Probability Seminar: Michael Lin

Let $P(s,A)$ be a transition probability on a general measurable space $(\S,\Sigma)$ with invariant probability $m$, and let $\Omega_1 =\S^{\mathbb N}$ be the space of trajectories with $\sigma$-field $\B_1 = \Sigma^{\otimes \mathbb N}$, with coordinate projections $\{\xi_n\}$. Let $\P_s$ be the probability on $\B:=\Sigma \times \B_1$ defined by the transition probability $P$ and initial distribution $\delta_s$. The probability $\P_m:= \int_\S \P_s dm(s)$ is shift invariant on $\S \times \Omega_1$.

The Markov operator $Pf(s):=\int_S f(t)P(s,dt)$ ia contraction of all the $L_p(\S,m)$ spaces ($1 \le p \le \infty$). We assume $P$ to be ergodic: $Pf=f \in L_\infty$ implies $f$ is constant a.e. This implies ergodicity of the shift, and for any $f \in L_1(\S,m)$ the ergodic theorem yields the SLLN for the chain $\{\xi_n\}$: $\frac1n \sum_{k=1}^n f(\xi_k) \to \int_\S f\,dm$ $\P_m$ a.e. Given $f \in L_2(\S,m)$ with zero integral, we look for conditions on $f$ for the CLT: when does $\frac1{\sqrt{n}} \sum_{k=1}^n f(\xi_k)$ converge in distribution (in $(\Omega,\B,\P_m)$ ) to a normal random variable?

To view the equations please view the abstract on the Math website here .