Limit theorems of continuous-time random walks with tails

Abstract

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.

Keywords

Weak convergence / J₁-topology / M₁-topology / stable Lévy process

References

[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 Verw 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, 1967 3rd ed. New York: Wiley

[7] Gihman I, Skorokhod A. The Theory of Stochastic Processes I, 1974, New York: Springer

[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, 1994, New York: Chapman and Hall

[11] Sato K. Lévy processes and Infinitely Divisible Distributions, 1999, Cambridge: Cambridge University Press

[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, 1982, Berlin: Springer-Verlag

[14] Weiss G. Aspects and Applications of the Random Walk, 1994, Amsterdam: North-Holland

[15] Whitt W. Stochastic Processes Limits, 2002, New York: Springer

[16] Woodroofe M. Nonlinear Renewal Theory in Sequential Analysis, 1982, Philadelphia: SIAM