both a positive result (achievability of a rather low coding rate of 1 bit per letter) and a negative result (unachievability of negligible redundancy).

In Sections \[sec:lossy\] and \[sec:universalLossy\], we turn to lossy coding. In Section \[sec:RDzero\] it is shown that, for a large class of sources, a natural rate-distortion function is the trivial zero-zero point. This inspires restriction to finite-sized blocks in Sections \[sec:finitefinite\] and \[sec:finiteuncountable\]. These sections discuss the rate-distortion functions for the discrete- and uncountable-alphabet cases, respectively. Section \[sec:finiteuncountable\] also presents several high-rate quantization analyses. Universal lossy coding is considered in Section \[sec:universalLossy\]. Finally, Section \[sec:seq2set\] concludes the paper with a discussion of intermediates between full relevance and full irrelevance of order; additional connections to related work; and a summary of the main results.

Many results presented here appeared first in [@VarshneyG2006b; @VarshneyG2006; @Varshney2006; @VarshneyG2007].

Separating Order and Value {#sec:ordervalue}
==========================

Consider source variables $X_1,\,X_2,\,\ldots,\,X_n$ drawn from a common alphabet $\mathcal{X}$ according to any joint distribution. A realization $(x_i)_{i=1}^n$ can be decomposed into a multiset of values $\{x_i\}_{i=1}^n$ and an order $j$. This can be expressed as $$\label{eq:transform}
(x_1,\,x_2,\,\ldots,\,x_n)
 = 
(y_{i_1},\,y_{i_2},\,\ldots,\,y_{i_n})
\longrightarrow
\left( {\begin{array}{*{20}c}
   {i_1 } & {i_2 } &  \cdots  & {i_n }  \\
   {x_1 } 