An Application of Mean Square Calculus to Sliding Wear

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Cláudio R. Ávila da Silva ◽  
Giuseppe Pintaude ◽  
Hazim Ali Al-Qureshi ◽  
Marcelo Alves Krajnc
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Sliding Wear ◽  
Random Variables ◽  
Random Variable ◽  
Expected Value ◽  
Mean Square ◽  
Cauchy Problems ◽  
Numerical Examples ◽  
Correlation Models

In this paper the Archard model and classical results of mean square calculus are used to derive two Cauchy problems in terms of the expected value and covariance of the worn height stochastic process. The uncertainty is present in the wear and roughness coefficients. In order to model the uncertainty, random variables or stochastic processes are used. In the latter case, the expected value and covariance of the worn height stochastic process are obtained for three combinations of correlation models for the wear and roughness coefficients. Numerical examples for both models are solved. For the model based on a random variable, a larger dispersion in terms of worn height stochastic process was observed.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

scholarly journals Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1417 ◽  
Author(s):  
Julia Calatayud ◽  
Juan Carlos Cortés ◽  
Marc Jornet ◽  
Francisco Rodríguez
Keyword(s):  
Stochastic Process ◽  
Differential Equations ◽  
Random Variables ◽  
Mean Square ◽  
Discrete Delay ◽  
Numerical Schemes ◽  
Recent Contribution ◽  
Order Of Accuracy ◽  
Euler’S Method ◽  
Random Differential Equations

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented.

Existence and positivity of the limit in processes with a branching structure

1999 ◽  
Vol 36 (1) ◽  
pp. 132-138
Author(s):  
M. P. Quine ◽  
W. Szczotka
Keyword(s):  
Stochastic Process ◽  
Branching Process ◽  
Random Variables ◽  
Random Variable ◽  
Natural Generalization ◽  
Partial Sums ◽  
Urn Model ◽  
Special Case ◽  
Independent And Identically Distributed

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

Imperfect repair

1983 ◽  
Vol 20 (04) ◽  
pp. 851-859 ◽  
Author(s):  
Mark Brown ◽  
Frank Proschan
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Random Variables ◽  
Life Distribution ◽  
Imperfect Repair ◽  
Monotonicity Properties ◽  
New States

A device is repaired at failure. With probability p, it is returned to the ‘good-as-new' state (perfect repair), with probability 1 – p, it is returned to the functioning state, but it is only as good as a device of age equal to its age at failure (imperfect repair). Repair takes negligible time. We obtain the distribution Fp of the interval between successive good-as-new states in terms of the underlying life distribution F. We show that if F is in any of the life distribution classes IFR, DFR, IFRA, DFRA, NBU, NWU, DMRL, or IMRL, then Fp is in the same class. Finally, we obtain a number of monotonicity properties for various parameters and random variables of the stochastic process. The results obtained are of interest in the context of stochastic processes in general, as well as being useful in the particular imperfect repair model studied.

scholarly journals Hermite-Hadamard, Jensen, and Fractional Integral Inequalities for Generalized P -Convex Stochastic Processes

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Fangfang Ma ◽  
Waqas Nazeer ◽  
Mamoona Ghafoor
Keyword(s):  
Probability Theory ◽  
Convex Function ◽  
Stochastic Processes ◽  
Fractional Integral ◽  
Probabilistic Models ◽  
Random Variable ◽  
Expected Value ◽  
Integral Inequalities ◽  
Stochastic Convexity ◽  
Convex Stochastic Processes

The stochastic process is one of the important branches of probability theory which deals with probabilistic models that evolve over time. It starts with probability postulates and includes a captivating arrangement of conclusions from those postulates. In probability theory, a convex function applied on the expected value of a random variable is always bounded above by the expected value of the convex function of that random variable. The purpose of this note is to introduce the class of generalized p -convex stochastic processes. Some well-known results of generalized p -convex functions such as Hermite-Hadamard, Jensen, and fractional integral inequalities are extended for generalized p -stochastic convexity.

scholarly journals Hermite-Hadamard type inequalities for p-convex stochastic processes

