scholarly journals A Sample Path Proof of the Duality for Stochastically Monotone Markov Processes

1985 ◽  
Vol 13 (2) ◽  
pp. 558-565 ◽  
Author(s):  
Peter Clifford ◽  
Aidan Sudbury
Keyword(s):  
Markov Processes ◽  
Sample Path

Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

2013 ◽  
Vol 50 (04) ◽  
pp. 931-942
Author(s):  
Takayuki Fujii
Keyword(s):  
Local Time ◽  
Markov Processes ◽  
Nonparametric Estimation ◽  
Sample Path ◽  
Point Of View ◽  
Statistical Point ◽  
Jump Size ◽  
Stationary Density ◽  
Piecewise Deterministic Markov Processes ◽  
Estimation Problems

In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

Event coupling and performance sensitivity analysis of generalized semi-Markov processes

1995 ◽  
Vol 27 (03) ◽  
pp. 741-769
Author(s):  
Xi-Ren Cao
Keyword(s):  
Steady State ◽  
Markov Processes ◽  
Queueing Networks ◽  
Sample Path ◽  
Performance Measure ◽  
Coupling Property ◽  
Performance Sensitivity ◽  
Realization Factors ◽  
Semi Markov Processes ◽  
Steady State Performance

We study a fundamental feature of the generalized semi-Markov processes (GSMPs), called event coupling. The event coupling reflects the logical behavior of a GSMP that specifies which events can be affected by any given event. Based on the event-coupling property, GSMPs can be classified into three classes: the strongly coupled, the hierarchically coupled, and the decomposable GSMPs. The event-coupling property on a sample path of a GSMP can be represented by the event-coupling trees. With the event-coupling tree, we can quantify the effect of a single perturbation on a performance measure by using realization factors. A set of equations that specifies the realization factors is derived. We show that the sensitivity of steady-state performance with respect to a parameter of an event lifetime distribution can be obtained by a simple formula based on realization factors and that the sample-path performance sensitivity converges to the sensitivity of the steady-state performance with probability one as the length of the sample path goes to infinity. This generalizes the existing results of perturbation analysis of queueing networks to GSMPs.

scholarly journals Transient Probability Functions: A Sample Path Approach

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michael L. Green ◽  
Alan Krinik ◽  
Carrie Mortensen ◽  
Gerardo Rubino ◽  
Randall J. Swift
Keyword(s):  
Markov Processes ◽  
Solution Method ◽  
Sample Path ◽  
Dual Processes ◽  
New Approach ◽  
Probability Functions ◽  
Counting Approach ◽  
International Audience ◽  
Path Counting ◽  
Transient Probability

International audience A new approach is used to determine the transient probability functions of Markov processes. This new solution method is a sample path counting approach and uses dual processes and randomization. The approach is illustrated by determining transient probability functions for a three-state Markov process. This approach also provides a way to calculate transient probability functions for Markov processes which have specific sample path characteristics.

Markov Processes and Random Sampling

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Posterior Distribution ◽  
Random Sampling ◽  
Sample Path ◽  
Regular Structure ◽  
Philosophical Approach ◽  
Parameter Spaces ◽  
Regular Structures ◽  
Sampling Algorithms

The parameter spaces of natural patterns are so complex that inference must often proceed compositionally, successively building up more and more complex structures, as well as back-tracking, creating simpler structures from more complex versions. Inference is transformational in nature. The philosophical approach studied in this chapter is that the posterior distribution that describes the patterns contains all of the information about the underlying regular structure. Therefore, the transformations of inference are guided via the posterior in the sense that the algorithm for changing the regular structures will correspond to the sample path of a Markov process. The Markov process is constructed to push towards the posterior distribution in which the information about the patterns are stored. This provides the deepconnection between the transformational paradigm of regular structure creation, and random sampling algorithms.

scholarly journals Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

2013 ◽  
Vol 50 (4) ◽  
pp. 931-942
Author(s):  
Takayuki Fujii
Keyword(s):  
Local Time ◽  
Markov Processes ◽  
Nonparametric Estimation ◽  
Sample Path ◽  
Point Of View ◽  
Statistical Point ◽  
Jump Size ◽  
Stationary Density ◽  
Piecewise Deterministic Markov Processes ◽  
Estimation Problems

In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

Monotone Stochastic Recursions and their Duals

1996 ◽  
Vol 10 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Søren Asmussen ◽  
Karl Sigman
Keyword(s):  
Markov Processes ◽  
Branching Processes ◽  
Sample Path ◽  
Risk Process ◽  
Dual Function ◽  
Sample Path Analysis ◽  
Reservoir Models ◽  
Stability Issues ◽  
Barrier Problems ◽  
Steady State Probabilities

A duality is presented for real-valued stochastic sequences [Vn] defined by a general recursion of the form Vn+1 = f(Vn, Un), with [Un] a stationary driving sequence and f nonnegative, continuous, and monotone in its first variable. The duality is obtained by defining a dual function g of f, which if used recursively on the time reversal of [Un] defines a dual risk process. As a consequence, we prove that steady-state probabilities for Vn can always be expressed as transient probabilities of the dual risk process. The construction is related to duality of stochastically monotone Markov processes as studied by Siegmund (1976, The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes, Annals of Probability 4: 914–924). Our method of proof involves an elementary sample-path analysis. A variety of examples are given, including random walks with stationary increments and two reflecting barriers, reservoir models, autoregressive processes, and branching processes. Finally, general stability issues of the content process are dealt with by expressing them in terms of the dual risk process.

Sign in / Sign up

Export Citation Format

Share Document