Boundary Crossing Problems for Compound Renewal Processes

2020 ◽  
Vol 61 (1) ◽  
pp. 21-46
Author(s):  
A. A. Borovkov
Keyword(s):  
Boundary Crossing ◽  
Renewal Processes ◽  
Compound Renewal Processes

scholarly journals FROM BOUNDARY CROSSING OF NON-RANDOM FUNCTIONS TO BOUNDARY CROSSING OF STOCHASTIC PROCESSES

2015 ◽  
Vol 29 (3) ◽  
pp. 345-359 ◽  
Author(s):  
Mark Brown ◽  
Victor de la Peña ◽  
Tony Sit
Keyword(s):  
Stochastic Processes ◽  
Sequential Analysis ◽  
Form Solution ◽  
Boundary Crossing ◽  
Renewal Processes ◽  
Bessel Processes ◽  
Average Behavior ◽  
Random Functions ◽  
Close Form ◽  
Sharp Lower Bound

One problem of wide interest involves estimating expected crossing-times. Several tools have been developed to solve this problem beginning with the works of Wald and the theory of sequential analysis. Deriving the explicit close form solution for the expected crossing times may be difficult. In this paper, we provide a framework that can be used to estimate expected crossing times of arbitrary stochastic processes. Our key assumption is the knowledge of the average behavior of the supremum of the process. Our results include a universal sharp lower bound on the expected crossing times. Furthermore, for a wide class of time-homogeneous, Markov processes, including Bessel processes, we are able to derive an upper bound E[a(Tr)]≤2r, which implies that sup r>0|((E[a(Tr)]−r)/r)|≤1, where a(t)=E[sup tXt] with {Xt}t≥0 be a non-negative, measurable process. This inequality motivates our claim that a(t) can be viewed as a natural clock for all such processes. The cases of multidimensional processes, non-symmetric and random boundaries are handled as well. We also present applications of these bounds on renewal processes in Example 10 and other stochastic processes.

Invariance principles in queueing theory

1989 ◽  
Vol 26 (04) ◽  
pp. 845-857
Author(s):  
Michael Alex ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Stochastic Processes ◽  
Queueing Theory ◽  
Counting Process ◽  
Heavy Traffic ◽  
Renewal Processes ◽  
Light Traffic ◽  
Invariance Principles ◽  
Renewal Counting Process ◽  
Compound Renewal Processes

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk }and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

scholarly journals On the convergence of quadratic variation for compound fractional Poisson processes

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Enrico Scalas ◽  
Noèlia Viles
Keyword(s):  
Stochastic Process ◽  
Random Variable ◽  
Quadratic Variation ◽  
Poisson Processes ◽  
Renewal Processes ◽  
The Relationship ◽  
Compound Renewal Processes

AbstractThe relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a stochastic process.

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