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)].

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)].

Reviewer: Mikhail Moklyachuk (Kyïv)

##### 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 |