## Asymptotic behavior of trajectories of extreme sets of random functions and stochastic estimation.(English. Ukrainian original)Zbl 0923.60058

Theory Probab. Math. Stat. 55, 1-12 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 1-12 (1996).
The following problem is considered. Let $$F_n(q,t)$$, $$q \in Q$$, $$t \geq 0$$, be a sequence of random functions, and let $$\{ q_n(t) \} = \arg \min_{q \in Q} F_n(q,t).$$ For any $$t>0$$ the set $$\{ q_n(t)\}$$ is a random set. There are to find conditions under which $$\{ q_n(t)\}$$ $$U$$-converges to $$q_0(t)$$ as $$n \to \infty.$$ Note, that the sequence $$\{ q_n(t)\}$$ $$U$$-converges to $$q_0(t)$$ if any sequence $${\widetilde q}_n(t)$$ such that $$P\{ {\widetilde q}_n(t) \in \{ q_n(t)\}\} =1$$ $$U$$-converges to $$q_0(t).$$ It means that the sequence of measures generated by $${\widetilde q}_n(t)$$ in the Skorokhod space $$D_{[0,T]}$$ converges weakly to the measure generated by $$q_0(t).$$ The author finds conditions under which $$\{ q_n(t)\}$$ $$U$$-converges to $$q_0(t)$$, where $$F_0(q,t)$$ is a continuous random function. Moreover, the author finds conditions under which $$V_n( \{ q_n(t)\} -q_0(t))$$ $$U$$-converges, where $$V_n$$ is a sequence such that $$V_n \to \infty$$ as $$n \to \infty.$$ These results are used to find the statistical estimators of parameters. Let $$p(y,\theta_0,\alpha)$$ be the density of random variables $$\gamma_k(\alpha)$$ and let observations $$y_{nk} = \gamma_k(S_{n,k})$$ and $$S_{n,k}, {k/n} \leq T$$, are known, where $$S_{n,k}$$ is a trajectory of some stochastic or determined system. Denote $$\theta_n(t) =\arg \max_{\theta \in \Theta} L_n(\theta,t),$$ where $L_n(\theta,t) ={1 \over n} \sum_{k=0}^{[nt]} \ln p(y_{n_k},\theta,S_{n,k})$ is the function of maximum likelihood. The author finds conditions under which $$\theta_n(t)$$ $$U$$-converges to $$\theta_0$$ for $$t \in [t_0,T]$$, $$t_0 >0$$.

### MSC:

 60G70 Extreme value theory; extremal stochastic processes 62F12 Asymptotic properties of parametric estimators 60F15 Strong limit theorems 62M09 Non-Markovian processes: estimation