## The asymptotic normality of Baxter-type estimates for parameters of nonstationary processes.(English. Ukrainian original)Zbl 0934.62084

Theory Probab. Math. Stat. 53, 107-112 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 97-102 (1995).
Let $$x(t)$$ be a diffusion process that satisfies the equation $dx(t)=f(x(t),t)dt + g(t,\vartheta)dw(t),$ where $$f(x,t)$$ is an unknown function, $$g(t,\vartheta)$$ is a function that is known up to the parameter $$\vartheta\in\Theta\subset R^d$$. Observations of the diffusion process $$x(t)$$ at the points $$t_j^N=j/N$$ are used to construct the Baxter-type sums $B_N(x,a)=\sum_{j=1}^N a(t_{j-1}^N)(x(t_j^N)-x(t_{j-1}^N))^2$ with some weight function $$a(t)$$ and a criterion $R_N(\alpha)=\int_0^1 g^4(\cdot,\alpha)dt - 2B_N(x,g^2(\cdot,\alpha)).$ The estimate for $$\vartheta$$ is defined by the relation $$\hat\vartheta_N=\text{arg min}_{\alpha\in\Theta} R_N(\alpha).$$ Conditions for asymptotic normality of $\sqrt{N}(B_N(x,a)- E B_N(x,a))\;\text{and} \sqrt{N}(\hat\vartheta_N-\vartheta)$ are obtained.

### MSC:

 62M05 Markov processes: estimation; hidden Markov models 62G20 Asymptotic properties of nonparametric inference 60F05 Central limit and other weak theorems 60G35 Signal detection and filtering (aspects of stochastic processes)