Closed Form Summation for Classical Distributions: Variations on a Theme of De Moivre

Diaconis, Persi; Zabell, Sandy L.

doi:10.1214/ss/1177011699

Cited by 131 publications

(111 citation statements)

References 35 publications

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“…Then X for all continuous and piecewise differentiable functions f such that E[|f (X)|] < ∞ (see e.g. [4], [5], [21]). If the above expectation is non-zero but close to zero, Stein's method can give us a way to express how close the law of X might be to the standard normal law, in particular by using the concept of Stein equation.…”

Section: Stein's Methods and The Analysis Of Nourdin And Peccatimentioning

confidence: 99%

General Upper and Lower Tail Estimates Using Malliavin Calculus and Stein’s Equations

Eden

Viens

2013

Seminar on Stochastic Analysis, Random Fields and Applications VII

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Abstract. Following a strategy recently developed by Ivan Nourdin and Giovanni Peccati, we provide a general technique to compare the tail of a given random variable to that of a reference distribution, and apply it to all reference distributions in the so-called Pearson class. This enables us to give concrete conditions to ensure upper and/or lower bounds on the random variable's tail of various power or exponential types. The Nourdin-Peccati strategy analyzes the relation bewteen Stein's method and the Malliavin calculus, and is adapted to dealing with comparisons to the Gaussian law. By studying the behavior of the solution to general Stein equations in detail, we show that the strategy can be extended to comparisons to a wide class of laws, including all Pearson distributions.Mathematics Subject Classification (2010). Primary 60H07; Secondary 60G15, 60E15.

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Section: Stein's Methods and The Analysis Of Nourdin And Peccatimentioning

confidence: 99%

General Upper and Lower Tail Estimates Using Malliavin Calculus and Stein’s Equations

Eden

Viens

2013

Seminar on Stochastic Analysis, Random Fields and Applications VII

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“…It is known that Stein's identity characterizes the normal distribution in an appropriately precise sense; one may see Diaconis and Zabell (1991). The following two questions, therefore, emerge naturally: We shall now address these two questions.…”

Section: Analysis Of the Heat Equation Identitymentioning

confidence: 97%

“…Diaconis and Zabell (1991) showed that the identity E((X − µ)h(X)) = E(h (X)) cannot hold for all C 1 c (R) functions except when X ∼ N (µ, 1). Interestingly, the inequality µ, m) and if h(·) is monotone nondecreasing, as we show below.…”

Section: Application To Decision Theorymentioning

confidence: 99%

The heat equation and Stein's identity: Connections, applications

Brown

Dasgupta

Haff

et al. 2006

Journal of Statistical Planning and Inference

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This article presents two expectation identities and a series of applications. One of the identities uses the heat equation, and we show that in some families of distributions the identity characterizes the normal distribution. We also show that it is essentially equivalent to Stein's identity. The applications we have presented are of a broad range. They include exact formulas and bounds for moments, an improvement and a reversal ofJensen's inequality, linking unbiased estimation to elliptic partial differential equations, applications to decision theory and Bayesian statistics, and an application to counting matchings in graph theory. Some examples are also given.

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“…Refer to [6] and [26] for more information about Pearson distributions, and [8] for Stein's method applied to comparisons of probability tails with a Pearson Z. From Remark 4, if the support of Z is unbounded and g * is a polynomial, then Z is necessarily Pearson.…”

Section: Convergence When G * Is a Polynomialmentioning

confidence: 99%

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Eden

Víquez

2015

Stochastic Processes and their Applications

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Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet-Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener-Poisson space. In the former, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. As an example, we apply this tool to bilinear functionals of Gaussian subordinated fields where the underlying process is a fractional Brownian motion with Hurst parameter bigger than 1/2. In the latter space we give sufficient conditions for a sequence of multiple (WienerPoisson) integrals to converge to a Normal random variable.

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Closed Form Summation for Classical Distributions: Variations on a Theme of De Moivre

Cited by 131 publications

References 35 publications

General Upper and Lower Tail Estimates Using Malliavin Calculus and Stein’s Equations

General Upper and Lower Tail Estimates Using Malliavin Calculus and Stein’s Equations

The heat equation and Stein's identity: Connections, applications

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

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