# Full Randomness in the Higher Difference Structure of Two-state Markov Chains

## TL;DR

This paper investigates the limiting behavior of higher-order differences in binary Markov chains, revealing convergence properties and classifying chains based on their difference structures using novel capacity concepts.

## Contribution

It introduces new theorems describing the convergence of higher differences in binary Markov chains and classifies chains using discrete capacities.

## Key findings

- Higher-order differences converge to an equi-distributed binary process on specific subsets
- Chains are classified into two types with different convergence behaviors
- New capacity concepts are used to describe the subsets where convergence occurs

## Abstract

The paper studies the higher-order absolute differences taken from progressive terms of time-homogenous binary Markov chains. Two theorems presented are the limiting theorems for these differences, when their order $k$ converges to infinity. Theorems 1 and 2 assert that there exist some infinite subsets $E$ of natural series such that $k$th order differences of every such chain converge to the equi-distributed random binary process as $k$ growth to infinity remaining on $E$. The chains are classified into two types and $E$ depend only on the type of a given chain. Two kinds of discrete capacities for subsets of natural series are defined, and in their terms such sets $E$ are described.