On the smoothness of an invariant measure of a Markov chain with respect to a parameter.
The authors consider an \(\mathbb{R}^d\)-valued Markov chain \((X_n\), \(n\geq 0)\) with transition density \(p(x,x';y)\) depending on a parameter \(y\in\mathbb{R}^d\). Let be the marginal distribution of \(X_n\) with initial value \(X_0= x\) and \(\mu_\infty\) be the (unique) invariant probability measure. Then, under some minor assumptions on the function \(\mu_\infty\), the measure \(p(x,x';y)\) has the density \[ p_\infty(x',y)= \int p(x,x'; y)\mu_\infty(dx).\tag{1} \] The authors study a general situation. The result obtained is supposed to be applied to the study of the Poisson equation with a parameter in \(\mathbb{R}^d\) [see \textit{A. Benveniste} et al. (1987) and \textit{E. Pardoux} and \textit{A. Yu. Veretennikov} (1998)]. Examples of Markov processes with a rate of convergence are given by \textit{A. Yu. Veretennikov} [Theory Probab. Appl. 43, No. 4, 664--666 (1998), translation from Theor. Veroyatn. Primen. 43, No. 4, 765--767 (1998; Zbl 0958.60030); ibid. 43, No. 2, 335--337 (1998), resp. ibid. 43, No. 2, 349--351 (1998; Zbl 0953.60042); and Stochastic Processes Appl. 70, No. 1, 115--127 (1997; Zbl 0911.60042)]. The authors use conditions of the following two types: (i) on the rate of convergence and the ``tails'' of marginal distributions of the process and (ii) on the smoothness of transition densities and on their ``tails''; (A\(^1_{\text{E}})\) condition on ergodicity, \((\text{A}^1_{\text{T}})\) condition on transition density. Theorem 1. Let conditions \((\text{A}^1_{\text{E}})\) and \((\text{A}^1_{\text{T}})\) be fulfilled. Then \(p_\infty(x_1',\dots)\in C^1_b\). Let us describe the method that is the main result of this paper. It comprises several steps. Equality (1), the ergodicity conditions, and the Chapman-Kolmogorov equation imply an estimate of the form \(\sup_{1\leq t\leq\infty} p_t(x,x')\leq C<\infty\). The transition density \(p_t(x,x')\) satisfies the inverse Chapman-Kolmogorov equation. The following step is the basic step of the method. It consists in proving the convergence \(q_t\to q_\infty\). Suppose that, in addition to the conditions of Theorem 1, the following condition in fulfiled: \((\text{A}^2_{\text{T}})\) condition on the second derivatives, then \(p_\infty(x_1',\dots)\in C^2_b\).
scientific article; zbMATH DE number 2074488