It is demonstrated how to represent asymptotically mean stationary (AMS) random sources with values in standard spaces as mixtures of ergodic AMS sources. This an extension of the well known decomposition of stationary sources which has facilitated the generalization of prominent source coding theorems to arbitrary, not necessarily ergodic, stationary sources. Asymptotic mean stationarity generalizes the definition of stationarity and covers a much larger variety of real-world examples of random sources of practical interest. It is sketched how to obtain source coding and related theorems for arbitrary, not necessarily ergodic, AMS sources, based on the presented ergodic decomposition.
The setting is a stationary, ergodic time series. The challenge is to construct a sequence of functions, each based on only finite segments of the past, which together provide a strongly consistent estimator for the conditional probability of the next observation, given the infinite past. Ornstein gave such a construction for the case that the values are from a finite set, and recently Algoet extended the scheme to time series with coordinates in a Polish space. The present study relates a different solution to the challenge. The algorithm is simple and its verification is fairly transparent. …
In this paper, we examine the Renyi entropy rate of stationary ergodic processes. For a special class of stationary ergodic processes, we prove that the Renyi entropy rate always exists and can be polynomially approximated by its defining sequence; moreover, using the Markov approximation method, we show that the Renyi entropy rate can be exponentially approximated by that of the Markov approximating sequence, as the Markov order goes to infinity. For the general case, by constructing a counterexample, we disprove the conjecture that the Renyi entropy rate of a general stationary ergodic proce…
We proposed a learning algorithm for nonparametric estimation and on-line prediction for general stationary ergodic sources. We prepare histograms each of which estimates the probability as a finite distribution, and mixture them with weights to construct an estimator. The whole analysis is based on measure theory. The estimator works whether the source is discrete or continuous. If it is stationary ergodic, then the measure theoretically given Kullback-Leibler information divided by the sequence length $n$ converges to zero as $n$ goes to infinity. In particular, for continuous sources, the m…
Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times $λ_n$ along which we will be able to estimate the conditional probability $P(X_{λ_n+1}=x|X_0,...,X_{λ_n})$ from data segment $(X_0,...,X_{λ_n})$ in a pointwise 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 statio…
Let $\{(X_i,Y_i)\}$ be a stationary ergodic time series with $(X,Y)$ values in the product space $\R^d\bigotimes \R .$ This study offers what is believed to be the first strongly consistent (with respect to pointwise, least-squares, and uniform distance) algorithm for inferring $m(x)=E[Y_0|X_0=x]$ under the presumption that $m(x)$ is uniformly Lipschitz continuous. Auto-regression, or forecasting, is an important special case, and as such our work extends the literature of nonparametric, nonlinear forecasting by circumventing customary mixing assumptions. The work is motivated by a time series…
This paper is concerned with the following problem: Given a stochastic non-linear system controlled over a noisy channel, what is the largest class of channels for which there exist coding and control policies so that the closed loop system is stochastically stable? Stochastic stability notions considered are stationarity, ergodicity or asymptotic mean stationarity. We do not restrict the state space to be compact, for example systems considered can be driven by unbounded noise. Necessary and sufficient conditions are obtained for a large class of systems and channels. A generalization of Bode…
In the online prediction framework, we use generalized entropy of to study the loss rate of predictors when outcomes are drawn according to stationary ergodic distributions over the binary alphabet. We show that the notion of generalized entropy of a regular game \cite{KVV04} is well-defined for stationary ergodic distributions. In proving this, we obtain new game-theoretic proofs of some classical information theoretic inequalities. Using Birkhoff's ergodic theorem and convergence properties of conditional distributions, we prove that a classical Shannon-McMillan-Breiman theorem holds for a r…
In this paper we consider the rate distortion problem of discrete-time, ergodic, and stationary sources with feed forward at the receiver. We derive a sequence of achievable and computable rates that converge to the feed-forward rate distortion. We show that, for ergodic and stationary sources, the rate {align} R_n(D)=\frac{1}{n}\min I(\hat{X}^n\rightarrow X^n){align} is achievable for any $n$, where the minimization is taken over the transition conditioning probability $p(\hat{x}^n|x^n)$ such that $\ex{}{d(X^n,\hat{X}^n)}\leq D$. The limit of $R_n(D)$ exists and is the feed-forward rate disto…
Given a heterogeneous time-series sample, the objective is to find points in time (called change points) where the probability distribution generating the data has changed. The data are assumed to have been generated by arbitrary unknown stationary ergodic distributions. No modelling, independence or mixing assumptions are made. A novel, computationally efficient, nonparametric method is proposed, and is shown to be asymptotically consistent in this general framework. The theoretical results are complemented with experimental evaluations.
