A proof of the data compression theorem of Slepian and Wolf for ergodic sources (Corresp.)

Abstract

If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{(X_i, Y_i)\}_{i=1}^{\infty}</tex> is a sequence of independent identically distributed discrete random pairs with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(X_i, Y_i) ~ p(x,y)</tex> , Slepian and Wolf have shown that the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> process and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> process can be separately described to a common receiver at rates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_X</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_Y</tex> hits per symbol if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_X + R_Y &gt; H(X,Y), R_X &gt; H(X\midY), R_Y &gt; H(Y\midX)</tex> . A simpler proof of this result will be given. As a consequence it is established that the Slepian-Wolf theorem is true without change for arbitrary ergodic processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{(X_i,Y_i)\}_{i=1}^{\infty}</tex> and countably infinite alphabets. The extension to an arbitrary number of processes is immediate.