Quasi-stationary distributions for Lévy processes. (English) Zbl 1130.60054

Kyprianou, A. E.; Palmowski, Z.

Bernoulli 12, No. 4, 571-581 (2006).

Let \(X=\{ X_{t}:t\geq 0\} \) be a Lévy process defined on a filtered space \(( \Omega ,\mathcal{F},\mathbf{F},P) \) where the filtration \(\mathbf{F}=\{ \mathcal{F}_{t}:t\geq 0\} \) is assumed to satisfy the usual assumptions of right continuity and completion. Consider the probabilities \(\{ P_{x}:x\in \mathbb{R}\} \) such that \(P_{x}( X_{0}=x) =1\) and \(P_{0}=P\). Define the first passage time into the lower half-line \(( -\infty ,0) \) by \(\tau =\inf \{t>0:X_{t}<0\} \). The main object of the authors’ interest is the existence and characterization of the so-called limiting quasi-stationary distribution (or Yaglom’s limit) \(\lim_{t\uparrow \infty}P_{x}( X_{t}\in B| \tau >t) =\mu ( B) \) for \(B\in \mathcal{B}( [0,\infty )) \) where, in particular, \(\mu \) does not depend on the initial state \(x\geq 0\). The authors consider this limit for \(x\geq 0\) in the case of Lévy processes for which \(( -\infty ,0) \) is irregular for \(0\), and for \(x>0\) in the case of Lévy processes for which \(( -\infty ,0) \) is regular for \(0\). The existence and characterization result obtained improves the analogous result for spectrally positive compound Poisson processes with negative drift and jump distribution with rational characteristic function proved (within the context of the \(M/G/1\) queue) by E. K. Kyprianou [J. Appl. Prob. 8, 494–507 (1971; Zbl 0234.60121)] as well as the result for Brownian motion with drift proved by S. Martinez and J. San Martin [J. Appl. Probab. 31, 911–920 (1994; Zbl 0818.60071)], and complements an analogous result for random walks obtained by D. L. Iglehart as well as related results due to R. A. Doney.

Reviewer: Marius Iosifescu (Bucureşti)

MSC:

60G51	Processes with independent increments; Lévy processes
60F05	Central limit and other weak theorems

Keywords:

Levy process; quasi-stationary distribution; random walk

Citations:

Zbl 0234.60121; Zbl 0818.60071

References:

[1]	Bertoin, J. (1993) Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl., 47, 17-35. · Zbl 0786.60101
[2]	Bertoin, J. (1996) Lévy Processes. Cambridge: Cambridge University Press. · Zbl 0861.60003
[3]	Bertoin, J. and Doney, R.A. (1994) On conditioning a random walk to stay nonnegative. Ann. Probab., 22, 2152-2167. · Zbl 0834.60079
[4]	Bertoin, J. and Doney, R.A. (1996) Some asymptotic results for transient random walks. Adv. in Appl. Probab., 28, 207-226. JSTOR: · Zbl 0854.60069
[5]	Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge: Cambridge University Press. · Zbl 0617.26001
[6]	Chaumont, L. (1996) Conditioning and path decompositions for Lévy processes. Stochastic Process. Appl., 64, 39-54. · Zbl 0879.60072
[7]	Chaumont, L. and Doney, R.A. (2005) On Lévy processes conditioned to stay positive. Electron. J. Probab., 10, 948-961. · Zbl 1109.60039
[8]	Croft, H.T. (1957) A question of limits. Eureka, 20, 11-13.
[9]	Doney, R.A. (1989) On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Related Fields, 81, 239-246. · Zbl 0643.60053
[10]	Embrechts, P., Goldie, C.M. and Veraverbeke, N. (1979) Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie. Verw. Geb., 49, 335-347. · Zbl 0397.60024
[11]	Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol II, 2nd edn. New York: Wiley. · Zbl 0219.60003
[12]	Getoor, R.K. and Sharpe, M.J. (1994) On the arc-sine laws for Lévy processes. J. Appl. Probab., 31, 76-89. · Zbl 0802.60070
[13]	Iglehart, D.L. (1974) Random walks with negative drift conditioned to stay positive. J. Appl. Probab., 11, 742-751. JSTOR: · Zbl 0302.60038
[14]	Jacka, S.D. and Roberts, G.O. (1995) Weak convergence of conditioned processes on a countable state space. J. Appl. Probab., 32, 902-916. JSTOR: · Zbl 0839.60069
[15]	Jacod, J. and Shiryaev, A.N. (2003) Limit Theorems for Stochastic Processes, 2nd edn. Berlin: Springer-Verlag. · Zbl 1018.60002
[16]	Kingman, J.F.C. (1963) Ergodic properties of continuous-time Markov processes and their discrete skeletons. Proc. London Math. Soc., 13(3), 593-604. · Zbl 0154.43101
[17]	Kyprianou, E.K. (1971) On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals. J. Appl. Probab., 8, 494-507. JSTOR: · Zbl 0234.60121
[18]	Martínez, S. and San Martín, J. (1994) Quasi-stationary distributions for a Brownian motion with drift and associated limit laws. J. Appl. Probab., 31, 911-920. · Zbl 0818.60071
[19]	Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab., 3, 403-434. JSTOR: · Zbl 0147.36603
[20]	Tweedie, R.L. (1974) Quasi-stationary distributions for Markov chains on a general state space. J. Appl. Probab., 11, 726-741. JSTOR: · Zbl 0309.60040