scholarly journals Analytical properties of sample paths of some stochastic processes

2020 ◽  
pp. 11-15
Author(s):  
I. V. Rozora
Keyword(s):  
Covariance Function ◽  
Sufficient Conditions ◽  
Random Processes ◽  
Continuity Module ◽  
Classic Result ◽  
Sample Paths ◽  
Analytical Properties ◽  
Relevant Topic ◽  
Local Properties ◽  
Kolmogorov's Theorem

The study of the analytical properties of random processes and their functionals, without a doubt, was and remains the relevant topic of the theory of random processes. The first result from which the study of the local properties of random processes began is Kolmogorov’s theorem on sample continuity with probability one. The classic result for Gaussian random processes is Dudley’s theorem. This paper is devoted to the study of local properties of sample paths of random processes that can be represented as a sum of squares of Gaussian random processes. Such processes are called square-Gaussian. We investigate the sufficient conditions of sample continuity with probability 1 for square-Gaussian processes based on the convergence of entropy Dudley type integrals. The estimation of the distribution of the continuity module is studied for square-Gaussian random processes. It is considered in detail an example with an estimator (correlogram) of the covariance function of a Gaussian stationary random process. The conditions on continuity of correlogram’s trajectories with probability one are found and the distribution of the continuity module is also estimated.

scholarly journals On a Necessary Condition for the Sample Path Continuity of Weakly Stationary Processes

1970 ◽  
Vol 38 ◽  
pp. 103-111 ◽  
Author(s):  
Izumi Kubo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Spectral Distribution ◽  
Covariance Function ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Sample Path ◽  
Necessary Condition ◽  
Stationary Processes ◽  
Spectral Distribution Function

We shall discuss the sample path continuity of a stationary process assuming that the spectral distribution function F(λ) is given. Many kinds of sufficient conditions have been given in terms of the covariance function or the asymptotic behavior of the spectral distribution function.

First-passage-time densities and avoiding probabilities for birth-and-death processes with symmetric sample paths

1998 ◽  
Vol 35 (2) ◽  
pp. 383-394 ◽  
Author(s):  
Antonio Di Crescenzo
Keyword(s):  
Queueing System ◽  
Transition Probabilities ◽  
First Passage Time ◽  
Sufficient Conditions ◽  
Passage Time ◽  
First Passage ◽  
Birth And Death Processes ◽  
Sample Paths ◽  
Reflecting Boundaries ◽  
Necessary And Sufficient

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.

P-th moment growth bounds of infinite-dimensional stochastic evolution equations

1998 ◽  
Vol 128 (1) ◽  
pp. 107-121 ◽  
Author(s):  
Kai Liu
Keyword(s):  
Evolution Equations ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Infinite Dimensional Space ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Infinite Dimensional ◽  
Sample Paths ◽  
Logarithmic Growth

The aim of this paper is to investigate the p-th moment growth bounds wilh a general rate function λ(t) of the strong solution for a class of stochastic differential equations in infinite dimensional space under various sufficient hypotheses. The results derived here extend the usual situations to some extent, containing for example the polynomial or iterated logarithmic growth cases studied by many authors. In particular, more generalised sufficient conditions, ensuring the p-th moment upper-bound of sample paths given by solutions of a class of nonlinear stochastic evolution equations, are captured. Applications to parabolic itô equations are also considered.

Generalized semi-Markov decision processes

1979 ◽  
Vol 16 (03) ◽  
pp. 618-630
Author(s):  
Bharat T. Doshi
Keyword(s):  
Markov Decision Processes ◽  
Sufficient Conditions ◽  
Decision Processes ◽  
The State ◽  
Service Rate ◽  
Storage Model ◽  
Sample Paths ◽  
Markov Decision ◽  
Sufficient Conditions For Optimality ◽  
Necessary And Sufficient

Various authors have derived the necessary and sufficient conditions for optimality in semi-Markov decision processes in which the state remains constant between jumps. In this paper similar results are presented for a generalized semi-Markov decision process in which the state varies between jumps according to a Markov process with continuous sample paths. These results are specialized to a general storage model and an application to the service rate control in a GI/G/1 queue is indicated.

