The probability of finding a fixed pattern in random data depends monotonically on the bifix indicator

TL;DR

This paper proves that the probability of detecting a fixed pattern in random data decreases monotonically as the pattern's bifix indicator increases, complementing known results about search time.

Contribution

It establishes a new theoretical relationship between pattern bifix structure and detection probability in random sequences.

Findings

Probability decreases with bifix length

Supports monotonic relationship in pattern detection

Enhances understanding of pattern occurrence in random data

Abstract

We consider the problem of finding a fixed L-ary sequence in a stream of random L-ary data. It is known that the expected search time is a strictly increasing function of the lengths of the bifices of the pattern. In this paper we prove the related statement that the probability of finding the pattern in a finite random word is a strictly decreasing function of the lengths of the bifices of the pattern.