Two strong limit theorems for processes with independent increments

Two related almost sure limit theorems are obtained in connection with a stochastic process {ξ(t), ?∞ < t < ∞} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that $\text{(ξ(t) ? ξ(0))}\text{H(t)}\text{→ 0}$ for almost all sample functions of the process. The second result deals with a wide-sense stationary process whose random spectral distributions is ξ. It addresses the question: Under what conditions does $\text{(2T)}{}^{\mathrm{?1}}\text{∫}{}_{\mathrm{?T}}{}^{\mathrm{T}}\text{X(t)}\text{X(t + τ)}\text{dt}$ converge as T → ∞ for all τ for almost all sample functions?     

The distribution of the likelihood ratio for additive processes

Let {X(t), 0 ≤ t ≤ T} and {Y(t), 0 ≤ t ≤ T} be two additive processes over the interval [0, T] which, as measures over D[0, T], are absolutely continuous with respect to each other. Let μ_X and μ_Y be the measures over D[0, T] determined by the two processes. The characteristic function of $\text{ln}\text{(}\text{dμ}{}_{\mathrm{X}}\text{dμ}{}_{\mathrm{Y}}\text{)}$ with respect to μ_Y is obtained in terms of the determining parameters of the two processes.     

Stochastic processes with independent increments,taking values in a Hilbert space

Let $\text{H}$ be a real separable Hilbert space; let X(t), t?[0, 1], be a separable, stochastically continuous, $\text{H}$-valued stochastic process with independent increments. Then a decomposition of X(t) into a uniformly convergent sum of independent processes is found. In this decomposition one of the processes is Gaussian with continuous sample functions, and the remainder of the processes have sample functions whose discontinuities correspond to those of certain real-valued Poisson processes. The decomposition of X(t) leads to a Levy-Khintchine representation of the characteristic functional of X(t). In addition, the case when X(t) has finite variance is explored, and, as a consequence of the above decomposition, a Kolmogorov-type representation of the characteristic functional of X(t) is derived.     

A law of the iterated logarithm for processes with independent increments

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Stochastic processes with independent increments for random mappings

Using additive functions defined on the combinatorial structure of all mappings of anN set into itself, we define paths in the space endowed with the Skorokhod topology. Taking a mapping with equal probability, we get a sequence of random processes. Necessary and sufficient conditions for the weak convergence of this sequence to a stochastic process with independent increments are established. It is shown that the class of such processes contains all possible limits, provided that, on the components of a mapping, the additive functions have values small in average. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 498–516, October–December, 1999. Translated by E. Manstavičius     

A class of stochastic processes with a law generalizing a nondecomposable law on the real line

In this paper we define a class of stochastic processes where law can be considered as a natural generalization of a nondecomposable law. In particular case, we express the processes thus defined as semimartingales with a Brownian martingale part, and compute the likelihood for detecting a signal immersed in additive noise which looks like Brownian motion, but has different independence properties.     

A sampling theorem for multivariate stationary processes

The notion of sampling for second-order q-variate processes is defined. It is shown that if the components of a q-variate process (not necessarily stationary) admits a sampling theorem with some sample spacing, then the process itself admits a sampling theorem with the same sample spacing. A sampling theorem for q-variate stationary processes, under a periodicity condition on the range of the spectral measure of the process, is proved in the spirit of Lloy's work. This sampling theorem is used to show that if a q-variate stationary process admits a sampling theorem, then each of its components will admit a sampling theorem too.     

Nonparametric estimation and testing time-homogeneity for processes with independent increments

We consider a nonparametric estimation problem for the Lévy measure of time-inhomogeneous process with independent increments. We derive the functional asymptotic normality and efficiency, in an ℓ^∞-space, of generalized Nelson–Aalen estimators. Also we propose some asymptotically distribution free tests for time-homogeneity of the Lévy measure. Our result is a fruit of the empirical process theory and the martingale theory.     

Moderate deviations and functional limits for random processes with stationary and independent increments

We first give a functional moderate deviation principle for random processes with stationary and independent increments under the Ledoux's condition. Then we apply the result to the functional limits for increments of the processes and obtain some Csorgo-Revesz type functional laws of the iterated logarithm.     

Spherically invariant processes: Their nonlinear structure,discrimination, and estimation

The structure of the nonlinear space of a spherically invariant process is studied and the problem of discriminating between two spherically invariant processes as well as the problem of nonlinear estimation for spherically invariant processes are solved.     

Limit theorems for stochastic measures of the accuracy of density estimators

Stochastic measures of the distance between a density f and its estimate f_n have been used to compare the accuracy of density estimators in Monte Carlo trials. The practice in the past has been to select a measure largely on the basis of its ease of computation, using only heuristic arguments to explain the large sample behaviour of the measure. Steele [11] has shown that these arguments can lead to incorrect conclusions. In the present paper we obtain limit theorems for the stochastic processes derived from stochastic measures, thereby explaining the large sample behaviour of the measures.