From Triangular Array Progression to Near-Log-Concave State Transitions

Abstract

The paper introduces near-log-concave random variables and the state-transitions of a family of Markov chains derived from an algorithmic integer sequence progression. We propose a combinatorial rule that generates an asymmetric triangular array whose $n$-th row sums to $2^{n}$ for all $n \geq 0$, where every row is the degree sequence of a multigraph without loops. The structure hosts infinitely many zero-free unimodal near-log-concave sequences whose log-concavity deviation converges to $\log(4/3)$ under logarithmic scaling. From a probabilistic perspective, the construct defines a sequence of near-log-concave probability mass functions, from which, a structural decoupling of moments emerges: the mean diverges while the variance asymptotically converges to a steady-state limit. We study the Shannon entropy dynamics of the triangular array, and generate infinitely many right-stochastic matrices via coordinate transformations over the Tychonoff cube. Our specific framework indicates a broader methodology for modelling asymmetric stochastic systems, where distributions propagate as non-dispersive discrete wave packets, advancing indefinitely without flattening.