Conference Agenda

This result is devoted to establishing upper bounds for a difference of $n$-step transition probabilities for two different, time-inhomogeneous, discrete-time Markov chains with values in a locally compact space when their one-step transition probabilities are close. This stability result is applied to the functional autoregression in $\mathbb{R}^n$.

We study a pair of independent, discrete-time Markov chains $\{X^{(i)}_n, n\ge 0\}$, $i\in \{1,2\}$ with values in a locally compact space $E$ equipped with a $\sigma$-field $\mathfrak{E}$. Since $E$ is locally compact, there exists a sequence of compact sets $\{K_n, n\ge 0\}$, such that $K_n \subset K_{n+1},\ n\ge 0$ and $\bigcup_{n\ge 0} K_n = E$. Let us introduce Markov kernels $$ P_{in}(x, A) = \mathbb{P}\left\{X^{(i)}_{n+1} \in A \middle|\ X^{(i)}_n = x\right\}, n\ge 1, x\in E, A\in \mathfrak{E}. $$ Since we are interested in stability, the kernels $P_{1n}$ and $P_{2n}$ should be close in some sense, which we will define next. To this end, we introduce substochastic kernels $$ Q_n(x, \cdot) = \left(P_{1n}\wedge P_{2n}\right)(x, \cdot), $$ where $\wedge$ should be understood as a minimum of two measures and put $$ \varepsilon := 1 - \inf_{n, x} Q_n(x, E) \le 1. $$ From now on, we assume that $\varepsilon < 1$. We denote the residual substochastic kernels by $$ R_{in}(x, A) = P_{in}(x, A) - Q_n(x, A), $$ so that $$ R_{in}(x, E) \le \varepsilon. $$

Assume the following conditions hold.

\textbf{Condition M} (Minorization condition) Assume that there exist a set $C\in \mathfrak{E}$, a sequence of real numbers $\{a_n, n\ge 1\}, a_n \in (0,1)$ and a sequence of probability measures $\nu_n$ on $(E, \mathfrak{E})$ such that: $$ \inf_{x\in C} P_{in}(x, A) \ge a_n \nu_n(A),\ i\in \{1,2\},$$ $$\inf_{n} \nu_n(C) > 0, $$ $$ 0 < a_* := \inf_n a_n \le a_n \le a^* := \sup_n a_n < 1,$$ for all $A\in \mathfrak{E}$ and $n\ge 1$. This condition can be understood as a local mixing condition, where mixing occurs on a set $C$.

Let us introduce the following condition of geometric recurrence of the pair of chains $\left(X^{(1)}, X^{(2)}\right)$.

\textbf{Condition GR} (Geometric Recurrence) Assume independent chains $X^{(1)}$ and $X^{(2)}$ satisfy \textbf{Condition M} and $C$ is a corresponding set. Then, there exist constant $\psi > 1$ such that $$ h(x,y) = \sup_n \mathbb{E}^n_{xy}\left[\psi^{\sigma_{C\times C}}\right] < \infty, $$ for all $x,y \in E$, where $$\sigma_{C\times C} = \inf\left\{ k \ge 1\ :\ \left(X^{(1)}_k, X^{(2)}_k\right) \in C\times C\right\}. $$ When used in the context of $\mathbb{P}^n_{xy}$ by $\sigma_{C\times C}$ we mean $$\sigma_{C\times C} = \inf\left\{ k \ge n+1\ :\ \left(X^{(1)}_{n+k}, X^{(2)}_{n+k}\right) \in C\times C\right\}. $$

\textbf{Condition T} (Tails condition). Denote by $A_m=K_{m+1}\setminus K_m$. Assume that there exist sequences $\{\hat S_n, n\ge 1\}$ and $\{\hat r_n, n\ge 1\}$, such that $$\hat m = \sum\limits{m\ge 1},\ \hat S_m < \infty,\ \Delta = \sum_{m\ge 1} \hat r_m \hat S_m < \infty, $$ and \begin{equation}\label{cond_t_1} \begin{array}{c} \left(\prod\limits_{k=1}^{n} Q_{t+k}\right) (x, A_m) \le \left(\prod\limits_{k=1}^n Q_{t+k}\right)(x, E)\hat S_m,\ x\in C,\\ \nu_t \left(\prod\limits_{k=1}^{n} Q^{t+k}\right) (A_m) \le \nu_t \left(\prod\limits_{k=1}^n Q_{t+k}\right)(E)\hat S_m,\ x\in C, \end{array} \end{equation} and $$ \sup_{x, t\in A_m} \int_{E^2\setminus C\times C} {R_{1t}(x, dy) R_{2t}(x, dz)\over 1-Q_t(x,E) } h(y,z) \le \hat r_m. $$ for all $t\ge 0$.

\begin{theorem}\label{thm_stability} Let $X^{(i)}$, $i\in \{1,2\}$, be two Markov chains defined above that satisfy \textbf{Condition M}, \textbf{Condition GR} and \textbf{Condition T}. Assume also that $\varepsilon < 1$. Then there exist constants $M_1, M_2\in \mathbb{R}$, such that for every $x \in C$ \begin{align}\label{main_estimate} \left|\left| \mathbb{P}^t_x\left\{X^{(1)}_{n} \in \cdot \right\} - \mathbb{P}^t_x\left\{X^{(2)}_{n} \in \cdot \right\} \right|\right| &\le\varepsilon \hat m M_1 + \Delta M_2, \\ \nonumber \end{align} where $\hat m$ and $\Delta$ are defined in \textbf{Condition T}.

For every $x\notin C$, the following inequality holds true \begin{align}\label{main_estimate2} \left|\left| \mathbb{P}^t_x\left\{X^{(1)}_{n} \in \cdot \right\} - \mathbb{P}^t_x\left\{X^{(2)}_{n} \in \cdot \right\} \right|\right| &\le\varepsilon ( 2 \hat m M_1 + \hat\mu(x))+ 2\Delta M_2, \\ \nonumber \end{align} where $$\hat\mu(x) = \sup_t \sum\limits_{k \ge 1} \left(\prod\limits_{j=0}^{k-1}Q_{t+j}\mathbb{1}_{E \setminus C}\right)(x, E\setminus C) \le 1.$$ \end{theorem}

In this talk we study the asymptotic capacity of the range of random walks. First, I will give a quick introduction to the concept of the capacity of the range, which has been investigated mostly on $\matbbb{Z}^d$. However, there are not many results going beyond. We will focus in this talk on random walks on groups having infinitely many ends, where we sketch the basic idea of the proof that the asymptotic capacity varies real-analytically in terms of probability measures of constant support.