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Journal articles on the topic 'Stationary random process'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

Chung, Jaeyoung, Dohan Kim, and Eun Gu Lee. "Stationary hyperfunctional random process." Complex Variables and Elliptic Equations 59, no. 11 (March 14, 2013): 1547–58. http://dx.doi.org/10.1080/17476933.2012.757309.

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2

Lacaze, Pr B. "A note about stationary process random sampling." Statistics & Probability Letters 31, no. 2 (December 1996): 133–37. http://dx.doi.org/10.1016/s0167-7152(96)00024-7.

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3

Menh, Nguyen Cao, and Tran Duong Tri. "Simulation of stationary, non-normal random process." Vietnam Journal of Mechanics 14, no. 4 (December 31, 1992): 19–26. http://dx.doi.org/10.15625/0866-7136/10240.

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Simulation of stationary random processes specified by spectral density function and NonNormal probability density function has been rarely studied, though these processes are often met in both theory and practice. In this paper we consider a simulation method which can apply to the mention-above processes. The main idea based on the simulation of the Normal process and approximation of functions in the space of quadratic ally integrable functions. The numerical program for illustration of the method is written by Turbo - Pascal language.

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4

Kella, Offer, Onno Boxma, and Michel Mandjes. "A Lévy Process Reflected at a Poisson Age Process." Journal of Applied Probability 43, no. 01 (March 2006): 221–30. http://dx.doi.org/10.1017/s0021900200001480.

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We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

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5

Kella, Offer, Onno Boxma, and Michel Mandjes. "A Lévy Process Reflected at a Poisson Age Process." Journal of Applied Probability 43, no. 1 (March 2006): 221–30. http://dx.doi.org/10.1239/jap/1143936255.

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We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

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6

Subba Rao, Suhasini. "On some nonstationary, nonlinear random processes and their stationary approximations." Advances in Applied Probability 38, no. 4 (December 2006): 1155–72. http://dx.doi.org/10.1017/s000186780000149x.

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In this paper our object is to show that a certain class of nonstationary random processes can locally be approximated by stationary processes. The class of processes we are considering includes the time-varying autoregressive conditional heteroscedastic and generalised autoregressive conditional heteroscedastic processes, amongst others. The measure of deviation from stationarity can be expressed as a function of a derivative random process. This derivative process inherits many properties common to stationary processes. We also show that the derivative processes obtained here have alpha-mixing properties.

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7

Daley, D. J., T. Rolski, and R. Vesilo. "Long-Range Dependence in a Cox Process Directed by a Markov Renewal Process." Journal of Applied Mathematics and Decision Sciences 2007 (November 22, 2007): 1–15. http://dx.doi.org/10.1155/2007/83852.

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A Cox process NCox directed by a stationary random measure ξ has second moment var NCox(0,t]=E(ξ(0,t])+var ξ(0,t], where by stationarity E(ξ(0,t])=(const.)t=E(NCox(0,t]), so long-range dependence (LRD) properties of NCox coincide with LRD properties of the random measure ξ. When ξ(A)=∫AνJ(u)du is determined by a density that depends on rate parameters νi(i∈𝕏) and the current state J(⋅) of an 𝕏-valued stationary irreducible Markov renewal process (MRP) for some countable state space 𝕏 (so J(t) is a stationary semi-Markov process on 𝕏), the random measure is LRD if and only if each (and then by irreducibility, every) generic return time Yjj(j∈X) of the process for entries to state j has infinite second moment, for which a necessary and sufficient condition when 𝕏 is finite is that at least one generic holding time Xj in state j, with distribution function (DF)\ Hj, say, has infinite second moment (a simple example shows that this condition is not necessary when 𝕏 is countably infinite). Then, NCox has the same Hurst index as the MRP NMRP that counts the jumps of J(⋅), while as t→∞, for finite 𝕏, var NMRP(0,t]∼2λ2∫0t𝒢(u)du, var NCox(0,t]∼2∫0t∑i∈𝕏(νi−ν¯)2ϖiℋi(t)du, where ν¯=∑iϖiνi=E[ξ(0,1]], ϖj=Pr{J(t)=j},1/λ=∑jpˇjμj, μj=E(Xj), {pˇj} is the stationary distribution for the embedded jump process of the MRP, ℋj(t)=μi−1∫0∞min(u,t)[1−Hj(u)]du, and 𝒢(t)∼∫0tmin(u,t)[1−Gjj(u)]du/mjj∼∑iϖiℋi(t) where Gjj is the DF and mjj the mean of the generic return time Yjj of the MRP between successive entries to the state j. These two variances are of similar order for t→∞ only when each ℋi(t)/𝒢(t) converges to some [0,∞]-valued constant, say, γi, for t→∞.

