Volume 15, Issue 4

We consider a general linear stochastic wave equation driven by fractional-in-time noise. We solve the equation and study its energy. We find asymptotic results for the expected energy for large and small times and as the Hurst parameter, H approaches 1/2. Finally we use analytic continuation to provide an alternate analysis for H < 1/2.

Let Ñ t be a standard compensated Poisson process on [0, 1]. We prove a new characterization of anticipating integrals of the Skorokhod type with respect to Ñ , and use it to obtain several counterparts to well established properties of semimartingale stochastic integrals. In particular we show that, if the integrand is sufficiently regular, anticipating Skorokhod integral processes with respect to Ñ admit a pointwise representation as usual Itô integrals in an independently enlarged filtration. We apply such a result to: (i) characterize Skorokhod integral processes in terms of products of backward and forward Poisson martingales, (ii) develop a new Itô-type calculus for anticipating integrals on the Poisson space, and (iii) write Burkholder-type inequalities for Skorokhod integrals.

Our main result is an infinitesimal characterization of Hilbert module module flows, not necessarily of inner type, in terms of stochastic derivations from the initial algebra into the Itô algebra. We prove that any such derivation is the difference of a *-homomorphism and the trivial embedding.

We present a new class of polynomials that are solutions of the generalized Laplace differential equation, written in spherical coordinates. Differentiation properties, orthogonality, Rodrigues like formula, recurrence relations, generating functions and an integral representation are discussed. As applications we recover the results associated with E n ( r ) polynomials and present the solution of Dirac like equation in terms of this new class of polynomials.

Multidimensional statistics, especially in the case where the sample size and the number of estimated parameters (or hypotheses tested) are comparable, is a natural area of application of the theory of random matrices (see V. L. Girko, Statistical Analysis of Observations of Increasing Dimension (Theory and Decision Library B). Springer, New York, 1995.). In this paper we consider procedures for multiple hypothesis testing that may be used in the case, which is even further from the classical statistics: the number of hypotheses may be much larger than the sample size. We consider general step-down multiple testing procedures – stopping-time-like decision rules that reject hypotheses sequentially, starting from those with the smallest (most significant) p -values. The most frequently used subclass of step-down procedures is that of threshold step-down (TSD) procedures exemplified by the classical Holm procedure. We obtain a formula that, for any step-down procedure satisfying a natural condition of monotonicity , provides the exact level at which the procedure controls the family-wise error rate – the probability of one or more false rejections. No assumptions are made about the joint distribution of p -values. The quantity is given by an easily computable expression. The result may be useful in all applications, where no parametric or other assumptions about the joint distribution of p -values can be legitimately made.