Growth and Structure of Stochastic Sequences

TL;DR

This paper introduces a new class of stochastic integer sequences where each term is a sum of two previous terms with at least one chosen randomly, revealing diverse growth behaviors and complex value structures.

Contribution

It defines and analyzes a novel class of stochastic sequences, exploring their growth patterns and the intricate structure of their possible values.

Findings

Sequences exhibit stretched exponential, log-normal, and algebraic growth.

The set of sequence values has an intricate, complex structure.

Randomness and memory interplay produce diverse behaviors.

Abstract

We introduce a class of stochastic integer sequences. In these sequences, every element is a sum of two previous elements, at least one of which is chosen randomly. The interplay between randomness and memory underlying these sequences leads to a wide variety of behaviors ranging from stretched exponential to log-normal to algebraic growth. Interestingly, the set of all possible sequence values has an intricate structure.