scholarly journals A characterization of the weak convergence of convolution powers

1990 ◽  
Vol 115 (1) ◽  
pp. 48-60
Author(s):  
Jan Rataj
Keyword(s):  
Weak Convergence ◽  
Convolution Powers

On weak convergence within the hnbue family of life distributions

1984 ◽  
Vol 21 (03) ◽  
pp. 654-660 ◽  
Author(s):  
Sujit K. Basu ◽  
Manish C. Bhattacharjee
Keyword(s):  
Weak Convergence ◽  
Exponential Distribution ◽  
Limiting Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moment Sequence ◽  
Positive Order ◽  
Life Distributions ◽  
Necessary And Sufficient

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

scholarly journals Rapid variation with remainder and rates of convergence

1990 ◽  
Vol 22 (04) ◽  
pp. 787-801 ◽  
Author(s):  
J. Beirlant ◽  
E. Willekens
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Rates Of Convergence ◽  
Convergence Result ◽  
Hill Estimator ◽  
Rapid Variation ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
The Hill

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

scholarly journals Rapid variation with remainder and rates of convergence

1990 ◽  
Vol 22 (4) ◽  
pp. 787-801 ◽  
Author(s):  
J. Beirlant ◽  
E. Willekens
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Rates Of Convergence ◽  
Convergence Result ◽  
Hill Estimator ◽  
Rapid Variation ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
The Hill

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

scholarly journals Amenability of groupoids and asymptotic invariance of convolution powers

2021 ◽  
pp. 69-92
Author(s):  
Theo Bühler ◽  
Vadim Kaimanovich
Keyword(s):  
Random Walk ◽  
Locally Compact Group ◽  
Probability Measures ◽  
Locally Compact ◽  
Von Neumann ◽  
Original Definition ◽  
Convolution Powers ◽  
Measure Class ◽  
Definition Of

The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich–Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.

Weak convergence of conditioned processes on a countable state space

1995 ◽  
Vol 32 (4) ◽  
pp. 902-916 ◽  
Author(s):  
S. D. Jacka ◽  
G. O. Roberts
Keyword(s):  
Weak Convergence ◽  
State Space ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Original Process ◽  
Countable State ◽  
Positive Time ◽  
Countably Infinite ◽  
Infinite State

We consider the problem of conditioning a continuous-time Markov chain (on a countably infinite state space) not to hit an absorbing barrier before time T; and the weak convergence of this conditional process as T → ∞. We prove a characterization of convergence in terms of the distribution of the process at some arbitrary positive time, t, introduce a decay parameter for the time to absorption, give an example where weak convergence fails, and give sufficient conditions for weak convergence in terms of the existence of a quasi-stationary limit, and a recurrence property of the original process.

scholarly journals A characterization of weak∗convergence

1964 ◽  
Vol 14 (3) ◽  
pp. 1059-1067
Author(s):  
Maurice Sion
Keyword(s):  
Weak Convergence
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