New rates for exponential approximation and the theorems of Rényi and Yaglom

By Erol A. Peköz, Adrian Röllin. Annals of Probability, 2011

87 citations recorded by the source corpus.

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Renyi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton―Watson process conditioned on nonextinction. The primary tools are an adaptation of Stein's method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f6dbfa6bc300bd840d1cf3a72974bd49fbb7d11f

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Citations in the corpus

Fundamentals of Stein's method (2011, background): “A more general theorem can be proved with a little added technicality which allows for the Xi to have different means and for the Xi to have a certain dependence - see [43].”
Chi-square approximation by Stein's method with application to Pearson's statistic (2015, methodology): “Over the years, Stein’s method has been extended to many other distributions, such as the Poisson [5], multinomial [20], exponential [4], [27], Laplace [29], variance-gamma [14], and the gamma distribution [21], [25], which we develop further in this paper.”
Stein's Method of Exchangeable Pairs with Application to the Curie-Weiss Model (2009, background): “We refer to Chatterjee, Fulman and Röllin (2008) and Peköz and Röllin (2009) for other general results for the exponential approximation.”
Degree asymptotics with rates for preferential attachment random graphs (2011, methodology): “Letting U be a uniform (0,1) random variable independent of all else, Goldstein (2009) and Peköz and Röllin (2011) develop Stein’s method for the exponential distribution using the fact that W and UW ′ have the same distribution if and only if W has an exponential distribution [Pakes and Khattree…”
On Stein's method for products of normal random variables and zero bias couplings (2013, methodology): “Since then, Stein’s method has been been extended to many other distributions, such as the Poisson [3], Gamma [14], [15] and Exponential [2], [17]; for an overview see Reinert [19].”
Variance-Gamma approximation via Stein's method (2013, methodology): “Since then, this method has been extended to many other distributions, such as the Poisson [10], Gamma [27], [29], Exponential [9], [31] and Laplace [13], [34].”
Stein's method for comparison of univariate distributions (2014, methodology): “Outside of the Gaussian and Poisson frameworks, for univariate distributions the method has now also been shown to be effective for: exponential approximation [19, 79], Gamma approximation [68, 81, 70], binomial approximation [31], Beta approximation [43, 28], the asymptotics of rank distributions [”
A short survey of Stein's method (2014, methodology): “Further advances were made in the recent manuscript of Röllin [75].”
Probability on Trees and Networks (2017, background): “The first result of this kind was due to Pemantle (1991), who proved it for Z with d = 3, 4 and also showed that there are at most two ends per tree for d ≥ 5.”