytopes: Central limit theorems for intrinsic volumes

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein’s method from probability theory.

A note on gaps in proofs of central limit theorems

A note on gaps in proofs of central limit theorems

A note on gaps in proofs of central limit theorems

We fill two gaps in the literature on central limit theorems. First we state and prove a generalization of the Cramér–Wold device which is useful for establishing multivariate central limit theorems without the need for assuming the existence of a limiting covariance matrix. Second we extend and provide a detailed proof of a very useful result for establishing univariate central limit theorems.

Finite-length analysis on tail probability for Markov chain and application to simple hypothesis testing

Using terminologies of information geometry, we derive upper and lower bounds of the tail probability of the sample mean. Employing these bounds, we obtain upper and lower bounds of the minimum error probability of the 2nd kind of error under the exponential constraint for the error probability of the 1st kind of error in a simple hypothesis testing for a finite-length Markov chain, which yields the Hoeffding type bound. For these derivations, we derive upper and lower bounds of cumulant generating function for Markov chain. As a byproduct, we obtain another simple proof of central limit theorem for Markov chain.

A New Proof of Central Limit Theorem for i.i.d. Random Variables

Central limit theorem (CLT) has long and widely been known as a fundamental result in probability theory. In this note, we give a new proof of CLT for independent identically distributed (i.i.d.) random variables. Our main tool is the viscosity solution theory of partial differential equation (PDE).

Statistical properties of skew-endomorphisms with Bernoulli base

We analyse numerically the statistical properties of a class of mappings of the 2-torus T 2 onto itself, whose investigation is suggested by some models of modulated diffusion. These transformations can be written as a skew-product of the endomorphism B p ( x) = px mod 1, p ϵ Z ⧹ {−1, 0, 1} , on the 1-torus T 1 := [0, 1] and a translation on T 1. Under suitable assumptions the skew-product can be proven to be mixing w.r.t. the Lebesgue measure. Central Limit (CL) and Functional Central Limit (FCL) properties are numerically checked for analytic observables. The result is remarkable because the mappings show no hyperbolic or quasi-hyperbolic structure, crucial for the proof of Central Limit Theorem and Donsker's Invariance Principle in all of the dynamical systems where these properties have been established up to now. Moreover, CL and FCL behaviours seem to hold also in the case of purely ergodic endomorphisms and even for observables whose correlations do not decay at infinity.

The Bracketing Condition for Limit Theorems on Stationary Linear Processes

This paper aims at improvement in regularity conditions on spectral densities for the limit theorems of the quasi maximum-likelihood estimator and the quasi likelihood-ratio statistic. The approach parallels the Daniels-Huber-Pollard proofs of central limit theorems under nonstandard conditions for i.i.d. situations. The results of the paper enable one to dispense with excessive regularity conditions on the spectral density.

Sequential Confidence Intervals for the Mean of a Subpopulation of a Finite Population

Abstract Sequential confidence intervals for the parameters of a subpopulation of a finite population are studied. Procedures are suggested using the mean, median, and trimmed means; these procedures are shown to be more efficient than their counterparts which are based on independent, identically distributed observations from infinite populations. The major technique used is to “linearize” the estimates, yielding as a simple consequence proofs of central limit theorems. A Monte Carlo study shows that the small-sample performance is very good.

Sequential Confidence Intervals for the Mean of a Subpopulation of a Finite Population

Abstract Sequential confidence intervals for the parameters of a subpopulation of a finite population are studied. Procedures are suggested using the mean, median, and trimmed means; these procedures are shown to be more efficient than their counterparts which are based on independent, identically distributed observations from infinite populations. The major technique used is to “linearize” the estimates, yielding as a simple consequence proofs of central limit theorems. A Monte Carlo study shows that the small-sample performance is very good.