On exponential bounds for mixing and the rate of convergence for Student processes. (Ukrainian, English) Zbl 1224.60127

Abourashchi, N.; Veretennikov, A. Yu.

Teor. Jmovirn. Mat. Stat. 81, 1-12 (2009); translation in Theory Probab. Math. Stat. 81, 1-13 (2010).

The authors established exponential bounds for the \(\beta\)-mixing coefficient for the Student diffusion process that is a solution to the stochastic differential equation \[ dX_t=-\theta(X_t-\mu)dt+\left( \frac{2\theta\delta^2}{\nu-1}\left[ 1+\left( \frac{X_t-\mu}{\delta}\right)^2 \right] \right)^{1/2}dB_t,\quad X_0=x, \] where \(\nu>2\), \(\theta>0\), \(\delta>0\), \(\mu\in\mathbb R\), and \(B = (B_t, t\geq 0)\) is a standard Brownian motion. Due to the classical Itô’s theorem, this stochastic differential equation has a unique strong solution which possesses the Markov and strong Markov property. The Student distribution \(T(\nu,\delta,\mu)\) with the density \[ f(x)=\frac{c(\nu)}{\delta}\left[ 1+\left( \frac{x-\mu}{\delta} \right)^2 \right]^{-(\nu+1)/2},\quad c(\nu)=\frac{\Gamma((\nu+1)/2)}{\sqrt{\pi}\Gamma(\nu/2)} \] is stationary for this Markov process. This density is denoted by \(st_{\nu}(x)\) for given \(\delta\) and \(\mu\). The main results proved in this article are as follows.
Theorem 1. For any \(0 < m < \nu/2\) and any \(0 <\alpha < 2m\theta(1-(2m-1)/(\nu-1))\), there exists a constant \(C > 0\) such that, for the \(\beta\)-mixing coefficient defined as \[ \beta^x(t) =\sup_{s\geq0}\operatorname{E}_x\text{var}_{B\in{\mathcal F}^X_{\geq t+s}} (\text{P}(B | {\mathcal F}^X_{\leq s})-\text{P}(B)), \] the following inequality holds true: \[ \beta^x(t)\leq C(x)\exp(-\alpha t),\quad C(x) = C\left( 1+| (x-\mu)/\delta| ^{2m}\right). \] Also, there exists a constant \(C > 0\) such that \[ \| \mu^x(t)-\mu_{\infty}\| _{TV} \leq C(x)\exp(-\alpha t),\quad C(x) = C\left( 1+| (x-\mu)/\delta| ^{2m}\right), \] where \(\mu_{\infty}\) denotes a unique stationary measure of the process \(\mu^x(t)\) denotes a marginal distribution of \(X_t\) given an initial value \(X_0 = x\) for the Markov process \(X\).
Corollary 1. If \(\nu > 4\), \(2 < m < \nu/2\) and \(f\) is bounded, then \( \frac{S_t-t\operatorname{E}_{st}f(X_0)}{\sqrt{t}}\) converges weakly to \(Z\sim N(0,\sigma^2),\) for \(S_t=\int_0^tf(X_s) \,ds\) in the stationary regime as well as in the nonstationary one.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60	Diffusion processes
60F05	Central limit and other weak theorems

Keywords:

Student diffusion; exponential mixing; heavy tails