My research interests and contributions mostly are in the areas of prediction, estimation
and hypotheses testing problems for second order discrete- or continuous-time stationary
stochastic processes, and related analytical problems:
Toeplitz and Wiener-Hopf operators, orthogonal polynomials on the unit circle and their
continual analogs (Krein's functions), and approximations in weighted Lebesgue spaces.
The basic problems are:
1. Parametric estimation of unknown spectral parameters
(approximation of the likelihood function,local asymptotic normality).
2. Nonparametric estimation of spectral functionals
(construction of asymptotically efficient estimators, bounding the minimax risks of estimators).
3. Limit theorems and the large deviation principle for Toeplitz type random quadratic forms and functionals.
4. Prediction of discrete- and continuous-time stationary stochastic processes
(asymptotic behavior of the prediction error: direct and inverse problems).
5. Testing of simple and composite hypotheses about the spectrum of stationary processes
(goodness-of-fit tests).
6. Asymptotic behavior of Toeplitz and Fredholm determinants, and
traces of products of truncated Toeplitz and Wiener-Hopf operators.
For description of the problems and some results see Research Statement (in pdf format).