Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem

We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.Comment: Published at http://dx.doi.org/10.1214/074921706000000266 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The context-tree weighting method: extensions

First we modify the basic (binary) context-tree weighting method such that the past symbols x1-D, x2-D, …, x 0 are not needed by the encoder and the decoder. Then we describe how to make the context-tree depth D infinite, which results in optimal redundancy behavior for all tree sources, while the number of records in the context tree is not larger than 2T-1. Here T is the length of the source sequence. For this extended context-tree weighting algorithm we show that with probability one the compression ratio is not larger than the source entropy for source sequence length T¿8 for stationary and ergodic source

Kodierung von Gaußmaßen

Es sei $gamma$ ein Gaußmaß auf der Borelschen $sigma$-Algebra $mathcal B$ des separablen Banachraums $B$. Für $X:Omega o B$ gelte $P_X=gamma$. Wir untersuchen den mittleren Fehler, der bei Kodierung von $gamma$ respektive $X$ mit $Ninmathbb N$ Punkten entsteht, und bestimmen untere und obere Abschätzungen für die Asymptotik ($N oinfty$) dieses Fehlers. Hierbei betrachten wir zu $r>0$ Gütekriterien wie folgt: Deterministische Kodierung $delta_2(N,r) := inf_{y_1,ldots,y_Nin B}Emin_{k=1,ldots,N}X-y_k^r.$ Zufällige Kodierung $delta_3(N,r) := inf_ u Emin_{k=1,ldots,N}X-Y_k^r.$ Die $(Y_k)$ seien hierbei i.i.d., unabhängig von $X$, und nach $u$ verteilt. Das Infimum wird über alle Wahrscheinlichkeitsmaße $u$ gebildet. Für das Gütekriterium $delta_4(cdot,r)$ wird ausgehend von der Definition von $delta_3(cdot,r)$ $u$ nicht optimal, sondern $u=gamma$ gewählt. Das Gütekriterium $delta_1(cdot,r)$ ergibt sich aus der Quellkodierungstheorie nach Shannon. Es gilt $delta_1(cdot,r) le delta_2(cdot,r) le delta_3(cdot,r) le delta_4(cdot,r).$ Wir stellen folgenden Zusammenhang zwischen der Asymptotik von $delta_4(cdot,r)$ und den logarithmischen kleinen Abweichungen von $gamma$ her: Es gebe $kappa,a>0$ und $binR$ mit psi(varepsilon) := -log P{X1.Let $gamma$ be a Gaussian measure on the Borel $sigma$-algebra $mathcal B$ of the separable Banach space $B$. Let $X:Omega o B$ with $P_X=gamma$. We investigate the average error in coding $gamma$ resp. $X$ with $Ninmathbb N$ points and obtain lower and upper bounds for the error asymptotics ($N oinfty$). We consider, given $r>0$, fidelity criterions as follows: Deterministic Coding $delta_2(N,r) := inf_{y_1,ldots,y_Nin B}Emin_{k=1,ldots,N}X-y_k^r.$ Random Coding $delta_3(N,r) := inf_ u Emin_{k=1,ldots,N}X-Y_k^r.$ The $(Y_k)$ above are i.i.d., independent of $X$, and distributed according to $u$. The infimum is taken with respect to all probability measures $u$. For the fidelity criterion $delta_4(cdot,r)$, starting from the definition of $delta_3(cdot,r)$, $u$ is not chosen optimal, but as $u=gamma$. The fidelity criterion $delta_1(cdot,r)$ is given according to the source coding theory of Shannon. The fidelity criterions are connected through $delta_1(cdot,r) le delta_2(cdot,r) le delta_3(cdot,r) le delta_4(cdot,r).$ We obtain the following connection between the asymptotics of $delta_4(cdot,r)$ and the den logarithmic small deviations of $gamma$: Let $kappa,a>0$ and $binR$ with psi(varepsilon) := -log P{X1