Almost Sure Approximation of the Superposition of the Random Processes

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• 作者：Nadiia Zinchenko (1)
  
  1. Department of Informatics and Applied Mathematics ; Nizhyn State Mukola Gogol University ; 16600 ; Nizhyn ; Ukraine
  
• 关键词：Strong invariance principle ; Superposition of random processes ; Randomly stopped sums ; Queuing system ; Risk process ; Total claim amount ; Stable process ; 60F15 ; 60F17 ; 60G50 ; 60G52 ; 91B30 ; 60K10
• 刊名：Methodology and Computing in Applied Probability
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：17
• 期：1
• 页码：235-250
• 全文大小：314 KB
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  12. Silvestrov D (2004) Limit theorems for randomly stopped stochastic processes. Springer, London lass="external" href="http://dx.doi.org/10.1007/978-0-85729-390-9" target="_blank" title="It opens in new window">CrossRef
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  14. Strassen V (1964) An invariance principle for the law of the iterated logarithm. Z Wahrscheinlichkeitstheor Verw Geb 3:211鈥?26 lass="external" href="http://dx.doi.org/10.1007/BF00534910" target="_blank" title="It opens in new window">CrossRef
  15. Strassen V (1965) Almost sure behavior for sums of independent randon variables and martingales. In: Proseedings of 5th Berkeley Symposium on Mathematical and Probability, vol. 2, pp. 315鈥?43
  16. Whitt W (2002) Stochastic-processes limits: an introduction to stochastic-process limits and their application to queues. Springer, New York
  17. Wu W (2007) Strong invariance principle for dependent random variables. Ann Probab 35:2294鈥?330 lass="external" href="http://dx.doi.org/10.1214/009117907000000060" target="_blank" title="It opens in new window">CrossRef
  18. Zinchenko N (1985) The strong invariance principle for sums of random variables in the domain of attraction of a stable law. Teor Verojatnost i Primenen 30:131鈥?35
  19. Zinchenko N (2000) The Skorokhod representation and strong invariance principle. Teor Jmovirn Mat Stat 60:51鈥?3
  20. Zinchenko N (2007) The strong invariance principle for renewal and randomly stopped processes. Theor Stoch Process 13(29):233鈥?45
  21. Zinchenko N (2010) Strong invariance principle for a superposition of random processes. Theor Stoch Process 16(32):130鈥?38
  22. Zinchenko N, Safonova M (2008) Erd枚s-Renyi type law for random sums with applications to claim amount process. J Numer Appl Math 1(96):246鈥?64
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• 刊物主题：Statistics, general; Life Sciences, general; Electrical Engineering; Economics general; Business/Management Science, general;
• 出版者：Springer US
• ISSN：1573-7713

文摘

This article presents sufficient conditions, which provide almost sure (a.s.) approximation of the superposition of the random processes S(N(t)), when c脿d-l脿g random processes S(t) and N(t) themselves admit a.s. approximation by a Wiener or stable L茅vy processes. Such results serve as a source of numerous strong limit theorems for the random sums under various assumptions on counting process N(t) and summands. As a consequence we obtain a number of results concerning the a.s. approximation of the Kesten鈥揝pitzer random walk, accumulated workload input into queuing system, risk processes in the classical and renewal risk models with small and large claims and use such results for investigation the growth rate and fluctuations of the mentioned processes.