Moments of the last exit times

Takeuchi, Junji

doi:10.3792/pja/1195521602

Cited by 16 publications

(13 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The process {X t } is called strongly transient if 1 ∈ T; it is called weakly transient if 1 ∈ T. Hawkes [11] and Takeuchi [37] are early works on the moment of the last exit time for a symmetric Lévy process. Strong transience is important in the theory of the range of random walks.…”

Section: Applications To Stable Processesmentioning

confidence: 99%

Asymptotic estimates of multi-dimensional stable densities and their applications

Watanabe

2007

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed. In particular, the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of Pruitt and Taylor in 1969 are solved. The proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere, or discrete with the support being a finite set, or a mixture of such cases are obtained. Those results are applied to the moment of the last exit time from a ball and the Spitzer type limit theorem involving capacity for a multi-dimensional transient stable process.

show abstract

Section: Applications To Stable Processesmentioning

confidence: 99%

Asymptotic estimates of multi-dimensional stable densities and their applications

Watanabe

2007

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Last exit times for stable processes and Markov chains were discussed by Takeuchi, Yamada and Watanabe [23], Port [11][12][13], and Takeuchi [22] in 1960s in connection with probabilistic treatment of equilibrium measures. Then, for general transient Lévy processes they were discussed by Port and Stone [14].…”

Section: Introductionmentioning

confidence: 99%

Moments of last exit times for Lévy processes

Sato¹

2004

Annales de l'Institut Henri Poincare (B) Probability and Statis

View full text Add to dashboard Cite

Let L B a be the last exit time from the ball B a = {|x| < a} for a transient Lévy process {X t } on R d. It is proved that, for each η 0, either E[L B a η ] < ∞ for all a > 0 or E[L B a η ] = ∞ for all a > 0. Let T be the set of η 0 having the former property. The size of T gives an order of transience of {X t }. A criterion for η ∈ T is given in terms of the logarithm of the characteristic function of X 1. The set T is determined when d = 1 and E[|X t |] < ∞. Examples and related results are given.

show abstract

“…In the special case when f (t) = t κ for some κ > 0, we use the terminology κ-weak and κ-strong transience introduced in [SW04]. In this context, we first discuss the κ-weak and κ-strong transience of {F t } t≥0 with respect to the dimension of the state space and Pruitt indices and, in particular, the κ-weak and κ-strong transience of elliptic diffusion and stable-like processes, thus generalizing the results obtained in [LL06] and [Tak67]. Finally, in the case when the symbol of {F t } t≥0 is radial in the co-variable we provide a series of conditions for the κ-weak and κ-strong transience in terms of the corresponding Lévy measure.…”

Section: Introductionmentioning

confidence: 95%

On transience of Lévy-type processes

Sandrić

2016

Stochastics

View full text Add to dashboard Cite

In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called Lévy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for Lévy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable Lévy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the Lévy measures.for a (transient) Lévy process {L t } t≥0 and any κ > 0, E[L κ B(0,r) ] is either finite or infinite for every r > 0 (see also [Haw77] for the case of symmetric Lévy processes). Accordingly, a (transient) Lévy process {L t } t≥0 is said to be κ-weakly transient if E[L κ B(0,r) ] = ∞ for all r > 0, and κ-strongly transient if E[L κ B(0,r) ] < ∞ for all r > 0. Also, in the same reference, the authors have proved that the above conditions are equivalent to ∞ 0 t κ P(L t ∈ B(0, r))dt = ∞ for all r > 0 and ∞ 0

show abstract

Moments of the last exit times

Cited by 16 publications

References 4 publications

Asymptotic estimates of multi-dimensional stable densities and their applications

Asymptotic estimates of multi-dimensional stable densities and their applications

Moments of last exit times for Lévy processes

On transience of Lévy-type processes

Contact Info

Product

Resources

About