On a class of reflected AR(1) processes

2016 ◽  
Vol 53 (3) ◽  
pp. 818-832
Author(s):  
Onno Boxma ◽  
Michel Mandjes ◽  
Josh Reed
Keyword(s):  
Heavy Traffic ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Steady State Distribution ◽  
State Distribution ◽  
Uhlenbeck Process ◽  
Stationary Analysis ◽  
Ornstein Uhlenbeck Process ◽  
Phase Type

AbstractIn this paper we study a reflected AR(1) process, i.e. a process (Zn)n obeying the recursion Zn+1= max{aZn+Xn,0}, with (Xn)n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Zn (in terms of transforms) in case Xn can be written as Yn−Bn, with (Bn)n being a sequence of independent random variables which are all Exp(λ) distributed, and (Yn)n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (Bn)n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

Asymptotic tail behaviour of phase-type scale mixture distributions

2017 ◽  
Vol 12 (2) ◽  
pp. 412-432 ◽  
Author(s):  
Leonardo Rojas-Nandayapa ◽  
Wangyue Xie
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Mixture Distributions ◽  
Scale Mixture ◽  
Domains Of Attraction ◽  
Heavy Tailed Distributions ◽  
Tail Behaviour ◽  
Phase Type ◽  
Heavy Tailed

AbstractWe consider phase-type scale mixture distributions which correspond to distributions of a product of two independent random variables: a phase-type random variable Y and a non-negative but otherwise arbitrary random variable S called the scaling random variable. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction. Particular focus is given to phase-type scale mixture distributions where the scaling random variable S has discrete support – such a class of distributions has been recently used in risk applications to approximate heavy-tailed distributions. Our results are complemented with several examples.

scholarly journals A generalized class of correlated run shock models

2018 ◽  
Vol 6 (1) ◽  
pp. 131-138 ◽  
Author(s):  
Femin Yalcin ◽  
Serkan Eryilmaz ◽  
Ali Riza Bozbulut
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Random Variable ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Shock Models ◽  
Phase Type ◽  
Over Time

AbstractIn this paper, a generalized class of run shock models associated with a bivariate sequence {(Xi, Yi)}i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X1, X2, ... over time, let the random variables Y1, Y2, ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑Nt=1 Yt , where N is a stopping time for the sequence {Xi}i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {Xi, 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.

Convergence rates in the law of large numbers. II

1968 ◽  
Vol 64 (2) ◽  
pp. 485-488 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Convergence Rates ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Large Numbers ◽  
Present Context ◽  
Image Position ◽  
Uniformly Bounded

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

scholarly journals ASYMPTOTIC CRAMÉR TYPE DECOMPOSITION FOR WIENER AND WIGNER INTEGRALS

2013 ◽  
Vol 16 (01) ◽  
pp. 1350005
Author(s):  
SOLESNE BOURGUIN ◽  
JEAN-CHRISTOPHE BRETON
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Independent Random Variables ◽  
Asymptotic Decomposition ◽  
Type Decomposition

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (01) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

scholarly journals On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

2009 ◽  
Vol 46 (3) ◽  
pp. 721-731 ◽  
Author(s):  
Shibin Zhang ◽  
Xinsheng Zhang
Keyword(s):  
Stable Distribution ◽  
Stochastic Integral ◽  
Random Variables ◽  
Transition Density ◽  
Random Variable ◽  
Independent Random Variables ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Tempered Stable Distribution ◽  
Transition Distribution

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C∞-function.

Maximal branching processes and ‘long-range percolation’

1970 ◽  
Vol 7 (01) ◽  
pp. 89-98
Author(s):  
John Lamperti
Keyword(s):  
Probability Distribution Function ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Transition Matrices ◽  
A Chain ◽  
Image Position ◽  
The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

scholarly journals On some tractable growth-collapse processes with renewal collapse epochs

2011 ◽  
Vol 48 (A) ◽  
pp. 217-234 ◽  
Author(s):  
Onno Boxma ◽  
Offer Kella ◽  
David Perry
Keyword(s):  
Steady State ◽  
Stationary Distribution ◽  
Unit Interval ◽  
General Distribution ◽  
Phase Type Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Positive Real ◽  
Ratio Distribution ◽  
Phase Type

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.

scholarly journals Probabalistic modelling of the results of economic activity as a function of random values

2020 ◽  
pp. 102-112
Author(s):  
Olesya Martyniuk ◽  
Stepan Popina ◽  
Serhii Martyniuk
Keyword(s):  
Mathematical Modeling ◽  
Distribution Function ◽  
Probability Distribution ◽  
Economic Activity ◽  
Probability Distribution Function ◽  
Random Variables ◽  
Economic Structure ◽  
Random Variable ◽  
Independent Random Variables ◽  
Research Results

Introduction. Mathematical modeling of economic processes is necessary for the unambiguous formulation and solution of the problem. In the economic sphere this is the most important aspect of the activity of any enterprise, for which economic-mathematical modeling is the tool that allows to make adequate decisions. However, economic indicators that are factors of a model are usually random variables. An economic-mathematical model is proposed for calculating the probability distribution function of the result of economic activity on the basis of the known dependence of this result on factors influencing it and density of probability distribution of these factors. Methods. The formula was used to calculate the random variable probability distribution function, which is a function of other independent random variables. The method of estimation of basic numerical characteristics of the investigated functions of random variables is proposed: mathematical expectation that in the probabilistic sense is the average value of the result of functioning of the economic structure, as well as its variance. The upper bound of the variation of the effective feature is indicated. Results. The cases of linear and power functions of two independent variables are investigated. Different cases of two-dimensional domain of possible values of indicators, which are continuous random variables, are considered. The application of research results to production functions is considered. Examples of estimating the probability distribution function of a random variable are offered. Conclusions. The research results allow in the probabilistic sense to estimate the result of the economic structure activity on the basis of the probabilistic distributions of the values of the dependent variables. The prospect of further research is to apply indirect control over economic performance based on economic and mathematical modeling.

Stochastic comparisons of random minima and maxima

1997 ◽  
Vol 34 (02) ◽  
pp. 420-425 ◽  
Author(s):  
Moshe Shaked ◽  
Tityik Wong
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Stochastic Comparison ◽  
Stochastic Comparisons ◽  
Comparison Results

Let X 1, X 2,… be a sequence of independent random variables and let N be a positive integer-valued random variable which is independent of the Xi. In this paper we obtain some stochastic comparison results involving min {X 1, X 2,…, XN ) and max{X 1, X 2,…, XN }.

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