An Introduction to Stein's Method

Barbour, A. D.; Chen, Louis H. Y.

doi:10.1142/9789812567680

Cited by 115 publications

(152 citation statements)

References 4 publications

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“…Let us recall briefly some facts on the Stein method to be used in the proofs. For more details, the reader can consult the excellent reviews [10,23] or the more complete book [9]. The main idea is to describe the closeness of the law of a r.v.…”

Section: 1mentioning

confidence: 99%

Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment

Grama

Liu

Miqueu

2017

Stochastic Processes and their Applications

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Abstract. Let (Z n ) be a supercritical branching process in a random environment ξ = (ξ n ). We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for log Z n under the annealed law P. We also improve some earlier results about the harmonic moments of the limit variable W = lim n→∞ W n , where W n = Z n /E ξ Z n is the normalized population size. Introduction and main resultsA branching process in a random environment (BPRE) is a natural and important generalisation of the Galton-Watson process, where the reproduction law varies according to a random environment indexed by time. It was introduced for the first time in Smith and Wilkinson [24] to modelize the growth of a population submitted to an environment. For background concepts and basic results concerning a BPRE we refer to Athreya and Karlin [4,3]. In the critical and subcritical regime the process goes out and the research interest is concentrated mostly on the survival probability and conditional limit theorems for the branching process, see e.g. [21]. In this article, we complete on these results by giving the Berry-Esseen bound and asymptotics of large deviations of Cramér's type for a supercritical BPRE.A BPRE can be described as follows. The random environment is represented by a sequence ξ = (ξ 0 , ξ 1 , ...) of independent and identically distributed random variables (i.i.d. r.v.'s); each realization of ξ n corresponds to a probability law {p i (ξ n ) : i ∈ N} on N = {0, 1, 2, . . . }, whose probability generating function is

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Section: 1mentioning

confidence: 99%

Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment

Grama

Liu

Miqueu

2017

Stochastic Processes and their Applications

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“…Substituting θ = −tσ 2 /(Cµ) into (11) completes the proof of (2). Moving on to the proof of (3), taking expectation in (6) with θ > 0, we obtain…”

Section: Proof Of the Main Resultsmentioning

confidence: 76%

Concentration of measures via size-biased couplings

Ghosh

Goldstein

2009

Probab. Theory Relat. Fields

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Let Y be a nonnegative random variable with mean µ and finite positive variance σ 2 , and let Y s , defined on the same space as Y , have the Y size biased distribution, that is, the distribution characterized byfor all functions f for which these expectations exist.Under a variety of conditions on the coupling of Y and Y s , including combinations of boundedness and monotonicity, concentration of measure inequalities such asfor all t ≥ 0 hold for some explicit A and B. Examples include the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of m-runs in a sequence of coin tosses, the number of local maximum of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, the volume covered by the union of n balls placed uniformly over a volume n subset of R d , the number of bulbs switched on at the terminal time in the so called lightbulb process, and the infinitely divisible and compound Poisson distributions that satisfy a bounded moment generating function condition.

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“…In 1972, Stein [1] gave a new technique to find a bound in the normal approximation to a distribution of a sum of dependent random variables. His technique was relied instead on the elementary differential equation.…”

Section: Stein-chen and Coupling Methodsmentioning

confidence: 99%