# Koval V. A.

### Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices

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.

### On the Strong Law of Large Numbers for Multivariate Martingales with Continuous Time

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.

### Law of the Iterated Logarithm for Unstable Gaussian Autoregressive Models

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.

### On One Sufficient Condition for the Validity of the Strong Law of Large Numbers for Martingales

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.

### On the Asymptotic Properties of Solutions of Linear Stochastic Differential Equations in $R^d$

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.

### Limit theorem for the maximum of dependent Gaussian random elements in a Banach space

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.

### Weak invariance principle for solutions of stochastic recurrence equations 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 equation*X* _{ n }=*A* _{ n } *X* _{ n+1}+*V* _{ n }, n=1, 2,..., converge in distribution to a solution of the corresponding operator equation.

### On the rate of convergence of stochastic approximation procedures

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.

### Asymptotic behavior of solutions of stochastic recurrence equations in *R*^{ d}

^{ d}

Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 829-833