Convergence in Distribution

2021 ◽  
pp. 505-528
Author(s):  
Olav Kallenberg
Keyword(s):  
Convergence In Distribution

An L 2 convergence theorem for random affine mappings

1995 ◽  
Vol 32 (01) ◽  
pp. 183-192 ◽  
Author(s):  
Robert M. Burton ◽  
Uwe Rösler
Keyword(s):  
Hilbert Space ◽  
Lyapunov Exponent ◽  
Bilinear Form ◽  
Convergence Theorem ◽  
Convergence In Distribution ◽  
Wasserstein Metric ◽  
Affine Mappings ◽  
Affine Maps ◽  
Second Moments ◽  
Contraction Condition

We consider the composition of random i.i.d. affine maps of a Hilbert space to itself. We show convergence of thenth composition of these maps in the Wasserstein metric via a contraction argument. The contraction condition involves the operator norm of the expectation of a bilinear form. This is contrasted with the usual contraction condition of a negative Lyapunov exponent. Our condition is stronger and easier to check. In addition, our condition allows us to conclude convergence of second moments as well as convergence in distribution.

scholarly journals Limit Theorems for a Generalized ST Petersburg Game

2010 ◽  
Vol 47 (3) ◽  
pp. 752-760 ◽  
Author(s):  
Allan Gut
Keyword(s):  
Limit Theorems ◽  
Convergence In Distribution ◽  
Stable Law ◽  
Geometric Size

The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X = sr(k-1)/α) = pqk-1, k = 1, 2,…, where p + q = 1, s = 1 / p, r = 1 / q, and 0 < α ≤ 1. For the case in which α = 1, we extend Feller's classical weak law and Martin-Löf's theorem on convergence in distribution along the 2n-subsequence. The analog for 0 < α < 1 turns out to converge in distribution to an asymmetric stable law with index α. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.

Poisson theorem for non-homogeneous Markov chains

1989 ◽  
Vol 26 (03) ◽  
pp. 637-642 ◽  
Author(s):  
Janusz Pawłowski
Keyword(s):  
Markov Chains ◽  
Poisson Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Compound Poisson Distribution ◽  
Convergence In Distribution ◽  
Compound Poisson ◽  
Necessary And Sufficient

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.

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