Small deviations of stable processes and entropy of the associated random operators (original) (raw)

Small Deviations of Stable Processes via Metric Entropy

Journal of Theoretical Probability, 2000

Let X=(X(t)) t ¥ T be a symmetric a-stable, 0 < a < 2, process with paths in the dual E g of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some L a (S, s) generating X in a canonical way. The aim of this paper is to compare the degree of compactness of u with the small deviation (ball) behavior of f(e)=−log P(||X|| E* < e) as e Q 0. In particular, we prove that a lower bound for the metric entropy of u implies a lower bound for f(e) under an additional assumption on E. As applications we obtain upper small deviation estimates for weighted a-stable Levy motions, linear fractional a-stable motions and d-dimensional a-stable sheets. Our results rest upon an integral representation of L a -valued operators as well as on small deviation results for Gaussian processes due to Kuelbs and Li and to the authors.

SMALL DEVIATIONS OF RIEMANN–LIOUVILLE PROCESSES IN L q SPACES WITH RESPECT TO FRACTAL MEASURES

Proceedings of the London Mathematical Society, 2006

We investigate Riemann-Liouville processes R H , H > 0, and fractional Brownian motions B H , 0 < H < 1, and study their small deviation properties in the spaces Lq([0, 1], µ). Of special interest are hereby thin (fractal) measures µ, i.e., those which are singular with respect to the Lebesgue measure. We describe the behavior of small deviation probabilities by numerical quantities of µ, called mixed entropy numbers, characterizing size and regularity of the underlying measure. For the particularly interesting case of self-similar measures the asymptotic behavior of the mixed entropy is evaluated explicitly. We also provide two-sided estimates for this quantity in the case of random measures generated by subordinators.

Completely operator-selfdecomposable distributions and operator-stable distributions

Nagoya Mathematical Journal, 1985

Urbanik introduces in [16] and [17] the classes Lm and L∞ of distributions on R1 and finds relations with stable distributions. Kumar-Schreiber [6] and Thu [14] extend some of the results to distributions on Banach spaces. Sato [7] gives alternative definitions of the classes Lm and L∞ and studies their properties on Rd . Earlier Sharpe [12] began investigation of operator-stable distributions and, subsequently, Urbanik [15] considered the operator version of the class L on Rd . Jurek [3] generalizes some of Sato’s results [7] to the classes associated with one-parameter groups of linear operators in Banach spaces. Analogues of Urbanik’s classes Lm (or L∞ ) in the operator case are called multiply (or completely) operator-selfdecomposable. They are studied in relation with processes of Ornstein-Uhlenbeck type or with stochastic integrals based on processes with homogeneous independent increments (Wolfe [18], [19], Jurek-Vervaat [5], Jurek [2], [4], and Sato-Yamazato [9], [10]). The ...

Some Spectral Properties of Operators which are Related to One-Dimensional MARKOV Processes

Mathematische Nachrichten, 1986

In the study of (temporally homogeneous) MARKOV processes spectral theory is sometimes useful in order to prove statements about the limit behavior of the transition probabilities for t + w . Suppose, e.g., that the state space of the process is compact and its transition function is FELLERian and stochastically continuous. With this transition function there is associated a strongly continuous nonnegative contraction (8.c.n.c.) semigroup of linear operators T,, t S O , in a space of continuous functions over the state space. Then the transition probabilities and their quotients converge in a fairly strong sense for t + -, if only the spectrum a(Tt) has the following properties :

Projective surjectivity of quadratic stochastic operators L_1 and its application

2021

A nonlinear Markov chain is a discrete time stochastic process whose transitions depend on both the current state and the current distribution of the process. The nonlinear Markov chain over a infinite state space can be identified by a continuous mapping (the so-called nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex). In the present paper, we consider a continuous analogue of the mentioned mapping acting on L^1-spaces. Main aim of the current paper is to investigate projective surjectivity of quadratic stochastic operators (QSO) acting on the set of all probability measures. To prove the main result, we study the surjectivity of infinite dimensional nonlinear Markov operators and apply them to the projective surjectivity of a QSO. Furthermore, the obtained result has been applied for the existence of positive solution of some Hammerstein integral equations.

On stochastic integral representation of stable processes with sample paths in Banach spaces

Journal of Multivariate Analysis, 1986

Certain path properties of a symmetric a-stable process X(r) = Is h(t, s) d&f(s), to T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double a-stable integral. Also, necessary and suflicient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to R. LePage, M. Woodroofe, and J. Zinn (1981, Ann. Probab. 9, 624632) is extensively used and the relationship between these two representations is discussed.