### Portfolio choice based on the empirical distribution

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Let 0 denote the class of all real valued i.i.d. processes and 1 all other ergodic real valued stationary processes. In spite of the fact that these classes are not countably tight we give a strongly consistent sequential test for distinguishing between them.

Let $\left\{{X}_{n}\right\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set $\mathcal{X}$ and that $f\left(X\right)$ 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 ${\lambda}_{n}$ along which we will be able to estimate the conditional expectation $E\left(f\left({X}_{{\lambda}_{n}+1}\right)\right|{X}_{0},\cdots ,{X}_{{\lambda}_{n}})$ from the observations $({X}_{0},\cdots ,{X}_{{\lambda}_{n}})$ in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series...

For a binary stationary time series define ${\sigma}_{n}$ 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 ${\sigma}_{n}$ 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...

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.

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 ${g}_{t}$, 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...

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