Fixed-length operational rate-distortion functions

Let $X_0,X_1,\dots$ be a source over a finite nonempty alphabet $\mathcal{X}$, let $\widehat{\mathcal{X}}$ be finite nonempty, and let $d:\mathcal{X}\times \widehat{\mathcal{X}}\to [0,\mathrm{\infty })$ be finite-valued. For positive integers $n$ and $M_n$, write $[M_n]=\{1,\dots ,M_n\}$. An $n$-block fixed-length code with $M_n$ messages is a deterministic encoder $f_n:{\mathcal{X}}^n\to [M_n]$ and decoder $g_n:[M_n]\to {\widehat{\mathcal{X}}}^n$; its reconstruction is ${\widehat{X}}^n=g_n(f_n(X^n))$, its block distortion is Single-letter and block distortion, and its rate is $n^{-1}{\log }_2{M}_n$ bits per source symbol.

For each $D\ge 0$, define the expected-distortion operational rate $R_{\mathrm{o}\mathrm{p}}^{\mathrm{a}\mathrm{v}\mathrm{g}}(D)$ as the infimum of ${\mathrm{lim\hspace{0.17em}sup}}_n{n}^{-1}{\log }_2{M}_n$ over code sequences satisfying ${\mathrm{lim\hspace{0.17em}sup}}_n\mathbb{E}{d}_n(X^n,{\widehat{X}}^n)\le D$. Define the positive-slack excess-distortion operational rate $R_{\mathrm{o}\mathrm{p}}^{\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{k}}(D)$ by the same rate infimum over code sequences satisfying

$\text{for every }\epsilon >0,\mathbb{P}\text{\hspace{0.17em}⁣}(d_n(X^n,{\widehat{X}}^n)>D+\epsilon )⟶0.$

The infimum of an empty set of admissible code sequences is $+\mathrm{\infty }$. The expected and positive-slack excess criteria are separate definitions. An exact-threshold condition $\mathbb{P}(d_n>D)\to 0$ is not part of either definition.