ICTP Mathematics Seminar - Wednesday 26 February, at 14:00 (Douglas Coates)

[ICTP Math Section]

M A T H E M A T I C S   S E M I N A R S  2020


Wednesday 26 February, at 14:00 hrs.


Douglas Coates (Exeter University, UK)


Semi-stable laws for wobbly intermittent maps


Abstract: A classical problem in probability theory is to understand the 
behaviour of centred and scaled sums of independent identically 
distributed (iid) random variables. A well known result in this area is 
the central limit theorem which loosely states that if X_1, X_2, ... is 
an iid sequence of mean 0 random variables of finite variance then 
(1/\sqrt{n}) \sum_{j=1}^{n} X_j will converge in distribution to a 
Gaussian random variable. Generalisations of the central limit theorem 
exists for the case that the variance of the X_i is no longer finite, in 
this case the scaling 1/\sqrt{n} is altered and the limiting 
distribution may no longer be Gaussian. One may generalise even further 
and consider all weak limit points of the sequence (1/A_n)(\sum_{j=1}^n 
X_j). It turns out that under some very mild restrictions all the limit 
points of this sequence have distribution belonging to the same class - 
namely they are semi-stable. In this talk we will be interested in what 
happens when the iid sequence X_1, X_2, ...  is replaced by a 
deterministic one. We will take a particular map T:[0,1] \to [0,1] (a so 
called wobbly intermittent map) and a function u:[0,1] \to R and replace 
the sequence X_1, X_2, ... with the sequence u, u \circ T, u \circ T^2, 
.... I will present recent work joint with M. Holland and D. Terhesiu in 
which we show that for all functions u belonging to wide class the 
sequence u, u \circ T, u \circ T^2, ... will satisfy semi-stable limit 
theorems.


VENUE:  Luigi Stasi Seminar Room (ICTP Leonardo da Vinci Building, first 
floor)