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8

Pleshakov, Ruslan V. "Simulation of non-stationary event flow with a nested stationary component." Russian Family Doctor 28, no. 1 (December 15, 2020): 35–48. http://dx.doi.org/10.17816/rfd10640.

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A method for constructing an ensemble of time series trajectories with a nonstationary flow of events and a non-stationary empirical distribution of the values of the observed random variable is described. We consider a special model that is similar in properties to some real processes, such as changes in the price of a financial instrument on the exchange. It is assumed that a random process is represented as an attachment of two processes - stationary and non-stationary. That is, the length of a series of elements in the sequence of the most likely event (the most likely price change in the sequence of transactions) forms a non-stationary time series, and the length of a series of other events is a stationary random process. It is considered that the flow of events is non-stationary Poisson process. A software package that solves the problem of modeling an ensemble of trajectories of an observed random variable is described. Both the values of a random variable and the time of occurrence of the event are modeled. An example of practical application of the model is given.

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9

Pleshakov, Ruslan V. "Simulation of non-stationary event flow with a nested stationary component." Russian Family Doctor 28, no. 1 (December 15, 2020): 35–48. http://dx.doi.org/10.17816/rfd10645.

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A method for constructing an ensemble of time series trajectories with a nonstationary flow of events and a non-stationary empirical distribution of the values of the observed random variable is described. We consider a special model that is similar in properties to some real processes, such as changes in the price of a financial instrument on the exchange. It is assumed that a random process is represented as an attachment of two processes - stationary and non-stationary. That is, the length of a series of elements in the sequence of the most likely event (the most likely price change in the sequence of transactions) forms a non-stationary time series, and the length of a series of other events is a stationary random process. It is considered that the flow of events is non-stationary Poisson process. A software package that solves the problem of modeling an ensemble of trajectories of an observed random variable is described. Both the values of a random variable and the time of occurrence of the event are modeled. An example of practical application of the model is given.

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10

Pleshakov, Ruslan V. "Simulation of non-stationary event flow with a nested stationary component." Discrete and Continuous Models and Applied Computational Science 28, no. 1 (December 15, 2020): 35–48. http://dx.doi.org/10.22363/2658-4670-2020-28-1-35-48.

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A method for constructing an ensemble of time series trajectories with a nonstationary flow of events and a non-stationary empirical distribution of the values of the observed random variable is described. We consider a special model that is similar in properties to some real processes, such as changes in the price of a financial instrument on the exchange. It is assumed that a random process is represented as an attachment of two processes - stationary and non-stationary. That is, the length of a series of elements in the sequence of the most likely event (the most likely price change in the sequence of transactions) forms a non-stationary time series, and the length of a series of other events is a stationary random process. It is considered that the flow of events is non-stationary Poisson process. A software package that solves the problem of modeling an ensemble of trajectories of an observed random variable is described. Both the values of a random variable and the time of occurrence of the event are modeled. An example of practical application of the model is given.

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11

Di Crescenzo, Antonio, and Barbara Martinucci. "A Damped Telegraph Random Process with Logistic Stationary Distribution." Journal of Applied Probability 47, no. 01 (March 2010): 84–96. http://dx.doi.org/10.1017/s0021900200006410.

