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

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.

### 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
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