Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

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

Science China Mathematics - We consider a branching Wiener process in ?d, in which particles reproduce as a super-critical Galton-Watson process and disperse according to a Wiener process....     

Asymptotic expansions based on smooth functions in the central limit theorem

Summary Stein's method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables, together with Lyapounov estimates of the error in the approximation.     

The central limit theorem for additive functionals of Markov and semi-Markov processes

One presents the results obtained recently by the collaborators of the Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, regarding limit theorems for additive functionals of Markov and semi-Markov processes. One makes use of the theory of inversion of singularly perturbed semigroups of operators in the phase extension scheme and of the methods of asymptotic analysis of singularly perturbed Markov renewal equations.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 229–246, 1986.     

Asymptotic expansions for Laplace transforms of Markov processes

Let ${\mu }^{?}$ be the probability measures on $D[0,T]$ of suitable Markov processes ${\{{\xi }_t^{?}\}}_{0\le t\le T}$ (possibly with small jumps) depending on a small parameter $?>0$, where $D[0,T]$ denotes the space of all functions on $[0,T]$ which are right continuous with left limits. In this paper we investigate asymptotic expansions for the Laplace transforms ${\int }_{D[0,T]}\exp ?\{{?}^{?1}F(x)\}{\mu }^{?}(dx)$ as $?\to 0$ for smooth functionals F on $D[0,T]$. This study not only recovers several well-known results, but more importantly provides new expansions for jump Markov processes. Besides several standard tools such as exponential change of measures and Taylor's expansions, the novelty of the proof is to implement the expectation asymptotic expansions on normal deviations which were recently derived in [13].     

Asymptotic expansions in limit theorems for stochastic processes. I

Summary. For some families of locally infinitely divisible Markov processes η ^ɛ (t), 0≦ t≦ T, with frequent small jumps, limit theorems for expectations of functionals F(η ^ɛ [0,T]) are proved of the form | E ^ɛ F(η ^ɛ [0,T])−E ⁰ F(η ⁰ [0,T])|≦ const ⋅ k(ɛ) , E ^ɛ F(η ^ɛ [0,T])=E ⁰ [F(η ⁰ [0,T])+ k(ɛ) ⋅ A ₁ F(η ⁰ [0,T])]+o(k(ɛ))  (ɛ↓ 0) , where A ₁ is a linear differential operator acting on functionals, and the constant is expressed in terms of the local characteristics of the processes η ^ɛ (t) and the norms of the derivatives of the functional F. Received: 1 April 1994 / In revised form: 30 September 1995     

Asymptotic expansions in limit theorems for stochastic processes. II

. For a certain class of families of stochastic processes η^ε(t), 0≤t≤T, constructed starting from sums of independent random variables, limit theorems for expectations of functionals F(η^ε[0,T]) are proved of the form

Asymptotic expansion in the central limit theorem for quadratic forms

We consider statistics of the form

On the functional central limit theorem and the law of the iterated logarithm for Markov processes

Summary Let X _tt0 be an ergodic stationary Markov process on a state space S. If  is its infinitesimal generator on L ²(S, dm), where m is the invariant probability measure, then it is shown that for all f in the range of } } 0)$$ " align="middle" border="0"> converges in distribution to the Wiener measure with zero drift and variance parameter ² =–2f, g=–2Âg, g where g is some element in the domain of  such that Âg=f (Theorem 2.1). Positivity of ² is proved for nonconstant f under fairly general conditions, and the range of  is shown to be dense in 1. A functional law of the iterated logarithm is proved when the (2+)th moment of f in the range of  is finite for some >0 (Theorem 2.7(a)). Under the additional condition of convergence in norm of the transition probability p(t, x, d y) to m(dy) as t , for each x, the above results hold when the process starts away from equilibrium (Theorems 2.6, 2.7 (b)). Applications to diffusions are discussed in some detail.This research was partially supported by NSF Grants MCS 79-03004, CME 8004499     

Asymptotic expansions for Markov processes with lévy generators

This paper considers a deterministic flow inn-dimensional space, perturbed by a Markov jump process with small variance. Asymptotic expansions are obtained for certain functionals of Feynman—Kac type, in powers of a small parameter representing a noise intensity. The methods are analytical rather than probabilistic.The research of the first author was partly supported by AFOSR under Contract No. 91-0116-0, by ONR under Contract No. N0014-83-K-0542, and by the Institute for Mathematics and Its Applications with funds provided by the NSF and ONR. The second author's research was partly supported by NSF under Contract No. DMS-8702537, and by the Institute for Mathematics and Its Applications with funds provided by the NSF and ONR.     

On the central limit theorem for geometrically ergodic Markov chains

Let X₀,X₁,... be a geometrically ergodic Markov chain with state space and stationary distribution . It is known that if h: R satisfies (|h|²⁺)< for some >0, then the normalized sums of the X_is obey a central limit theorem. Here we show, by means of a counterexample, that the condition (|h|²⁺)< cannot be weakened to only assuming a finite second moment, i.e., (h²)<.Reasearch supported by the Swedish Research Council.