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We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

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12

Di Crescenzo, Antonio, and Barbara Martinucci. "A Damped Telegraph Random Process with Logistic Stationary Distribution." Journal of Applied Probability 47, no. 1 (March 2010): 84–96. http://dx.doi.org/10.1239/jap/1269610818.

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We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

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13

Franke, Brice, and Tatsuhiko Saigo. "The extremes of random walks in random sceneries." Advances in Applied Probability 41, no. 02 (June 2009): 452–68. http://dx.doi.org/10.1017/s0001867800003372.

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In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

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14

Franke, Brice, and Tatsuhiko Saigo. "The extremes of random walks in random sceneries." Advances in Applied Probability 41, no. 2 (June 2009): 452–68. http://dx.doi.org/10.1239/aap/1246886619.

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In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

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15

Nicolau, João. "STATIONARY PROCESSES THAT LOOK LIKE RANDOM WALKS— THE BOUNDED RANDOM WALK PROCESS IN DISCRETE AND CONTINUOUS TIME." Econometric Theory 18, no. 1 (February 2002): 99–118. http://dx.doi.org/10.1017/s0266466602181060.

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Several economic and financial time series are bounded by an upper and lower finite limit (e.g., interest rates). It is not possible to say that these time series are random walks because random walks are limitless with probability one (as time goes to infinity). Yet, some of these time series behave just like random walks. In this paper we propose a new approach that takes into account these ideas. We propose a discrete-time and a continuous-time process (diffusion process) that generate bounded random walks. These paths are almost indistinguishable from random walks, although they are stochastically bounded by an upper and lower finite limit. We derive for both cases the ergodic conditions, and for the diffusion process we present a closed expression for the stationary distribution. This approach suggests that many time series with random walk behavior can in fact be stationarity processes.

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16

Chakrabarty, Arijit, Rajat Subhra Hazra, and Deepayan Sarkar. "From random matrices to long range dependence." Random Matrices: Theory and Applications 05, no. 02 (April 2016): 1650008. http://dx.doi.org/10.1142/s2010326316500088.

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Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this work. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process. This is similar to the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues after a different scaling. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.

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17

Miyazawa, Masakiyo, Rolf Schassberger, and Volker Schmidt. "On the structure of an insensitive generalized semi-Markov process with reallocation and point-process input." Advances in Applied Probability 27, no. 01 (March 1995): 203–25. http://dx.doi.org/10.1017/s0001867800046310.

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A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).

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18

Miyazawa, Masakiyo, Rolf Schassberger, and Volker Schmidt. "On the structure of an insensitive generalized semi-Markov process with reallocation and point-process input." Advances in Applied Probability 27, no. 1 (March 1995): 203–25. http://dx.doi.org/10.2307/1428104.

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A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).

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19

Ethier, S. N., and R. C. Griffiths. "The neutral two-locus model as a measure-valued diffusion." Advances in Applied Probability 22, no. 04 (December 1990): 773–86. http://dx.doi.org/10.1017/s0001867800023120.

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The neutral two-locus model in population genetics is reformulated as a measure-valued diffusion process and is shown under certain conditions to have a unique stationary distribution and be weakly ergodic. The limits of the process and its stationary distribution as the recombination parameter tends to infinity are found. Genealogies are incorporated into the model, and it is shown that a random sample of size n from the population at stationarity has a common ancestor.

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20

Ethier, S. N., and R. C. Griffiths. "The neutral two-locus model as a measure-valued diffusion." Advances in Applied Probability 22, no. 4 (December 1990): 773–86. http://dx.doi.org/10.2307/1427561.

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The neutral two-locus model in population genetics is reformulated as a measure-valued diffusion process and is shown under certain conditions to have a unique stationary distribution and be weakly ergodic. The limits of the process and its stationary distribution as the recombination parameter tends to infinity are found. Genealogies are incorporated into the model, and it is shown that a random sample of size n from the population at stationarity has a common ancestor.

