Koval V. A.
Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 1006–1008
We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of martingale differences in $R^d$. We consider the normalization of matrices inverse to the covariance matrices of these sums by square roots. This result is used for the proof of the bounded law of the iterated logarithm for martingales with arbitrary matrix normalization.
Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1287-1291
We prove the strong law of large numbers for vector martingales with arbitrary operator normalizations. From the theorem proved, we deduce several known results on the strong law of large numbers for martingales with continuous time.
Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 428-432
We investigate the asymptotic properties of one-dimensional Gaussian autoregressive processes of the second order. We prove the law of the iterated logarithm in the case of an unstable autoregressive model.
Ukr. Mat. Zh. - 2000. - 52, № 10. - pp. 1357-1362
We prove a theorem on the strong law of large numbers for martingales. The existence of higher moments is not assumed. From the theorem proved, we deduce numerous well-known results on the strong law of large numbers both for martingales and for sequences of sums of independent random variables.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1166-1175
We investigate necessary and sufficient conditions for the almost-sure boundedness of normalized solutions of linear stochastic differential equations in $R^d$ their almost-sure convergence to zero. We establish an analog of the bounded law of iterated logarithm.
Strong Law of Large Numbers with Operator Normalizations for Martingales and Sums of Orthogonal Random Vectors
Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1045-1061
We establish the strong law of large numbers with operator normalizations for vector martingales and sums of orthogonal random vectors. We describe its applications to the investigation of the strong consistency of least-squares estimators in a linear regression and the asymptotic behavior of multidimensional autoregression processes.
Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 1005–1008
The well-known Nisio result on the asymptotie equality for the maximum of real-valued Gaussian random variables is generalized to the case of Gaussian random variables taking values in a Banach space.
Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 114–117
We show that, in a Banach space, continuous random processes constructed by using solutions of the difference equationX n =A n X n+1+V n , n=1, 2,..., converge in distribution to a solution of the corresponding operator equation.
Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 997–1002
The rate of convergence of a linear stochastic approximation procedure in $R^d$ is studied under fairly general assumptions on the coefficients of the equation.
Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 829-833