# On typical encodings of multivariate ergodic sources

## TL;DR

This paper demonstrates that typical coordinate-wise encodings of multivariate ergodic sources produce entropy profiles close to a convolution of the source's entropy profile and a polymatroid, with the exceptional cases being exceedingly rare.

## Contribution

It establishes the typical behavior of coordinate-wise encodings for a broad class of multivariate ergodic sources and provides explicit bounds on the rarity of atypical encodings.

## Key findings

- Typical encodings have entropy profiles close to the convolution of source and polymatroid profiles.
- The proportion of atypical encodings diminishes doubly exponentially.
- Typical encodings satisfy the asymptotic equipartition property for output variables.

## Abstract

We show that the typical coordinate-wise encoding of multivariate ergodic source into prescribed alphabets has the entropy profile close to the convolution of the entropy profile of the source and the modular polymatroid that is determined by the cardinalities of the output alphabets. We show that the proportion of the exceptional encodings that are not close to the convolution goes to zero doubly exponentially. The result holds for a class of multivariate sources that satisfy asymptotic equipartition property described via the mean fluctuation of the information functions. This class covers asymptotically mean stationary processes with ergodic mean, ergodic processes, irreducible Markov chains with an arbitrary initial distribution. We also proved that typical encodings yield the asymptotic equipartition property for the output variables. These asymptotic results are based on an explicit lower bound of the proportion of encodings that transform a multivariate random variable into a variable with the entropy profile close to the suitable convolution.