On estimating the memory for finitarily markovian processes

Finitarily Markovian processes are those processes $\{X_n\}_{n=-\infty}^{\infty}$ for which there is a finite $K$ ($K = K(\{X_n\}_{n=-\infty}^0$) such that the conditional distribution of $X_1$ given the entire past is equal to the conditional distribution of $X_1$ given only $\{X_n\}_{n=1-K}^0$. The least such value of $K$ is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of $K$, both in the backward sense that we have just described and in the forward sense, where one observes successive values of $\{X_n\}$ for $n \geq 0$ and asks for the least value $K$ such that the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=n-K+1}^n$ is the same as the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=-\infty}^n$. We allow for finite or countably infinite alphabet size

Intermittent estimation for finite alphabet finitarily Markovian processes with exponential tails

summary:We give some estimation schemes for the conditional distribution and conditional expectation of the the next output following the observation of the first $n$ outputs of a stationary process where the random variables may take finitely many possible values. Our schemes are universal in the class of finitarily Markovian processes that have an exponential rate for the tail of the look back time distribution. In addition explicit rates are given. A necessary restriction is that the scheme proposes an estimate only at certain stopping times, but these have density one so that one rarely fails to give an estimate

Nonparametric sequential prediction for stationary processes

We study the problem of finding an universal estimation scheme $h_n:\mathbb{R}^n\to \mathbb{R}$, $n=1,2,...$ which will satisfy \lim_{t\rightarrow\infty}{\frac{1}{t}}\sum_{i=1}^t|h_ i(X_0,X_1,...,X_{i-1})-E(X_i|X_0,X_1,...,X_{i-1})|^p=0 a.s. for all real valued stationary and ergodic processes that are in $L^p$. We will construct a single such scheme for all $1<p\le\infty$, and show that for $p=1$ mere integrability does not suffice but $L\log^+L$ does.Comment: Published in at http://dx.doi.org/10.1214/10-AOP576 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org