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.

scholarly journals Trait anxiety and unplanned delivery mode enhance the risk for childbirth-related post-traumatic stress disorder symptoms in women with and without risk of preterm birth: A multi sample path analysis

PLoS ONE ◽  
2021 ◽  
Vol 16 (8) ◽  
pp. e0256681
Author(s):  
Sarah Sommerlad ◽  
Karin Schermelleh-Engel ◽  
Valentina Lucia La Rosa ◽  
Frank Louwen ◽  
Silvia Oddo-Sommerfeld
Keyword(s):  
Preterm Birth ◽  
Pain Intensity ◽  
Trait Anxiety ◽  
Sample Path ◽  
Delivery Mode ◽  
Ptsd Symptoms ◽  
Post Traumatic Stress ◽  
Sample Path Analysis ◽  
Childbirth Experience ◽  
Post Traumatic

Childbirth-related post-traumatic stress disorder (CB-PTSD) occurs in 3–7% of all pregnancies and about 35% of women after preterm birth (PTB) meet the criteria for acute stress reaction. Known risk factors are trait anxiety and pain intensity, whereas planned delivery mode, medical support, and positive childbirth experience are protective factors. It has not yet been investigated whether the effects of anxiety and delivery mode are mediated by other factors, and whether a PTB-risk alters these relationships. 284 women were investigated antepartum and six weeks postpartum (risk-group with preterm birth (RG-PB) N = 95, risk-group with term birth (RG-TB) N = 99, and control group (CG) N = 90). CB-PTSD symptoms and anxiety were measured using standardized psychological questionnaires. Pain intensity, medical support, and childbirth experience were assessed by single items. Delivery modes were subdivided into planned vs. unplanned delivery modes. Group differences were examined using MANOVA. To examine direct and indirect effects on CB-PTSD symptoms, a multi-sample path analysis was performed. Rates of PTS were highest in the RG-PB = 11.58% (RG-TB = 7.01%, CG = 1.1%). MANOVA revealed higher values of CB-PTSD symptoms and pain intensity in RG-PB compared to RG-TB and CG. Women with planned delivery mode reported a more positive birth experience. Path modeling revealed a good model fit. Explained variance was highest in RG-PB (R2 = 44.7%). Direct enhancing effects of trait anxiety and indirect reducing effects of planned delivery mode on CB-PTSD symptoms were observed in all groups. In both risk groups, CB-PTSD symptoms were indirectly reduced via support by medical staff and positive childbirth experience, while trait anxiety indirectly enhanced CB-PTSD symptoms via pain intensity in the CG. Especially in the RG-PB, a positive birth experience serves as protective factor against CB-PTSD symptoms. Therefore, our data highlights the importance of involving patients in the decision process even under stressful birth conditions and the need for psychological support antepartum, mainly in patients with PTB-risk and anxious traits. Trial registration number: NCT01974531 (ClinicalTrials.gov identifier).

scholarly journals Continuous-State Branching Processes and Self-Similarity

2008 ◽  
Vol 45 (04) ◽  
pp. 1140-1160 ◽  
Author(s):  
A. E. Kyprianou ◽  
J. C. Pardo
Keyword(s):  
Markov Processes ◽  
Branching Processes ◽  
Self Similarity ◽  
Lamperti Transformation ◽  
Continuous State ◽  
Self Similar

In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

A sample path analysis of the M/M/1 queue

1989 ◽  
Vol 26 (02) ◽  
pp. 418-422 ◽  
Author(s):  
Francois Baccelli ◽  
William A. Massey
Keyword(s):  
Generating Functions ◽  
Queueing Networks ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Sample Path ◽  
Laplace Transforms ◽  
Sample Path Analysis ◽  
Transient Distribution ◽  
The Past ◽  
Novel Approach

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.

On two classes of interacting stochastic processes arising in cancer modeling

1983 ◽  
Vol 15 (4) ◽  
pp. 695-712 ◽  
Author(s):  
Robert Bartoszyński ◽  
Prem S. Puri
Keyword(s):  
Markov Processes ◽  
Branching Processes ◽  
The Other ◽  
Time Interval ◽  
Positive Probability ◽  
Martingale Convergence Theorem ◽  
Limiting Behavior ◽  
State Dependent ◽  
Cancer Modeling ◽  
Martingale Convergence

The processes X and Y are said to interact if the laws governing the changes of either of them at time t depend on the values of the other process at times up to t. For bivariate interacting Markov processes, their limiting behavior is analysed by means of an approximation suggested by Fuhrmann, consisting of discretizing time, and assuming that in each time interval the processes develop independently, according to the laws obtained by fixing the value of the other process at its level attained at the beginning of the interval.In this way the conditions for almost sure extinction, escape to ∞ with positive probability, etc., are obtained (by using the martingale convergence theorem) for state-dependent branching processes studied by Roi, and for bivariate processes with one component piecewise determined.

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

The age distribution of Markov processes

1977 ◽  
Vol 14 (3) ◽  
pp. 492-506 ◽  
Author(s):  
Benny Levikson
Keyword(s):  
Differential Equations ◽  
Markov Processes ◽  
Difference Equations ◽  
Diffusion Processes ◽  
Genetic Model ◽  
Branching Processes ◽  
Age Distribution ◽  
Renewal Theory ◽  
Gene Frequencies ◽  
Birth And Death Processes

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

A sample path analysis of M/GI/1 queues with workload restrictions

1993 ◽  
Vol 14 (1-2) ◽  
pp. 203-213 ◽  
Author(s):  
Jian-Qiang Hu ◽  
Michael A. Zazanis
Keyword(s):  
Path Analysis ◽  
Sample Path ◽  
Sample Path Analysis

A sample path analysis of the delay in the M/G/C system

1996 ◽  
Vol 33 (1) ◽  
pp. 256-266 ◽  
Author(s):  
Sridhar Seshadri
Keyword(s):  
Path Analysis ◽  
Service Time ◽  
Sample Path ◽  
Service Discipline ◽  
Service Time Distribution ◽  
Sample Path Analysis ◽  
New Device ◽  
Mean Delay ◽  
The Mean ◽  
General Service

Using sample path analysis we show that under the same load the mean delay in queue in the M/G/2 system is smaller than that in the corresponding M/G/1 system, when the service time has either the DMRL or NBU property and the service discipline is FCFS. The proof technique uses a new device that equalizes the work in a two server system with that in a single sterver system. Other interesting quantities such as the average difference in work between the two servers in the GI/G/2 system and an exact alternate derivation of the mean delay in the M/M/2 system from sample path analysis are presented. For the same load, we also show that the mean delay in the M/G/C system with general service time distribution is smaller than that in the M/G/1 system when the traffic intensity is less than 1/c.

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