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21

Bellizzi, Sergio, and Rubens Sampaio. "Analysis of Stationary Random Vibrating Systems Using Smooth Decomposition." Shock and Vibration 20, no. 3 (2013): 493–502. http://dx.doi.org/10.1155/2013/205162.

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A modified Karhunen-Loève Decomposition/Proper Orthogonal Decomposition method, named Smooth Decomposition (SD) (also named smooth Karhunen-Loève decomposition), was recently introduced to analyze stationary random signal. It is based on a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time-derivative random process. The SD appears to be an interesting tool in terms of modal analysis. In this paper, the SD will be described in case of stationary random processes and extended also to stationary random fields. The main properties will be discussed and illustrated on a randomly excited clamped-free beam.

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22

Gouet, Raúl, F. Javier López, and Gerardo Sanz. "Records from stationary observations subject to a random trend." Advances in Applied Probability 47, no. 4 (December 2015): 1175–89. http://dx.doi.org/10.1239/aap/1449859805.

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We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Yn = Xn + Tn, n ≥ 1, where (Xn)n ∈ Z is a stationary ergodic sequence of random variables and (Tn)n ≥ 1 is a stochastic trend process with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff's ergodic theorem. For the asymptotic normality we rely on the approach of Ballerini and Resnick (1987), coupled with a moment bound for stationary sequences, which is used to deal with the random trend process. Examples of applications are provided. In particular, we obtain strong convergence and asymptotic normality for the number of ladder epochs in a random walk with stationary ergodic increments.

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23

Karr, Alan F. "Inference for stationary random fields given Poisson samples." Advances in Applied Probability 18, no. 02 (June 1986): 406–22. http://dx.doi.org/10.1017/s0001867800015822.

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Given a d-dimensional random field and a Poisson process independent of it, suppose that it is possible to observe only the location of each point of the Poisson process and the value of the random field at that (randomly located) point. Non-parametric estimators of the mean and covariance function of the random field—based on observation over compact sets of single realizations of the Poisson samples—are constructed. Under fairly mild conditions these estimators are consistent (in various senses) as the set of observation becomes unbounded in a suitable manner. The state estimation problem of minimum mean-squared error reconstruction of unobserved values of the random field is also examined.

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24

Karr, Alan F. "Inference for stationary random fields given Poisson samples." Advances in Applied Probability 18, no. 2 (June 1986): 406–22. http://dx.doi.org/10.2307/1427306.

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Given a d-dimensional random field and a Poisson process independent of it, suppose that it is possible to observe only the location of each point of the Poisson process and the value of the random field at that (randomly located) point. Non-parametric estimators of the mean and covariance function of the random field—based on observation over compact sets of single realizations of the Poisson samples—are constructed. Under fairly mild conditions these estimators are consistent (in various senses) as the set of observation becomes unbounded in a suitable manner. The state estimation problem of minimum mean-squared error reconstruction of unobserved values of the random field is also examined.

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25

Sun, Zhi Li, Yun Feng Zhang, and Yu Tao Yan. "Experimental Research on Wear Random Process." Advanced Materials Research 126-128 (August 2010): 976–80. http://dx.doi.org/10.4028/www.scientific.net/amr.126-128.976.

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The wear volume is obtained by means of experiment and the wear random process model is established according to the result. The Carbon steel material is used and the samples are grouped five after surface treatment, each group tests six times under the same condition. The wear volume under each wear time shows big dispersion. The additional study indicates that the sample has the large wear volume is in the serious wear state from the beginning, and the wear of running-in phase is inflected by the work velocity and the state condition of the surface of the samples. The wear process which the mean value is a constant and the standard deviation is different is a normal process generally, it is a stationary normal process if the standard deviation has no relation to the start of the wear time, or a Wiener process if the standard deviation is liner with the wear time, it is valuable to forecast the wear reliability.

