Weak convergence of positive self-similar Markov processes and overshoots of levy processes

被引:35
作者
Caballero, M. E. [1 ]
Chaumont, L.
机构
[1] Univ Nacl Autonoma Mexico, Inst Matemat, Mexico City 04510, DF, Mexico
[2] Univ Paris 06, Lab Probabil & Modeles Aleatoires, F-75252 Paris 05, France
关键词
self-similar Markov process; levy process; Lamperti representation; overshoot; weak convergence; first passage time;
D O I
10.1214/009117905000000611
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
Using Lamperti's relationship between Levy processes and positive selfsimilar Markov processes (pssMp), we study the weak convergence of the law P., of a pssMp starting at x > 0, in the Skorchod space of cadlag paths, when x tends to 0. To do so, we first give conditions which allow us to construct a cadlag Markov process X-(0), starting from 0, which stays positive and verifies the scaling property. Then we establish necessary and sufficient conditions for the laws P-x to converge weakly to the law of X(0) as x goes to 0. In particular, this answers a question raised by Lamperti [Z Wahrsch. Verw. Gebiete 22 (1972) 205-225] about the Feller property for pssMp at X = 0.
引用
收藏
页码:1012 / 1034
页数:23
相关论文
共 14 条
[1]  
[Anonymous], LEVY PROCESSES
[2]  
Bertoin J, 2002, BERNOULLI, V8, P195
[3]   The entrance laws of self-similar Markov processes and exponential functionals of Levy processes [J].
Bertoin, J ;
Yor, M .
POTENTIAL ANALYSIS, 2002, 17 (04) :389-400
[4]   Renewal theory and level passage by subordinators [J].
Bertoin, J ;
van Harn, K ;
Steutel, FW .
STATISTICS & PROBABILITY LETTERS, 1999, 45 (01) :65-69
[5]  
CABALLERO ME, 2006, CONDITIONED STABLE P
[6]   On levy processes conditioned to stay positive. [J].
Chaumont, L ;
Doney, RA .
ELECTRONIC JOURNAL OF PROBABILITY, 2005, 10 :948-961
[7]   Conditionings and path decompositions for Levy processes [J].
Chaumont, L .
STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 1996, 64 (01) :39-54
[8]   ON MOMENTS OF LADDER HEIGHT VARIABLES [J].
CHOW, YS .
ADVANCES IN APPLIED MATHEMATICS, 1986, 7 (01) :46-54
[9]  
Doney RA, 2002, ANN PROBAB, V30, P188
[10]  
DONEY RA, 2004, COMMUNICATION