We give a universal discrimination procedure for determining if a sample point drawn from an ergodic and stationary simple point process on the line with finite intensity comes from a homogeneous Poisson process with an unknown parameter. Presented with the sample on the interval [0,t] the discrimination procedure gt, which is a function of the finite subsets of [0,t], will almost surely eventually stabilize on either POISSON or NOTPOISSON with the first alternative occurring if and only if the process is indeed homogeneous Poisson. The procedure is based on a universal discrimination procedure for the independence of a discrete time series based on the observation of a sequence of outputs of this time series.
Let {Xn} be a stationary and ergodic time series taking values from a finite or countably infinite set X and that f(X) is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times λn along which we will be able to estimate the conditional expectation E(f(Xλn+1)|X0,…,Xλn) from the observations (X0,…,Xλn) in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (in particular, all stationary and ergodic Markov chains are included in this class) then limn→∞nλn>0 almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λn is upper bounded by a polynomial, eventually almost surely.
There are two kinds of universal schemes for estimating residual waiting times, those where the error tends to zero almost surely and those where the error tends to zero in some integral norm. Usually these schemes are different because different methods are used to prove their consistency. In this note we will give a single scheme where the average error is eventually small for all time instants, while the error itself tends to zero along a sequence of stopping times of density one.
One of the basic estimation problems for continuous time stationary processes Xt, is that of estimating E{Xt+β|Xs:s∈[0,t]} based on the observation of the single block {Xs:s∈[0,t]} when the actual distribution of the process is not known. We will give fairly optimal universal estimates of this type that correspond to the optimal results in the case of discrete time processes.
For a binary stationary time series define σn to be the number of consecutive ones up to the first zero encountered after time n, and consider the problem of estimating the conditional distribution and conditional expectation of σn after one has observed the first n outputs. We present a sequence of stopping times and universal estimators for these quantities which are pointwise consistent for all ergodic binary stationary processes. In case the process is a renewal process with zero the renewal state the stopping times along which we estimate have density one.
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.
A simple renewal process is a stochastic process {Xn} taking values in {0,1} where the lengths of the runs of 1's between successive zeros are independent and identically distributed. After observing X0,X1,…Xn one would like to estimate the time remaining until the next occurrence of a zero, and the problem of universal estimators is to do so without prior knowledge of the distribution of the process. We give some universal estimates with rates for the expected time to renewal as well as for the conditional distribution of the time to renewal.