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Gouet, Raúl, F. Javier López, and Gerardo Sanz. "Records from stationary observations subject to a random trend." Advances in Applied Probability 47, no. 04 (December 2015): 1175–89. http://dx.doi.org/10.1017/s0001867800049065.

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We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Y n = X n + T n , n ≥ 1, where (X n ) n ∈ Z is a stationary ergodic sequence of random variables and (T n ) n ≥ 1 is a stochastic trend process with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff's ergodic theorem. For the asymptotic normality we rely on the approach of Ballerini and Resnick (1987), coupled with a moment bound for stationary sequences, which is used to deal with the random trend process. Examples of applications are provided. In particular, we obtain strong convergence and asymptotic normality for the number of ladder epochs in a random walk with stationary ergodic increments.

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27

Park, Jun-Bum, Kyung-Su Kim, Joon-Mo Choung, Jae-Woo Kim, Chang-Hyuk Yoo, and Yeong-Su Ha. "Data Acquisition of Time Series from Stationary Ergodic Random Process Spectrums." Journal of Ocean Engineering and Technology 25, no. 2 (April 30, 2011): 120–26. http://dx.doi.org/10.5574/ksoe.2011.25.2.120.

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28

Alain, Boudou, and Viguier-Pla Sylvie. "Structure of the random measure associated with an isotropic stationary process." Journal of Multivariate Analysis 123 (January 2014): 111–28. http://dx.doi.org/10.1016/j.jmva.2013.08.001.

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29

BARONE, PIERO. "ON THE UNIVERSALITY OF THE DISTRIBUTION OF THE GENERALIZED EIGENVALUES OF A PENCIL OF HANKEL RANDOM MATRICES." Random Matrices: Theory and Applications 02, no. 01 (January 2013): 1250014. http://dx.doi.org/10.1142/s2010326312500141.

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Universality properties of the distribution of the generalized eigenvalues of a pencil of random Hankel matrices, arising in the solution of the exponential interpolation problem of a complex discrete stationary process, are proved under the assumption that every finite set of random variables of the process have a multivariate spherical distribution. An integral representation of the condensed density of the generalized eigenvalues is also derived. The asymptotic behavior of this function turns out to depend only on stationarity and not on the specific distribution of the process.

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30

Zhu, W. Q., and M. X. Jiang. "Nonlinear Fatigue Damage Accumulation Under Random Loading." Journal of Pressure Vessel Technology 118, no. 2 (May 1, 1996): 168–73. http://dx.doi.org/10.1115/1.2842176.

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The analytical expressions for the probability densities of the cumulative fatigue damage and fatigue life and for the reliability function are obtained for a mechanical or structural component subject to stationary random stress process on the basis of a stochastic theory of fatigue damage accumulation proposed by the first author and his co-worker and the Morrow’s nonlinear damage rule. The comparison between the results from Morrow’s and Palmgren-Miner’s damage rules for the case when the stress is a narrow-band stationary Gaussian process with zero mean is made and some important conclusions are drawn.

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31

Kobayashi, Masahiro, Masakiyo Miyazawa, and Hiroshi Shimizu. "STRUCTURE-REVERSIBILITY OF A TWO-DIMENSIONAL REFLECTING RANDOM WALK AND ITS APPLICATION TO QUEUEING NETWORK." Probability in the Engineering and Informational Sciences 29, no. 1 (September 29, 2014): 1–25. http://dx.doi.org/10.1017/s0269964814000199.

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We consider a two-dimensional reflecting random walk on the non-negative integer quadrant. It is assumed that this reflecting random walk has skip-free transitions. We are concerned with its time-reversed process assuming that the stationary distribution exists. In general, the time-reversed process may not be a reflecting random walk. In this paper, we derive necessary and sufficient conditions for the time-reversed process also to be a reflecting random walk. These conditions are different from but closely related to the product form of the stationary distribution.

