On Information Processes in Statistical Experiments with Distributed Observations

Using stochastic integration theory in topological vector spaces general formulas for the Hellinger processes are derived. Feynman–Kac type formulas are obtained for the related Hellinger integrals in terms of the Hellinger processes and the geometric mean measures. The expected logarithmic utility from data, characterized as the Shannon information, is also considered.     

On the Wick theorem for mixtures of centered Gaussian distributions

We consider centered conditionally Gaussian d-dimensional vectors X with random covariance matrix Ξ having an arbitrary probability distribution law on the set of nonnegative definite symmetric d × d matrices M _d ⁺. The paper deals with the evaluation problem of mean values \( E\left[ {\prod\nolimits_{i = 1}^{2n} {\left( {{c_i},X} \right)} } \right] \) for c _i ∈ ? ^d , i = 1, …, 2n, extending the Wick theorem for a wide class of non-Gaussian distributions. We discuss in more detail the cases where the probability law ?(Ξ) is infinitely divisible, the Wishart distribution, or the inverse Wishart distribution. An example with Ξ \( = \sum\nolimits_{j = 1}^m {{Z_j}{\sum_j}} \), where random variables Z _j , j = 1, …, m, are nonnegative, and Σ _j ∈ M _d ⁺, j = 1, …, m, are fixed, includes recent results from Vignat and Bhatnagar, 2008.     

On statistical manifolds of solutions of martingale problems

We consider geometrically regular statistical models defined by local densities of probability measures corresponding to discrete or continuous time Markov processes and smoothly depending on a finite dimensional parameter. Evolution equations are derived in terms of the generators of the underlying Markov additive processes for the elements of the related Fisher information matrix and the skewness tensor defining the Riemannian metric and the Amari-Chentsov's affine α-connections as functions of time and starting points of Markov processes. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikes Rinkinys, Vol. 35, No. 4, pp. 456–468, October–December, 1995.     

Processes of Meixner type

Canonical form, self-decomposability, and the Esscher transforms of Meixner processes are discussed. Mixed Meixner processes are defined and characterized as Markov processes or semimartingales. Ornstein-Uhlenbeck and selfsimilar processes of Meixner type are also described. Partially supported by the Lithuanian State Science and Studies Foundation. Institute of Mathematics and Informatics, Akademijos 4; Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 40–51, January–March, 1999.     

On the Extreme-Value Theory for Stationary Diffusions Under Power Normalization

We consider regular positive recurrent diffusion processes on an open interval (l,r) R ¹ with inaccessible end points. Necessary and sufficient conditions are given in terms of the scale functions describing the domains of attraction of the nondegenerate limiting probability laws for the extreme-values under power normalization. Sufficient von Mises-type conditions and formulas for norming constants are also obtained.     

Generalized z-Distributions and Related Stochastic Processes

The class of generalized z–distributions is defined and their properties are investigated. Ornstein–Uhlenbeck–type and self–similar generalized z–processes are constructed and described. Esscher transforms of the generalized z–processes and the mixed generalized z–processes are characterized. Finally, construction and some properties of generalized z–diffusions are also discussed.