Some Long-Range Dependence Processes Arising from Fluctuations of Particle Systems
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Some Long-Range Dependence Processes Arising from Fluctuations of Particle Systems LUIS G. GOROSTIZA1 , REYLA NAVARRO2 and ELIANE R. RODRIGUES3 1 Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados, A.P. 14-740,
México 07000 D.F., México. e-mail: [email protected] 2 Departamento de Física y Matemáticas, Universidad de las Américas, Cholula, 72820 Puebla, México. e-mail: [email protected] 3 Instituto de Matemáticas, UNAM, Area de la Investigación Científica, Circuito Exterior, CU, México 04510 D.F., México. e-mail: [email protected] (Received: 12 May 2004; in final form: 18 November 2004; accepted: 29 November 2004) Abstract. Several long-range dependence, self-similar Gaussian processes arise from asymptotics of some classes of spatially distributed particle systems and superprocesses. The simplest examples are fractional Brownian motion and sub-fractional fractional Brownian motion, the latter being intermediate between Brownian motion and fractional Brownian motion. In this paper we focus mainly on long-range dependence processes that arise from occupation time fluctuations of immigration particle systems with or without branching, and we study their properties. Some long-range dependence non-Gaussian processes that appear in a similar way are also mentioned. Mathematics Subject Classifications (2000): Primary 60G15, 60G18; secondary 60F17, 60G20, 60J80. Key words: long-range dependence, long memory, self-similar Gaussian process, fractional Brownian motion, sub-fractional Brownian motion, branching particle system, immigration , superprocess, occupation time, fluctuation.
1. Introduction Long-range dependence processes (also called long-memory processes) and their statistics have many areas of application: finance, econometrics, hydrology, meteorology, turbulence, geophysics, statistical physics, communication networks, neuroscience, analysis of DNA sequences. The literature on the subject is vast and varied; we give only a sample of recent references in applied fields: [8–12, 15, 20, 22, 28, 31, 33, 36–38, 41, 44, 46, 47, 49, 50, 54, 57]; the book [16] surveys long-range dependence from its origins to several recent applications. The phenomenon of long-range dependence may occur in the temporal structure of a process and in the spatial structure of a random field, and the processes and fields may be Gaussian or nonGaussian, stationary or nonstationary (see, e.g., [15, 49] and references therein for long-range dependence random fields). Research partially supported by CONACyT grant 37130-E (Mexico).
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LUIS G. GOROSTIZA ET AL.
It is worthwhile to study long-range dependence processes that arise from specific stochastic systems which are also of interest by themselves. In this paper we study long-range dependence, self-similar Gaussian processes related to some spatially distributed particle systems. We focus mainly on occupation time fluctuations of branching and nonbranching systems with immigration. Our purpose is to show how various types of
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