Fundamentals of Stein's method

By Nathan Ross. 2011

479 citations recorded by the source corpus.

This survey article discusses the main concepts and techniques of Stein's method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its relation to concentration of measure inequalities. The material is presented at a level accessible to beginning raduate students studying probability with the main emphasis on the themes that are common to these topics and also to much of the Stein's method literature.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/af3465e5c0f54b6d2160e3a6fa18254357b21469

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Heavy traffic approximation for the stationary distribution of a generalized Jackson network: The BAR approach (2015, methodology): “To prove Proposition 4.1 (that ϕ(nk)(θ) and its boundary counterparts satisfy BAR (2.30) asymptotically), we work with a continuous time Markov process X(n) = {X(n)(t), t ≥ 0} that describes the dynamics of the nth GJN.”
On Stein's Method for Infinitely Divisible Laws with Finite First Moment (2017, methodology): “on case the method hasbeen developed for variousdistributionssuch as compoundPoisson, geometric, negative binomial, exponential, Laplace. (We refer the reader to Chen, Goldstein and Shao [10] or Ross [33] for good introductions to the method, as well as more precise and complete references.) The met”
Degree asymptotics with rates for preferential attachment random graphs (2011, methodology): “This transformation allows us to develop a new variation of Stein’s method; we refer to Chen, Goldstein and Shao (2011), Ross (2011) and Ross and Peköz (2007) for introductions to Stein’s method.”
Stein’s density approach and information inequalities (2012, background): “For the origins we refer the reader to [40, 38, 37]; for an overview of the more recent achievements in this field we refer to the monographs [28, 3, 4, 12] or the review articles [27, 31].”
Size biased couplings and the spectral gap for random regular graphs (2015, methodology): “See [18] or [48] for surveys of Stein’s method including size biased coupling.”
Second-order properties and central limit theorems for geometric functionals of Boolean models (2013, background): “act that the Kolmogorov distance to a standard Gaussian random variable is always bounded by the square root of the Wasserstein distance to a standard Gaussian random variable (see Proposition 1.2 in [29], for example), it follows from (9.4) that sup x∈R P ψ(Z∩W)−Eψ(Z∩W) p Varψ(Z∩W) ≤ x ! −P(N≤ x) ≤”
Stein's method for steady-state diffusion approximations of $M/Ph/n+M$ systems (2015, methodology): “n isolation. Stein’s method is a powerful method that has been widely used in probability, statistics, and their wide range of applications such as bioinformatics; see, for example, the survey papers [11, 45], the recent book [12] and the references within. The connection between Stein’s method and ”
Size bias for one and all (2013, background): “1An early draft of the present paper, with the title ‘Size biasing, when is the increment independent?’, has been circulated since 1998, and was cited in [72]; an update, ‘Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent?’ was cited in [70, ”
Spectral Gaps of Random Graphs and Applications (2012, methodology): “es in law to Poisson(e−c/d!). The proof of this lemma is standard and can be proved in the same manner as the Poisson convergence of the number of isolated vertices in G(n,p). See Proposition 4.13 of [42]. Using spectral techniques, we give a new proof of the Linial–Meshulam–Wallach theorem, althoug”
Stein's Method and the Laplace Distribution (2012, background): “Similar characterizing operators have since been worked out for several other distributions [6, 11, 1, 13, 15].”
Stein's method for steady-state diffusion approximations: an introduction through the Erlang-A and Erlang-C models (2015, background): “It is known, see for example [52], convergence under the Wasserstein metric implies convergence in distribution.”
Stein's method for comparison of univariate distributions (2014, background): “under unfavorable assumptions on W) including exchangeable pairs (as for example in [87, 48, 80, 19, 17, 26]), biasing mechanisms (as in [4, 41, 38, 74, 35]), and other couplings (as in [20, 9]); see [79, 83, 16] for overviews. Nourdin and Peccati [65] paved the way for many elegant results in the c”
Randomized numerical linear algebra: Foundations and algorithms (2020, methodology): “N. Ross (2011), ‘Fundamentals of Stein’s method’, Probab.”
Stein kernels and moment maps (2018, unknown): “We refer to the survey [34] for an overview.”