A Weak Convergence Result Relevant in Recurrent and Renewal Models

2000 ◽  
pp. 493-514 ◽  
Author(s):  
Edsel A. Peña ◽  
Robert L. Strawderman ◽  
Myles Hollander
Keyword(s):  
Weak Convergence ◽  
Convergence Result ◽  
Renewal Models

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.

On pair and tuple formation under independent Poisson or renewal arrival processes

2004 ◽  
Vol 41 (4) ◽  
pp. 1138-1144 ◽  
Author(s):  
K. Borovkov
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Formation Process ◽  
Short Proof ◽  
Probabilistic Approach ◽  
Convergence Result ◽  
Pair Formation ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Arrival Processes

We present several results refining and extending those of Neuts and Alfa on weak convergence of the pair-formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward, and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair-formation process to a Poisson process actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuple formation).

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.

On the weak convergence of U-statistic processes, and of the empirical process

1978 ◽  
Vol 83 (2) ◽  
pp. 269-272 ◽  
Author(s):  
R. M. Loynes
Keyword(s):  
Weak Convergence ◽  
Wiener Process ◽  
Empirical Process ◽  
Convergence Result ◽  
The Other ◽  
Direct Argument ◽  
Random Functions ◽  
Martingale Theorem ◽  
The Relationship ◽  
Special Case

1. Summary and introductionIn (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.

scholarly journals A Fast Image Restoration Algorithm Based on a Fixed Point and Optimization Method

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 378 ◽  
Author(s):  
Adisak Hanjing ◽  
Suthep Suantai
Keyword(s):  
Fixed Point ◽  
Weak Convergence ◽  
Image Restoration ◽  
Optimization Method ◽  
Convergence Result ◽  
Fixed Point Algorithm ◽  
Convergence Behavior ◽  
Lower Semi ◽  
Restoration Algorithm ◽  
Nonexpansive Operators

In this paper, a new accelerated fixed point algorithm for solving a common fixed point of a family of nonexpansive operators is introduced and studied, and then a weak convergence result and the convergence behavior of the proposed method is proven and discussed. Using our main result, we obtain a new accelerated image restoration algorithm, called the forward-backward modified W-algorithm (FBMWA), for solving a minimization problem in the form of the sum of two proper lower semi-continuous and convex functions. As applications, we apply the FBMWA algorithm to solving image restoration problems. We analyze and compare convergence behavior of our method with the others for deblurring the image. We found that our algorithm has a higher efficiency than the others in the literature.

Iterative algorithms for split common fixed points of demicontractive operators without priori knowledge of operator norms

2018 ◽  
Vol 34 (3) ◽  
pp. 459-466
Author(s):  
YONGHONG YAO ◽  
◽  
JEN-CHIH YAO ◽  
YEONG-CHENG LIOU ◽  
MIHAI POSTOLACHE ◽  
...  
Keyword(s):  
Weak Convergence ◽  
Fixed Points ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Convergence Result ◽  
Common Fixed Points ◽  
Operator Norms ◽  
Priori Knowledge

The split common fixed points problem for demicontractive operators has been studied in Hilbert spaces. An iterative algorithm is considered and the weak convergence result is given under some mild assumptions.

UNIFORM CONVERGENCE TO A LAW CONTAINING GAUSSIAN AND CAUCHY DISTRIBUTIONS

2012 ◽  
Vol 26 (3) ◽  
pp. 437-448
Author(s):  
Jose H. Blanchet ◽  
Carlos G. Pacheco-González
Keyword(s):  
Weak Convergence ◽  
Poisson Process ◽  
Uniform Convergence ◽  
Time Horizon ◽  
Convergence Result ◽  
Gaussian Distributions ◽  
Long Time ◽  
Time Average ◽  
Average Position

A source of light is placeddinches apart from the center of a detection bar of lengthL≥d. The source spins very rapidly, while shooting beams of light according to, say, a Poisson process with rate λ. The positions of the beams, relative to the center of the bar, are recorded for those beams that actually hit the bar. Which law best describes the time-average position of the beams that hit the bar given a fixed but long time horizont? The answer is given in this paper by means of a uniform weak convergence result inL, dast→ ∞. Our approximating law includes as particular cases the Cauchy and Gaussian distributions.

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