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32

Roy, Emmanuel. "Bartlett spectrum and mixing properties of infinitely divisible random measures." Advances in Applied Probability 39, no. 04 (December 2007): 893–97. http://dx.doi.org/10.1017/s0001867800002159.

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We prove that the Bartlett spectrum of a stationary, infinitely divisible (ID) random measure determines ergodicity, weak mixing, and mixing. In this context, the Bartlett spectrum plays the same role as the spectral measure of a stationary Gaussian process.

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33

Roy, Emmanuel. "Bartlett spectrum and mixing properties of infinitely divisible random measures." Advances in Applied Probability 39, no. 4 (December 2007): 893–97. http://dx.doi.org/10.1239/aap/1198177231.

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We prove that the Bartlett spectrum of a stationary, infinitely divisible (ID) random measure determines ergodicity, weak mixing, and mixing. In this context, the Bartlett spectrum plays the same role as the spectral measure of a stationary Gaussian process.

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34

Neuts, Marcel F., and H. Sitaraman. "The square-wave spectral density of a stationary renewal process." Journal of Applied Mathematics and Simulation 2, no. 2 (January 1, 1989): 117–30. http://dx.doi.org/10.1155/s1048953389000109.

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35

Baccelli, François. "Stochastic ordering of random processes with an imbedded point process." Journal of Applied Probability 28, no. 3 (September 1991): 553–67. http://dx.doi.org/10.2307/3214491.

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We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

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36

Baccelli, François. "Stochastic ordering of random processes with an imbedded point process." Journal of Applied Probability 28, no. 03 (September 1991): 553–67. http://dx.doi.org/10.1017/s0021900200042418.

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We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

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37

Jagers, Peter, and Zhunwei Lu. "Branching processes with deteriorating random environments." Journal of Applied Probability 39, no. 02 (June 2002): 395–401. http://dx.doi.org/10.1017/s0021900200022609.

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We introduce Galton-Watson-style branching processes in random environments which are deteriorating rather than stationary or independent. Some primary results on process growth and extinction probability are shown, and two simple examples are given.

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38

Jagers, Peter, and Zhunwei Lu. "Branching processes with deteriorating random environments." Journal of Applied Probability 39, no. 2 (June 2002): 395–401. http://dx.doi.org/10.1239/jap/1025131435.

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We introduce Galton-Watson-style branching processes in random environments which are deteriorating rather than stationary or independent. Some primary results on process growth and extinction probability are shown, and two simple examples are given.

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39

Chen, Hai Long, Jing Yang, Chun Li Liu, and Jun Ting Wang. "Parameters Estimations for Autoregressive Process." Applied Mechanics and Materials 543-547 (March 2014): 1711–16. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.1711.

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Autoregressive process is better than other stochastic processes, which models flexibility, and simulates some other stochastic processes by setting the parameters of model. This paper firstly describes in detail the basic definition of autoregressive process and its stationary conditions, and then studies estimation autoregressive process parameters with residuals briefly in order to research Hubors М-estimation for autoregressive process with symmetric stable residuals. Secondly, prove autoregressive processes with residuals, which are-stable random variables, are also stationary processes, and use Markov's inequality and related theories to discuss Hubors М-estimation for autoregressive process with symmetric stable residuals. Finally, prove the consistency and asymptotic normality of this estimate further.

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40

Serfozo, Richard F. "Heredity of stationary and reversible stochastic processes." Advances in Applied Probability 18, no. 02 (June 1986): 574–76. http://dx.doi.org/10.1017/s0001867800015895.

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When a stochastic process (a random measure, set, field, etc. on a group) is stationary, ergodic, or reversible, then certain functions of this process inherit these properties. We present sufficient conditions for this inheritance.

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41

Serfozo, Richard F. "Heredity of stationary and reversible stochastic processes." Advances in Applied Probability 18, no. 2 (June 1986): 574–76. http://dx.doi.org/10.2307/1427313.

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When a stochastic process (a random measure, set, field, etc. on a group) is stationary, ergodic, or reversible, then certain functions of this process inherit these properties. We present sufficient conditions for this inheritance.