We investigate the problem of Multiple Description (MD) coding of discrete ergodic processes. We introduce the notion of MD stationary coding, and characterize its relationship to the conventional block MD coding. In stationary coding, in addition to the two rate constraints normally considered in the MD problem, we consider another rate constraint which reflects the conditional entropy of the process generated by the third decoder given the reconstructions of the two other decoders. The relationship that we establish between stationary and block MD coding enables us to devise a universal algo…
The forward estimation problem for stationary and ergodic time series $\{X_n\}_{n=0}^{\infty}$ taking values from a finite alphabet ${\cal X}$ is to estimate the probability that $X_{n+1}=x$ based on the observations $X_i$, $0\le i\le n$ without prior knowledge of the distribution of the process $\{X_n\}$. We present a simple procedure $g_n$ which is evaluated on the data segment $(X_0,...,X_n)$ and for which, ${\rm error}(n) = |g_{n}(x)-P(X_{n+1}=x |X_0,...,X_n)|\to 0$ almost surely for a subclass of all stationary and ergodic time series, while for the full class the Cesaro average of the er…
This paper proposes an analysis of asymptotically mean stationary (AMS) communication channels. A hierarchy based on stability properties (stationarity, quasi-stationarity, recurrence and asymptotically mean stationarity) of channels is identified. Stationary channels are a subclass of quasi-stationary channels which are a subclass of recurrent AMS channels which are a subclass of AMS channels. These classes are proved to be stable under Markovian composition of channels (e.g., the cascade of AMS channels is an AMS channel). Characterizations of channels of each class are given. Some propertie…
Bailey showed that the general pointwise forecasting for stationary and ergodic time series has a negative solution. However, it is known that for Markov chains the problem can be solved. Morvai showed that there is a stopping time sequence $\{λ_n\}$ such that $P(X_{λ_n+1}=1|X_0,...,X_{λ_n}) $ can be estimated from samples $(X_0,...,X_{λ_n})$ such that the difference between the conditional probability and the estimate vanishes along these stoppping times for all stationary and ergodic binary time series. We will show it is not possible to estimate the above conditional probability along a sto…
Transfer entropy is a measure of the magnitude and the direction of information flow between jointly distributed stochastic processes. In recent years, its permutation analogues are considered in the literature to estimate the transfer entropy by counting the number of occurrences of orderings of values, not the values themselves. It has been suggested that the method of permutation is easy to implement, computationally low cost and robust to noise when applying to real world time series data. In this paper, we initiate a theoretical treatment of the corresponding rates. In particular, we cons…
A plug-in estimator of entropy is the entropy of the distribution where probabilities of symbols or blocks have been replaced with their relative frequencies in the sample. Consistency and asymptotic unbiasedness of the plug-in estimator can be easily demonstrated in the IID case. In this paper, we ask whether the plug-in estimator can be used for consistent estimation of the entropy rate $h$ of a stationary ergodic process. The answer is positive if, to estimate block entropy of order $k$, we use a sample longer than $2^{k(h+ε)}$, whereas it is negative if we use a sample shorter than $2^{k(h…
We consider transmission of stationary and ergodic sources over non-ergodic composite channels with channel state information at the receiver (CSIR). Previously we introduced alternate capacity definitions to Shannon capacity, including the capacity versus outage and the expected capacity. These generalized definitions relax the constraint of Shannon capacity that all transmitted information must be decoded at the receiver. In this work alternate end-to-end distortion metrics such as the distortion versus outage and the expected distortion are introduced to relax the constraint that a single d…
The problem of constructing effective statistical tests for random number generators (RNG) is considered. Currently, statistical tests for RNGs are a mandatory part of cryptographic information protection systems, but their effectiveness is mainly estimated based on experiments with various RNGs. We find an asymptotic estimate for the p-value of an optimal test in the case where the alternative hypothesis is a known stationary ergodic source, and then describe a family of tests each of which has the same asymptotic estimate of the p-value for any (unknown) stationary ergodic source.
We prove that the average smooth Renyi entropy rate will approach the entropy rate of a stationary, ergodic information source, which is equal to the Shannon entropy rate for a classical information source and the von Neumann entropy rate for a quantum information source.
In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distan…
Even simply-defined, finite-state generators produce stochastic processes that require tracking an uncountable infinity of probabilistic features for optimal prediction. For processes generated by hidden Markov chains the consequences are dramatic. Their predictive models are generically infinite-state. And, until recently, one could determine neither their intrinsic randomness nor structural complexity. The prequel, though, introduced methods to accurately calculate the Shannon entropy rate (randomness) and to constructively determine their minimal (though, infinite) set of predictive feature…
Currently, statistical tests for random number generators (RNGs) are widely used in practice, and some of them are even included in information security standards. But despite the popularity of RNGs, consistent tests are known only for stationary ergodic deviations of randomness (a test is consistent if it detects any deviations from a given class when the sample size goes to $ \infty $). However, the model of a stationary ergodic source is too narrow for some RNGs, in particular, for generators based on physical effects. In this article, we propose computable consistent tests for some classes…
Consider a stochastic process $\{X(t)\}$ on a finite state space $ {\sf X}=\{1,\dots, d\}$. It is conditionally Markov, given a real-valued `input process' $\{ζ(t)\}$. This is assumed to be small, which is modeled through the scaling, \[ ζ_t = \varepsilon ζ^1_t, \qquad 0\le \varepsilon \le 1\,, \] where $\{ζ^1(t)\}$ is a bounded stationary process. The following conclusions are obtained, subject to smoothness assumptions on the controlled transition matrix and a mixing condition on $\{ζ(t)\}$: (i) A stationary version of the process is constructed, that is coupled with a stationary version of …
The problem of prediction consists in forecasting the conditional distribution of the next outcome given the past. Assume that the source generating the data is such that there is a stationary ergodic predictor whose error converges to zero (in a certain sense). The question is whether there is a universal predictor for all such sources, that is, a predictor whose error goes to zero if any of the sources that have this property is chosen to generate the data. This question is answered in the negative, contrasting a number of previously established positive results concerning related but smalle…
