Abstmct-A coding theorem for the discrete memoryleas broadcast channel is proved for tbe case where no common message is to be transmitted. The theorem is a generalization of the results of Cover and van der Meulen on this problem. Tbe result is tight for broadcast channels having one deterministic component

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A broadcast channel with one sender and two receivers is considered. Three independent messages are to be transmitted over this channel: one common message which is meant for both receivers, and one private message for each qf them. The coding theorem and strong converse for this communication situation is proved for the case when one of the private messages has rate zero. I. INTRODUCTION W E CONSIDER a two-receiver broadcast channel defined by T. M. Cover [l] as a pair of discrete memoryless channels (V, W) with common input alphabet Y and respective output alphabets X and 2. (We use the same symbol for discrete memoryless channels and for their transition probability matrices, and we suppose that all alphabets are finite.) The nth memoryless extension of this broadcast channel is defined by the pair (VI", W"), where, e.g., foryn = YIYZ "'Yn E Y",x~=x1x2"'xn E X". An (n,t)-code for this channel is given by codewords yjnkl E Y n (1 _< j 5 M1, 1 _< k 5 M2, 1 5 1 < MO); and corresponding decoding sets 3Qjl c X",,@kl c 2" such that both (&jl{ and (@hll are disjoint families, and Vn(AjllYjnkl) 2 1-t, w"(@)klIy~~~) L 1-e for all j,/z,l. A triple of nonnegative numbers (R~,Rz,Ro) is called an E-achievable rate triple for this channel, if, for any 6 > 0 and large enough n, there exists an (n,t)-code (yj"kl, &jl, @kl;

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The concentration of measure phenomenon in product spaces means the following: if a subset A of the n'th power of a probability space A' does not have too small a probability then very large probability is concentrated in a small neighborhood of A. The neighborhood is in many cases understood in the sense of Hamming distance, and then measure concentration is known to occur for product probability measures, and also for the distribution of some processes with very fast and uniform decay of memory. Recently Talagrand introduced another notion of neighborhood of sets for which he proved a similar measure concentration inequality which in many cases allows more efficient applications than the one for a Hamming neighborhood. So far this inequality has only been proved for product distributions. The aim of this paper is to give a new proof of Talagrand's inequality, which admits an extension to contracting Markov chains. The proof is based on a new asymmetric notion of distance between probability measures, and bounding this distance by informational divergence.As an application, we analyze the bin packing problem for Markov chains.

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