Probabilistic properties of non-Gaussian piecewise-linear processes on Poisson flows with independent random values at points of flow

Abstract

Abstract:The paper is focused on study of non-stationary piecewise-linear processes on Poisson point flows with independent identically distributed random variables at support points. An approach to calculate the correlation function of the process on the base of the total probability formula is considered. A general expression for the correlation function of a non-stationary process is obtained. Particular cases are considered. Using the method of direct simulation, it is shown numerically that the correlation function of the process has a point of inflection.Keywords: Random process, Poisson flow, correlation function. MSC 2010: 60G55, 65C05Some approaches to modelling the piecewise-linear non-Gaussian processes on point flows were considered in [1] relative to simulation of price series and the study of various trading algorithms based on these models. The corresponding models use intervals between points of flow and the distributions of process values at support points as input characteristics. Various approaches to construction of processes were considered, in particular, the alternation of distributions for ascending and descending sections of the polyline was used with the help of a special Markov chain.Special piecewise-linear processes were used in [9] for construction of models for climate prediction. The typical peculiarity of such processes is that the segments of a piecewise-linear function form a discontinuous function. These researches have shown a prospect of using simulation algorithms for piecewise-linear processes to solve practical problems.The issues related to the study of different types of piecewise-linear processes on point flows were considered in [3][4][5][6][7][8]. Those processes are modifications of piecewise-constant processes on point flows proposed and studied previously in [2]. One has to specify probabilistic characteristics of point flows and distribution of random variables at those points to apply numerical stochastic modelling of piecewise-linear processes which, in their turn, may be used for description of some real processes, for example, for modelling solar radiation scattering processes in stochastic cloudy media, modelling climate series, etc. The way of specification of random values in Poisson processes essentially determines the properties of the process. Thus, for example, specifying additive random variables [3] with successively increasing number of summands at flow points, we get a non-stationary process, and specifying them as independent identically distributed random variables [4,5], we get an asymptotically stationary one. In particular, one-point characteristics were considered for such types of processes and their asymptotic properties were studied. An approach to the study of correlation structure of piecewise-linear processes on Poisson flows was proposed in [5].In this paper we present the results of study of a process on Poisson flows whose values are IIDRV with finite variance at Poisson support points. Considering processes of such type, we obtained...