Convergence to the maximum process of a fractional Brownian motion with shot noise

被引:4
作者
Wang, Yizao [1 ]
机构
[1] Univ Cincinnati, Dept Math Sci, Cincinnati, OH 45221 USA
关键词
Fractional Brownian motion; Perturbed random walk; Invariance principle; Point process; Continuous mapping theorem; Skorohod metric; EXTREMAL PROCESSES; ORDER-STATISTICS; WEAK-CONVERGENCE;
D O I
10.1016/j.spl.2014.03.014
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
We consider the maximum process of a random walk with additive independent noise in the form of max(i=1,...,n)(S-i + Y-i). The random walk may have dependent increments, but its sample path is assumed to converge weakly to a fractional Brownian motion. When the largest noise has the same order as the maximal displacement of the random walk, we establish an invariance principle for the maximum process in the Skorohod topology. The limiting process is the maximum process of the fractional Brownian notion with shot noise generated by Poisson point processes. (C) 2014 Elsevier B.V. All rights reserved.
引用
收藏
页码:33 / 41
页数:9
相关论文
共 20 条
[1]  
ALSMEYER G, 2014, J THEORET P IN PRESS
[2]  
[Anonymous], 1951, MEM AM MATH SOC
[3]   Tail asymptotics for the maximum of perturbed random walk [J].
Araman, Victor F. ;
Glynn, Peter W. .
ANNALS OF APPLIED PROBABILITY, 2006, 16 (03) :1411-1431
[4]   2 MOMENTS SUFFICE FOR POISSON APPROXIMATIONS - THE CHEN-STEIN METHOD [J].
ARRATIA, R ;
GOLDSTEIN, L ;
GORDON, L .
ANNALS OF PROBABILITY, 1989, 17 (01) :9-25
[5]   WEAK-CONVERGENCE OF SUMS OF MOVING AVERAGES IN THE ALPHA-STABLE DOMAIN OF ATTRACTION [J].
AVRAM, F ;
TAQQU, MS .
ANNALS OF PROBABILITY, 1992, 20 (01) :483-503
[6]  
Billingsley P, 1999, WILEY SERIES PROBABI, V2nd
[7]  
Davis B, 1996, ANN PROBAB, V24, P2007
[8]   Invariance principles for linear processes with application to isotonic regression [J].
Dedecker, Jerome ;
Merlevede, Florence ;
Peligrad, Magda .
BERNOULLI, 2011, 17 (01) :88-113
[9]   New techniques for empirical processes of dependent data [J].
Dehling, Herold ;
Durieu, Olivier ;
Volny, Dalibor .
STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 2009, 119 (10) :3699-3718
[10]   EXTREMAL PROCESSES [J].
DWASS, M .
ANNALS OF MATHEMATICAL STATISTICS, 1964, 35 (04) :1718-&