Subordinator

About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.

A note on the J1-convergence of a first-passage process

Abstract: This paper considers queues with server vacations, but departs from the traditional setting in two ways: (i) the queueing model is driven by Levy processes rather than just compound Poisson processes; (ii) the vacation lengths depend on the length of the server's preceding busy period. Regarding the former point: the Levy process active during the busy period is assumed to have no negative jumps, whereas the Levy process active during the vacation is a subordinator. Regarding the latter point: where in a previous study (Boxma et al. in Probab. Eng. Inf. Sci. 22:537---555, 2008) the durations of the vacations were positively correlated with the length of the preceding busy period, we now introduce a dependence structure that may give rise to both positive and negative correlations. We analyze the steady-state workload of the resulting queueing (or: storage) system, by first considering the queue at embedded epochs (viz. the beginnings of busy periods). We show that this embedded process does not always have a proper stationary distribution, due to the fact that there may occur an infinite number of busy-vacation cycles in a finite time interval; we specify conditions under which the embedded process is recurrent. Fortunately, irrespective of whether the embedded process has a stationary distribution, the steady-state workload of the continuous-time storage process can be determined. In addition, a number of ramifications are presented. The theory is illustrated by several examples.

On Lévy-driven vacation models with correlated busy periods and service interruptions

Abstract: This paper considers queues with server vacations, but departs from the traditional setting in two ways: (i) the queueing model is driven by Levy processes rather than just compound Poisson processes; (ii) the vacation lengths depend on the length of the server's preceding busy period. Regarding the former point: the Levy process active during the busy period is assumed to have no negative jumps, whereas the Levy process active during the vacation is a subordinator. Regarding the latter point: where in a previous study (Boxma et al. in Probab. Eng. Inf. Sci. 22:537---555, 2008) the durations of the vacations were positively correlated with the length of the preceding busy period, we now introduce a dependence structure that may give rise to both positive and negative correlations. We analyze the steady-state workload of the resulting queueing (or: storage) system, by first considering the queue at embedded epochs (viz. the beginnings of busy periods). We show that this embedded process does not always have a proper stationary distribution, due to the fact that there may occur an infinite number of busy-vacation cycles in a finite time interval; we specify conditions under which the embedded process is recurrent. Fortunately, irrespective of whether the embedded process has a stationary distribution, the steady-state workload of the continuous-time storage process can be determined. In addition, a number of ramifications are presented. The theory is illustrated by several examples.

On the definition, stationary distribution and second order structure of positive semidefinite Ornstein--Uhlenbeck type processes

Abstract: Several important properties of positive semidefinite processes of Ornstein--Uhlenbeck type are analysed. It is shown that linear operators of the form $X\mapsto AX+XA^{\mathrm{T}}$ with $A\in M_d(\mathbb{R})$ are the only ones that can be used in the definition provided one demands a natural non-degeneracy condition. Furthermore, we analyse the absolute continuity properties of the stationary distribution (especially when the driving matrix subordinator is the quadratic variation of a $d$-dimensional Levy process) and study the question of how to choose the driving matrix subordinator in order to obtain a given stationary distribution. Finally, we present results on the first and second order moment structure of matrix subordinators, which is closely related to the moment structure of positive semidefinite Ornstein--Uhlenbeck type processes. The latter results are important for method of moments based estimation.

Modeling Anomalous Diffusion by a Subordinated Integrated Brownian Motion

Abstract: We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times are correlated. In a continuum limit, the process can be defined by an integrated Brownian motion subordinated by an inverse -stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when , normal diffusion when , and superdiffusion when . The time-averaged mean square displacement is also employed to show weak ergodicity breaking occurring in the proposed process. An extension to the fractional case is also considered.

Berry-Esseen Bounds for Standardized Subordinators via Moduli of Smoothness

Abstract: We introduce moduli of smoothness techniques to deal with Berry? Esseen bounds, and illustrate them by considering standardized subordinators with finite variance. Instead of the classical Berry?Esseen smoothing inequality, we give an easy inequality involving the second modulus. Under finite third moment assumptions, such an inequality provides the main term of the approximation with small constants, even asymptotically sharp constants in the lattice case. Under infinite third moment assumptions, we show that the optimal rate of convergence can be simply written in terms of the first modulus of smoothness of an appropriate function, depending on the characteristic random variable of the subordinator. The preceding results are extended to standardized L?evy processes with finite variance.