2019 ◽  
Vol 9 (2) ◽  
pp. 148-153 ◽  
Author(s):  
Nurgul Okur ◽  
Imdat Işcan ◽  
Emine Yuksek Dizdar
Keyword(s):  
Stochastic Process ◽  
Mathematical Programming ◽  
Stochastic Processes ◽  
Mean Square ◽  
Important Place ◽  
Mathematical Programming Problems ◽  
Square Integrability ◽  
Convex Stochastic Processes ◽  
Convex Stochastic Process

In this study are investigated p-convex stochastic processes which are extensions of convex stochastic processes. A suitable example is also given for this process. In addition, in this case a p-convex stochastic process is increasing or decreasing, the relation with convexity is revealed. The concept of inequality as convexity has an important place in literature, since it provides a broader setting to study the optimization and mathematical programming problems. Therefore, Hermite-Hadamard type inequalities for p-convex stochastic processes and some boundaries for these inequalities are obtained in present study. It is used the concept of mean-square integrability for stochastic processes to obtain the above mentioned results.

scholarly journals Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach

2018 ◽  
Vol 23 (1) ◽  
pp. 79-100 ◽  
Author(s):  
Maria-Consuelo Casaban ◽  
Juan-Carlos Cortes ◽  
Lucas Jodar
Keyword(s):  
Stochastic Process ◽  
Fourier Transform ◽  
Standard Deviation ◽  
Stochastic Processes ◽  
Numerical Solution ◽  
Integral Form ◽  
Initial Condition ◽  
Mean Square ◽  
Improper Integrals ◽  
Gauss Hermite Quadrature

This paper deals with the construction of mean square analytic-numerical solution of parabolic partial differential problems where both initial condition and coefficients are stochastic processes. By using a random Fourier transform, an inf- nite integral form of the solution stochastic process is firstly obtained. Afterwards, explicit expressions for the expectation and standard deviation of the solution are obtained. Since these expressions depend upon random improper integrals, which are not computable in an exact manner, random Gauss-Hermite quadrature formulae are introduced throughout an illustrative example.

scholarly journals Contextuality Scenarios Arising from Networks of Stochastic Processes

2016 ◽  
Vol 23 (03) ◽  
pp. 1650012
Author(s):  
Rodrigo Iglesias ◽  
Fernando Tohmé ◽  
Marcelo Auday
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Empirical Model ◽  
Joint Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Empirical Models ◽  
Classical Source ◽  
Input And Output ◽  
Long Run

An empirical model is a generalization of a probability space. It consists of a simplicial complex of subsets of a class 𝒳 of random variables such that each simplex has an associated probability distribution. The ensuing marginalizations are coherent, in the sense that the distribution on a face of a simplex coincides with the marginal of the distribution over the entire simplex. An empirical model is called contextual if its distributions cannot be obtained by marginalizing a joint distribution over 𝒳. Contextual empirical models arise naturally in quantum theory, giving rise to some of its counter -intuitive statistical consequences. In this paper, we present a different and classical source of contextual empirical models: the interaction among many stochastic processes. We attach an empirical model to the ensuing network in which each node represents an open stochastic process with input and output random variables. The statistical behaviour of the network in the long run makes the empirical model generically contextual and even strongly contextual.

MAKING SIMULATIONS MORE EFFICIENT WHEN ANALYZING POISSON ARRIVAL SYSTEMS AND MEANS OF MONOTONE FUNCTIONS

2006 ◽  
Vol 20 (2) ◽  
pp. 251-256
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Random Variable ◽  
Expected Value ◽  
Stratified Sampling ◽  
Arrival Process ◽  
Monotone Functions ◽  
New Approach ◽  
Poisson Arrival ◽  
Increasing Function

For a system in which arrivals occur according to a Poisson process, we give a new approach for using simulation to estimate the expected value of a random variable that is independent of the arrival process after some specified time t. We also give a new approach for using simulation to estimate the expected value of an increasing function of independent uniform random variables. Stratified sampling is a key technique in both cases.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (2) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

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