Training-based equivalence of asymptotic symbol error probabilities

Consider the nonlinear input-output system with $B$ input-output training pairs $(X_B,Y_B)$ and its equivalent system, with input vectors ${\mathbf{x}}_{B+1}$ and $\bar{{\mathbf{x}}}$ and corresponding observations ${\mathbf{y}}_{B+1}$ and $\bar{{\mathbf{y}}}$. For each transmitter $m$, let $\hat{x}_{B+1,m}$ and $\hat{\bar{x}}_m$ be the respective maximum a posteriori estimates:

$$\hat{x}_{B+1,m}=\mathop{\mathrm{argmax}}_{x}P(x_{B+1,m}=x|X_B,Y_B,{\mathbf{y}}_{B+1}),\qquad \hat{\bar{x}}_m=\mathop{\mathrm{argmax}}_{x}P(\bar{x}_m=x|G,\bar{{\mathbf{y}}}).$$

Define the average error probabilities

$$P_{e,x}=\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}P(x_{B+1,m}\neq\hat{x}_{B+1,m}|X_B,Y_B,{\mathbf{y}}_{B+1}),$$ $$\bar{P}_{e,x}=\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}P(\bar{x}_m\neq\hat{\bar{x}}_m|G,\bar{{\mathbf{y}}}).$$

Training-based error-probability equivalence conjecture. For the system with $B$ input-output training pairs $(X_B,Y_B)$,

$$\lim_{T\to\infty}P_{e,x}=\lim_{T\to\infty}\bar{P}_{e,x}.$$

Here $x_{B+1,m}$ and $\bar{x}_m$ are the $m$th elements of the input vectors in the original and equivalent systems, respectively. The conjecture translates the mutual-information equivalence theorem into a prediction for symbol error probabilities; it is stated because it is not proved in the paper, although its consequences are reported to be accurate and useful.