Coding theorems for the compound semi-continuous memoryless channels
Ken-ichi Yoshihara · Kodai Mathematical Journal · 1965
The idea of " compound channels " was introduced by Blackwell, Breiman, and Thomasian [1] and Wolfowitz [4], independently, and coding theorems and their strong converses for the compound discrete memoryless channels were proved by Wolfowitz, mainly.In [3], Kesten considered coding theorems and their weak converses for the compound semi-continuous memoryless channels in the case where the output alphabet is the set of integers.In this paper, under some assumptions, we shall consider coding theorems and their strong converses for the compound semi-continuous memoryless channels in the case where the output space is the real line.In Section 2, we shall make assumptions and in Sections 3 and 4, we shall prove some lemmas by which coding theorems will be proved in the following sections.The results in this paper contain, as a special case, Wolfowitz's results in [6], Assumptions.In this paper, we shall consider the semi-continuous compound channels under the following assumptions:A 1: Let S be defined by ra (i) s= x [ Tl v }=ι where [r^\ γ^} denotes a (bounded) closed interval of the real line, and X denotes .7 = 1 the Cartesian products of these intervals.A2: (a) For each seS, the channel (D' n , D'ή, h( \ \s)) is the semi-continuous memoryless channel where (i) D' n = x {!,-,*}, J=l n (ii) D'ή= X DΪ, DΪ being the real line, ,7=1