OPTIMAL ORDERING POLICIES WITH STOCHASTIC DEMAND AND PRICE PROCESSES

2012 ◽  
Vol 29 (06) ◽  
pp. 1250037 ◽  
Author(s):  
KIMITOSHI SATO ◽  
KATSUSHIGE SAWAKI
Keyword(s):  
Optimal Policy ◽  
Impulse Control ◽  
Sufficient Conditions ◽  
Spot Market ◽  
Simultaneous Equation ◽  
Spot Price ◽  
Impulse Control Problem ◽  
Analytical Properties ◽  
Discounted Costs ◽  
Optimal Ordering

In this paper, we consider an inventory model in which a firm uses the spot market for procurement in order to accomplish the minimization of total discounted costs. The model can be formulated as impulse control problem where the demand and spot price follow diffusion stochastic processes. We explore sufficient conditions under which an optimal policy exists. Furthermore, we derive an optimal policy as an (s, S) policy where s and S are uniquely determined as a solution of simultaneous equation. Finally, we show some analytical properties of the optimal policy. Some numerical examples are also presented.

Notes regarding the local properties of the sample functions of random processes

1985 ◽  
Vol 37 (6) ◽  
pp. 506-509
Author(s):  
Yu. G. Balasanov ◽  
I. G. Zhurbenko
Keyword(s):  
Random Processes ◽  
Local Properties

scholarly journals On Duffing equation with random perturbations

2010 ◽  
Vol 15 (2) ◽  
pp. 129-138 ◽  
Author(s):  
A. Ambrazevičius ◽  
F. Ivanauskas ◽  
H. Pragarauskas
Keyword(s):  
Sufficient Conditions ◽  
Duffing Equation ◽  
Random Perturbations ◽  
Sample Paths ◽  
Long Time ◽  
Initial States

We consider a family of particles with different initial states and/or velocities whose dynamics is described by a modified Duffing equation with random perturbations. Sufficient conditions ensuring almost identical sample paths of the particles after a long time are given.

Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

2014 ◽  
Vol 16 (1) ◽  
pp. 75-95 ◽  
Author(s):  
Debraj Ghosh ◽  
Anup Suryawanshi
Keyword(s):  
Covariance Matrix ◽  
Covariance Function ◽  
Heat Conduction Problem ◽  
Random Processes ◽  
Single Parameter ◽  
Eigenvalue Decomposition ◽  
Spatial Covariance ◽  
Practical Applications ◽  
Temporal Process ◽  
Spatio Temporal

AbstractA new representation of spatio-temporal random processes is proposed in this work. In practical applications, such processes are used to model velocity fields, temperature distributions, response of vibrating systems, to name a few. Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations, for instance, in a computational mechanics problem. For a single-parameter process such as spatial or temporal process, the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Loève (KL) decomposition. However, for multiparameter processes such as a spatio-temporal process, the covariance function itself can be defined in multiple ways. Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants. Then the spatial covariance matrix at different time instants are considered to define the covariance of the process. This set of square, symmetric, positive semi-definite matrices is then represented as a third-order tensor. A suitable decomposition of this tensor can identify the dominant components of the process, and these components are then used to define a closed-form representation of the process. The procedure is analogous to the KL decomposition for a single-parameter process, however, the decompositions and interpretations vary significantly. The tensor decompositions are successfully applied on (i) a heat conduction problem, (ii) a vibration problem, and (iii) a covariance function taken from the literature that was fitted to model a measured wind velocity data. It is observed that the proposed representation provides an efficient approximation to some processes. Furthermore, a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL, both in terms of computer memory and execution time.

scholarly journals Weak convergence of random processes with immigration at random times

2020 ◽  
Vol 57 (1) ◽  
pp. 250-265
Author(s):  
Congzao Dong ◽  
Alexander Iksanov
Keyword(s):  
Weak Convergence ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Processes ◽  
Branching Random Walk ◽  
General Point ◽  
Shot Noise Process ◽  
Finite Dimensional ◽  
Strong Resemblance ◽  
Random Times

AbstractBy a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

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