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42

Yundari, Yundari, and Setyo Wira Rizki. "Invertibility of Generalized Space-Time Autoregressive Model with Random Weight." CAUCHY 6, no. 4 (May 30, 2021): 246–59. http://dx.doi.org/10.18860/ca.v6i4.11254.

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The generalized linear process accomplishes stationarity and invertibility properties. The invertibility property must be having a series of convergence conditions of the process parameter. The generalized Space-Time Autoregressive (GSTAR) model is one of the stationary linear models therefore it is necessary to reveal the invertibility through the convergence of the parameter series. This article studies the invertibility of model GSTAR(1;1) with kernel random weight. The result shows that the model GSTAR(1;1) under kernel random weight fulfills the invertibility property and obtains a finite order of Generalized Space-Time Moving Average (GSTMA) process. The other result obtained is the time order of the finite orde . On the Triangular kernel resulted in the relatively great value n, so that it does not apply to the kernel with a finite value n.

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43

Tanaka, Minoru. "Estimation of the correlogram for a stationary Gaussian process by random clipping." Kodai Mathematical Journal 9, no. 3 (1986): 385–400. http://dx.doi.org/10.2996/kmj/1138037266.

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Shevgunov, Timofey. "A comparative example of cyclostationary description of a non-stationary random process." Journal of Physics: Conference Series 1163 (February 2019): 012037. http://dx.doi.org/10.1088/1742-6596/1163/1/012037.

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Deijfen, Maria. "Stationary random graphs with prescribed iid degrees on a spatial Poisson process." Electronic Communications in Probability 14 (2009): 81–89. http://dx.doi.org/10.1214/ecp.v14-1448.

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Kulikova, T. M. "Linear-spline estimation of the correlation function of a stationary random process." Russian Physics Journal 39, no. 4 (April 1996): 308–14. http://dx.doi.org/10.1007/bf02068051.

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Bruno, G., A. Pankov, and T. Pankova. "Time averaging for random nonlinear abstract parabolic equations." Abstract and Applied Analysis 5, no. 1 (2000): 1–11. http://dx.doi.org/10.1155/s1085337500000075.

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Sharma, V., J. Virtamo, and P. Lassila. "PERFORMANCE ANALYSIS OF THE RANDOM EARLY DETECTION ALGORITHM." Probability in the Engineering and Informational Sciences 16, no. 3 (May 22, 2002): 367–88. http://dx.doi.org/10.1017/s0269964802163078.

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In this article we consider a finite queue with its arrivals controlled by the random early detection algorithm. This is one of the most prominent congestion avoidance schemes in the Internet routers. The aggregate arrival stream from the population of transmission control protocol sources is locally considered stationary renewal or Markov modulated Poisson process with general packet length distribution. We study the exact dynamics of this queue and provide the stability and the rates of convergence to the stationary distribution and obtain the packet loss probability and the waiting time distribution. Then we extend these results to a two traffic class case with each arrival stream renewal. However, computing the performance indices for this system becomes computationally prohibitive. Thus, in the latter half of the article, we approximate the dynamics of the average queue length process asymptotically via an ordinary differential equation. We estimate the error term via a diffusion approximation. We use these results to obtain approximate transient and stationary performance of the system. Finally, we provide some computational examples to show the accuracy of these approximations.

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Last, Günter. "Stationary partitions and Palm probabilities." Advances in Applied Probability 38, no. 3 (September 2006): 602–20. http://dx.doi.org/10.1239/aap/1158684994.

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A stationary partition based on a stationary point process N in ℝd is an ℝd-valued random field π={π(x): x∈ℝd} such that both π(y)∈N for each y∈ℝd and the random partition {{y∈ℝd: π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

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Lam, Hoang-Chuong. "A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z." Journal of Applied Probability 51, no. 04 (December 2014): 1051–64. http://dx.doi.org/10.1017/s0021900200011979.

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The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (P ω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

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