## Electronic Communications in Probability

### A tame sequence of transitive Boolean functions

#### Abstract

Given a sequence of Boolean functions $(f_{n})_{n \geq 1}$, $f_{n} \colon \{ 0,1 \}^{n} \to \{ 0,1 \}$, and a sequence $(X^{(n)})_{n\geq 1}$ of continuous time $p_{n}$-biased random walks $X^{(n)} = (X_{t}^{(n)})_{t \geq 0}$ on $\{ 0,1 \}^{n}$, let $C_{n}$ be the (random) number of times in $(0,1)$ at which the process $(f_{n}(X_{t}))_{t \geq 0}$ changes its value. In [7], the authors conjectured that if $(f_{n})_{n \geq 1}$ is non-degenerate, transitive and satisfies $\lim _{n \to \infty } \mathbb {E}[C_{n}] = \infty$, then $(C_{n})_{n \geq 1}$ is not tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.

#### Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 83, 8 pp.

Dates
Accepted: 8 December 2020
First available in Project Euclid: 23 December 2020

https://projecteuclid.org/euclid.ecp/1608714241

Digital Object Identifier
doi:10.1214/20-ECP366

Subjects
Primary: 60K99: None of the above, but in this section

Keywords
Boolean functions

#### Citation

Forsström, Palö Forsström. A tame sequence of transitive Boolean functions. Electron. Commun. Probab. 25 (2020), paper no. 83, 8 pp. doi:10.1214/20-ECP366. https://projecteuclid.org/euclid.ecp/1608714241

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