We establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in McLeish [Ann. In doing so we shall preserve the generality of the coefficients, including the long range dependence case, and we shall express the variance of partial sums in a form easy to apply. Ergodicity is not required.

The purpose of this Note is to prove a form of CLT (Theorem 1.4) that is used in Epstein and Seo (2011). More general central limit results and other applications will follow in later drafts. Let S = fB;Ng and K (S) = ffBg; fNg; fB;Ngg the set of nonempty subsets of S. Denote by s1 = (s1; s2; :::) the generic element of S1 and by n (s1) the empirical frequency of the outcome B in the …rst n exp...

The de Moivre–Laplace theorem is a special case of the central limit for Bernoulli random variables, and can be proved by direct computation. We deduce any variable with finite variance from theorem. Our proof does not use advanced notions such as characteristic functions, Brownian motion, or stopping times.

Hawkes process is a self-exciting point process with clustering effect whose jump rate depends on its entire past history. It has wide applications in neuroscience, finance and many other fields. Linear Hawkes process has an immigration-birth representation and can be computed more or less explicitly. It has been extensively studied in the past and the limit theorems are well understood. On the...

An Edgeworth-type expansion is established for the relative Fisher information distance to the class of normal distributions of sums of i.i.d. random variables, satisfying moment conditions. The validity of the central limit theorem is studied via properties of the Fisher information along convolutions.

Let Tl be a transformation on the interval [−1, 1] defined by Chebyshev polynomial of degree l (l ≥ 2), i.e., Tl(cos θ) = cos(lθ). In this paper, we consider Tl as a measure preserving transformation on [−1, 1] with an invariant measure 1 π √ 1−x2 dx. We show that If f(x) is a nonconstant step function with finite kdiscontinuity points with k < l − 1, then it satisfies the Central Limit Theorem...

1. Main theorem. In this paper, we consider parameters defined on random discrete processes. When the parameters change by only a small amount from one state in the process to the next, one often finds that the parameters satisfy a law of large numbers, that is, the parameters are sharply concentrated around certain values. Wormald [6] gives some general criteria which ensure that given paramet...

This expository article states and proves four, concrete, projective, central limit theorems. The results are known or suspected to be true by experts who are familiar with the more general central limit theorem for convex bodies, and related theory. Here we consider only four types of high dimensional geometric objects: spheres, balls, cubes, and boundaries of cubes. Each is capable of transfo...

Let Fn denote the distribution function of the normalized sum Zn = (X1+ · · ·+Xn)/σ √ n of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of Fn to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of Fn by the Edgeworth corrections (modulo logarithmically growing factors in n) are given in ter...