Theory of Stochastic Processes, 2008, № 1

The measure preserving and nonsingular transformations of the jump Levy processes

Tue, 01 Jan 2008 00:00:00 GMT

The measure preserving and nonsingular transformations of the jump Levy processes Smorodina, N.V. Let ξ(t), t belongs [0, 1], be a jump Levy process. By Pξ, we denote the law of ξ in the Skorokhod space D[0, 1]. Under some conditions on the Levy measure of the process, we construct the group of Pξ preserving transformations of D[0, 1]. For the Levy process that has only positive (or only negative) jumps, we construct the semigroup of nonsingular transformations.

Convergence in Skorokhod J-topology for compositions of stochastic processes

Tue, 01 Jan 2008 00:00:00 GMT

Convergence in Skorokhod J-topology for compositions of stochastic processes Silvestrov, D.S. A survey on functional limit theorems for compositions of stochastic processes is presented. Applications to stochastic processes with random scaling of time, random sums, extremes with random sample size, generalised exceeding processes, sum- and max-processes with renewal stopping, and shock processes are discussed.

Estimation in an implicit multivariate measurement error model with clustering in the regressor

Tue, 01 Jan 2008 00:00:00 GMT

Estimation in an implicit multivariate measurement error model with clustering in the regressor Polekha, M. An implicit linear multivariate model DZ ≈ 0 is considered, where the data matrix D is observed with errors, and Z is a parameter matrix. The error matrix is partitioned into two uncorrelated blocks, and the total covariance structure in each block is supposed to be known up to a corresponding scalar factor. Moreover, the row data are clustered into two groups. Based on the method of corrected objective function, the strongly consistent estimators of scalar factors and the kernel of the matrix D are constructed, as the numbers of rows in the clusters tend to infinity.

Asymptotic formulas for probabilities of large deviations of ladder heights

Tue, 01 Jan 2008 00:00:00 GMT

Asymptotic formulas for probabilities of large deviations of ladder heights Nagaev, S.V. Asymptotic formulas for large-deviation probabilities of a ladder height in a random walk generated by a sequence of sums of i.i.d. random variables are deduced. Two cases are considered: a) the distribution F(x) of summands is normal with a zero mean. b) F(x) belongs to the domain of the normal attraction of a stable law with the exponent 0 < α < 1. The method of Laplace transforms is applied in proofs.