# Universal parameterized family of distributions of runs

## TL;DR

This paper derives explicit formulas for probabilities related to runs and nonoverlapping words in i.i.d. finite-valued sequences, generalizing previous results and analyzing computational complexity.

## Contribution

It introduces a unified explicit formula for run probabilities in i.i.d. sequences, extending -overlapping probabilities and analyzing computational efficiency.

## Key findings

- Explicit formulas for run probabilities in i.i.d. sequences
- Linear computational complexity for fixed parameters
- Asymptotic analysis of integer partitions

## Abstract

We present explicit formulae for parameterized families of probabilities of the number of nonoverlapping words and increasing nonoverlapping words in independent and identically distributed (i.i.d.) finite valued random variables, respectively. Then we provide an explicit formula for a parameterized family of probabilities of the number of runs, which generalizes \(\mu\)-overlapping probabilities for \(\mu\geq 0\) in i.i.d.~binary valued random variables. We also demonstrate exact probabilities of the number of runs whose size are exactly given numbers (Mood 1940). The number of arithmetic operations required to compute our formula for generalized probabilities of runs is linear order of sample size for fixed number of parameters and range. To analyse these number of arithmetic operations for unbounded number of parameters, we show an asymptotic formula for the number of integer partitions that are less than or equal to given number as a special case of Meinardus's theorem.