Electronic Communications in Probability

A tame sequence of transitive Boolean functions

Palö Forsström Forsström

Full-text: Open access

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
Received: 11 June 2020
Accepted: 8 December 2020
First available in Project Euclid: 23 December 2020

Permanent link to this document
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

Rights
Creative Commons Attribution 4.0 International License.

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


Export citation

References

  • [1] Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y., Linial, N.: The influence of variables in product spaces. Israel J. Math. 77, (1992), 55–64.
  • [2] Benjamini, I., Kalai, G., Schramm, O.: Noise sensitivity of Boolean functions and applications to percolation. Publ. Math. Inst. Hautes Études Sci. 90, (1999), 5–43.
  • [3] Cerbone, G., Ricciardi, L. M., Sacerdote, L.: Mean variance and skewness of the first passage time for the Ornstein-Uhlenbeck process. Cybernet. Syst. 12, (1981), 395–429.
  • [4] Forsström, M. P.: Denseness of volatile and nonvolatile sequences of functions. Stoch. Proc. Appl. 128, (2018), no. 11, 3880–3896.
  • [5] Galicza, P.: Pivotality versus noise stability for monotone transitive functions. Electron. Commun. Probab. 25, (2020).
  • [6] Garban, C., Steif, J. E.: Noise sensitivity of Boolean functions and percolation. Cambridge University Press, New York, (2015).
  • [7] Jonasson, J., Steif, J.: Volatility of Boolean functions. Stoch. Proc. Appl. 126, (2006), no. 10, 2956–2975.
  • [8] Karlin, S., Taylor H. M.: A second course in stochastic processes, Academic press, Inc., New York-London, (1981).
  • [9] Luigi M. Ricciardi and Shunsuke Sato: First-Passage-Time Density and Moments of the Ornstein-Uhlenbeck Process. J. Appl. Probab. 25, (1988), no. 1, 43–57.
  • [10] Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: Invariance and optimality. Ann. Math. 171, (2010), no. 1, 295–341.
  • [11] O’Donnell, R.: Analysis of Boolean functions, Cambridge University Press, New York, (2014).