On coding theorems with modified length functions

Abstract

Let R=(r ₁, r₂,...) be an infinite sequence of real numbers (0<r _i<1). For a binary word (a bit sequence) w of length n, let ¦w¦_R denote the value “-log₂(the probability that n coin-flippings of biased coins generate the sequence w, where the probability that the i-th coinflipping generates 0 is r _i)”. The usual length ¦w¦ is the value ¦w ¦ _R for the special case R=(1/2, 1/2,...). Csiszar and Körner proved that, if there are u>0, v>0 such that u≤ r _i≤ 1-v for all i, then the coding theorem for memoryless sources holds even if ¦w¦r is used as the length of a code word w instead of the usual length ¦w¦. We prove that if lim_i→∞ r_i/2⁻ⁱ=0 then the coding thorem with this modified length ¦w¦r does not hold true.