# zbMATH — the first resource for mathematics

Exponential approximation of distributions. (English) Zbl 1029.60020
Theory Probab. Math. Stat. 66, 119-132 (2003) and Teor. Jmovirn. Mat. Stat. 66, 108-120 (2002).
The authors investigate the problem of exponential approximation of distributions of stochastic processes. Such exponential approximations were originated by L. Le Cam [Univ. California Publ. Stat. 3, 37-98 (1960; Zbl 0104.12701)]. This paper, where the concept of contiguity was first introduced, contains a wealth of asymptotic results. The first author presented results on exponential approximation in a Markovian framework in his book “Contiguity of probability measures. Some applications in statistics” (1972; Zbl 0265.60003). It is shown that under suitable regularity conditions a sequence of families of probability measures indexed by a $$k$$-dimensional parameter $$\theta$$ may be approximated in the neighborhood of $$\theta$$ by a sequence of exponential families of probability measures. The approximation is in the sup-norm sense when probability measures are used or in the $$L_1$$-norm sense when the measures are replaced by their probability densities. In the present paper such an approximation is obtained for discrete-time parameter stochastic processes.
Let $$\{X_n\}$$ be a stochastic process, let $$X_0,X_1,\dots,X_n$$ be the first $$n+1$$ random variables from the stochastic process $$\{X_n\}$$ defined on the probability space $$({\mathcal X},{\mathcal A},P_{\theta}),\theta\in\Theta\subset\mathbb R^k$$. It is assumed that the joint probability law of any finite number of random variables $$X_0,X_1,\dots,X_n$$ is of known functional form except that it depends on the parameter $$\theta$$. Let $$\{\tau_n\},n\geq 1,$$ be a sequence of positive real numbers ($$\tau_n\to\infty$$ as $$n\to\infty$$) and let $$\theta_{\tau_n}=\theta+h\tau_n^{-1/2},h\in\mathbb R^k$$. Let $$\{\nu_n\},n\geq 1,$$ be a sequence of stopping times defined on the sequence $$\{X_n\},n\geq 0,$$ and let $$\mathcal A_{\nu_n}$$ be the $$\sigma$$-field generated by the r.v.’s $$X_0,X_1,\dots,X_{\nu_n}$$. Denote by $$\tilde{P}_{n,\theta}$$ the restriction of $$P_{\theta}$$ to $$\mathcal A_{\nu_n}$$. The sole purpose of this paper is to show that in the neighborhood $$\theta_0$$, the probability measures $$\tilde{P}_{n,\theta_0+h\tau_n^{-1/2}}$$ may be approximated (in the sup-norm sense) by an exponential family of probability measures $$\tilde{P}_{n,h}$$, where $$h$$ is the parameter in the approximating exponential family. The novelty here is that the exponential approximation just described is also valid for a general discrete-time parameter stochastic process where the sample size is a random variable. The exponential measures $$\tilde{P}_{n,h}$$ are defined in terms of a suitably truncated version of the random vector $$\Delta_{\nu_n}$$. The assumptions under which the main theorem of this paper holds true are identical to the ones used by the authors [in: Semi-Markov models and applications, 119-147 (1999; Zbl 0962.62078)].

##### MSC:
 60F25 $$L^p$$-limit theorems 60G40 Stopping times; optimal stopping problems; gambling theory 62E20 Asymptotic distribution theory in statistics 62F12 Asymptotic properties of parametric estimators 62M09 Non-Markovian processes: estimation