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Register a new userIncrement stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization
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Alexandre Richard
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We are interested in the increment stationarity property for $L^2$-indexed stochastic processes, which is a fairly general concern since many random fields can be interpreted as the restriction of a more generally defined $L^2$-indexed process. We first give a spectral representation theorem in the sense of citet{Ito54}, and see potential applications on random fields, in particular on the $L^2$-indexed extension of the fractional Brownian motion. Then we prove that this latter process is characterized by its increment stationarity and self-similarity properties, as in the one-dimensional case.
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Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Levy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Holder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Levy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Holder regularity on general indexing collections.
This paper introduces the concept of random context representations for the transition probabilities of a finite-alphabet stochastic process. Processes with these representations generalize context tree processes (a.k.a. variable length Markov chains), and are proven to coincide with processes whose transition probabilities are almost surely continuous functions of the (infinite) past. This is similar to a classical result by Kalikow about continuous transition probabilities. Existence and uniqueness of a minimal random context representation are proven, and an estimator of the transition probabilities based on this representation is shown to have very good pastwise adaptativity properties. In particular, it achieves minimax performance, up to logarithmic factors, for binary renewal processes with bounded $2+gamma$ moments.
We study the asymptotic behavior of empirical processes generated by measurable bounded functions of an infinite source Poisson transmission process when the session length have infinite variance. In spite of the boundedness of the function, the normalized fluctuations of such an empirical process converge to a non-Gaussian stable process. This phenomenon can be viewed as caused by the long-range dependence in the transmission process. Completing previous results on the empirical mean of similar types of processes, our results on non-linear bounded functions exhibit the influence of the limit transmission rate distribution at high session lengths on the asymptotic behavior of the empirical process. As an illustration, we apply the main result to estimation of the distribution function of the steady state value of the transmission process.
This paper contributes to the study of class $(Sigma^{r})$ as well as the c`adl`ag semi-martingales of class $(Sigma)$, whose finite variational part is c`adl`ag instead of continuous. The two above-mentioned classes of stochastic processes are extensions of the family of c`adl`ag semi-martingales of class $(Sigma)$ considered by Nikeghbali cite{nik} and Cheridito et al. cite{pat}; i.e., they are processes of the class $(Sigma)$, whose finite variational part is continuous. The two main contributions of this paper are as follows. First, we present a new characterization result for the stochastic processes of class $(Sigma^{r})$. More precisely, we extend a known characterization result that Nikeghbali established for the non-negative sub-martingales of class $(Sigma)$, whose finite variational part is continuous (see Theorem 2.4 of cite{nik}). Second, we provide a framework for unifying the studies of classes $(Sigma)$ and $(Sigma^{r})$. More precisely, we define and study a new larger class that we call class $(Sigma^{g})$. In particular, we establish two characterization results for the stochastic processes of the said class. The first one characterizes all the elements of class $(Sigma^{g})$. Hence, we derive two corollaries based on this result, which provides new ways to characterize classes $(Sigma)$ and $(Sigma^{r})$. The second characterization result is, at the same time, an extension of the above mentioned characterization result for class $(Sigma^{r})$ and of a known characterization result of class $(Sigma)$ (see Theorem 2 of cite{fjo}). In addition, we explore and extend the general properties obtained for classes $(Sigma)$ and $(Sigma^{r})$ in cite{nik,pat,mult, Akdim}.
These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.
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