Frontiers of Mathematics in China >
Limit theorems of continuous-time random walks with tails
Received date: 08 Jan 2011
Accepted date: 24 Dec 2012
Published date: 01 Apr 2013
Copyright
We study the functional limits of continuous-time random walks (CTRWs) with tails under certain conditions. We find that the scaled CTRWs with tails converge weakly to an α-stable Lévy process in D([0, 1]) with M1-topology but the corresponding scaled CTRWs converge weakly to the same limit in D([0, 1]) with J1-topology.
Key words: Weak convergence; J1-topology; M1-topology; stable Lévy process
Yuqiang LI . Limit theorems of continuous-time random walks with tails[J]. Frontiers of Mathematics in China, 2013 , 8(2) : 371 -391 . DOI: 10.1007/s11464-013-0275-0
1 |
Bardina X, Jolis M. Weak approximation of the Brownian sheet from a Poisson process in the plane. Bernoulli, 2000, 6: 653-665
|
2 |
Becker-Kern P, Meerschaert M, Scheffler H. Limit theorems for coupled continuous time random walks. Ann Probab, 2004, 32: 730-756
|
3 |
Bingham N. Limit theorem for occupation times of Markov processes. Z Wahrs Ver Geb, 1971, 17: 1-22
|
4 |
Dai H, Li Y. Approximation to sub-Gaussian processes. Acta Math Sci Ser B Engl Ed, 2011, 31(5): 1945-1958
|
5 |
Delgado R, Jolis M. Weak approximation for a class of Gaussion processes. J Appl Probab, 2000, 37: 400-407
|
6 |
Feller W. An Introduce to Probability Theory and Its Applications. 3rd ed. New York: Wiley, 1967
|
7 |
Gihman I, Skorokhod A. The Theory of Stochastic Processes I. New York: Springer, 1974
|
8 |
Meerschaert M, Scheffler H. Limit theorems for continuous-time random walks with infinite mean waiting times. J Appl Probab, 2004, 41: 623-638
|
9 |
Montroll E, Weiss G. Random walks on lattices. II. J Math Phys, 1965, 6: 167-181
|
10 |
Samorodnitsky G, Taqqu M. Stable Non-Gaussian Random Processed. New York: Chapman and Hall, 1994
|
11 |
Sato K. Lévy processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999
|
12 |
Scalas E, Gorenflo R, Mainardi F. Fractional calculus and continuous-time finance. Physica A, 2000, 284: 376-384
|
13 |
Stroock D. Topics in Stochastic Differential Equations. Tata Institute of Fundamental Research, Bombay. Berlin: Springer-Verlag, 1982.
|
14 |
Weiss G. Aspects and Applications of the Random Walk. Amsterdam: North-Holland, 1994
|
15 |
Whitt W. Stochastic Processes Limits. New York: Springer, 2002
|
16 |
Woodroofe M. Nonlinear Renewal Theory in Sequential Analysis. Philadelphia: SIAM, 1982
|
/
〈 | 〉 |