Noise sensitivity and FK-type representations for Gaussian and stable processes
Doktorsavhandling, 2019
Paper A concerns the question of whether the exclusion sensitivity and exclusion stability of a sequence of Boolean functions are monotone with respect to adding edges to the underlying sequence of graphs.
In paper B, we use the tools developed in Paper A to give an elementary proof of the behaviour of the mixing time of a random interchange process on a complete graph.
In Paper C we discuss the relationship between the noise sensitivity, noise stability and volatility of sequences of Boolean functions. In particular, we show that the set of volatile such sequences is dense in the set of all sequences of Boolean functions. Moreover, we construct a noise stable and volatile sequence of Boolean functions which is not o(1)-close to any non-volatile sequence of Boolean functions.
Finally, in Paper D, we investigate which threshold Gaussian and threshold stable random vectors have color representations. We discuss this from many different perspectives, and results include formulae for the dimension of the kernel of the associated linear operator, geometric conditions on the Gaussian vectors whose threshold have color representations and explicit examples of stable vectors with phase transitions at any stability index for the corresponding threshold process to have a color representation for large thresholds .
color representation
exclusion process
color process
Noise sensitivity
threshold stable vector
volatility
noise stability
interchange process
threshold Gaussian vector
multivariate stable distribution
Bernoulli random vector
mixing time
Författare
Malin Palö Forsström
Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori
Monotonicity properties of exclusion sensitivity
Electronic Journal of Probability,;Vol. 21(2016)p. 45-
Artikel i vetenskaplig tidskrift
The spectrum and convergence rates of exclusion and interchange processes on the complete graph
Journal of Theoretical Probability,;Vol. 30(2017)p. 639-654
Artikel i vetenskaplig tidskrift
Denseness of volatile and nonvolatile sequences of functions
Stochastic Processes and their Applications,;Vol. 128(2018)p. 3880-3896
Artikel i vetenskaplig tidskrift
M. P. Forsström, J. E. Steif, Fortuin-Kastelyn representations for threshold Gaussian and stable vectors
En annan fråga som studeras i denna avhandling är följande. Antag att en kortlek blandas genom att man gång på gång byter plats på två slumpmässigt utvalda kort. Hur länge måste man fortsätta blanda innan kortleken kan anses vara väl blandad?
Slutligen undersöks i den sista artikeln när en slumpvektor är en så kallad färgprocess, dvs. när slumpvektor (X_1,X_2, \ldots, X_n) kan konstrueras genom att slumpvis och oberoende färglägga de olika delarna i en slumpvis vald partition av \{ 1,2,\ldots, n\}.
Fundament
Grundläggande vetenskaper
Ämneskategorier
Sannolikhetsteori och statistik
ISBN
978-91-7597-855-0
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4536
Utgivare
Chalmers
Pascal
Opponent: Gil Kalai, The Hebrew University of Jerusalem, Israel