abstract:
For over thirty
years, there has been a systematic method in theoretical physics for
calculating the detailed properties of scaling limits of models such as
self-avoiding walk in four dimensions. These calculations usually involve
the following steps
(1) write the model as a perturbation of a Gaussian integral
(2) apply the renormalisation group to determine a simplified form of the
perturbation which is equivalent in the scaling limit.
(3) calculate on the basis of this simplified form.
This gives rise to the important idea that only a limited class of simple
models needs to be studied in order to classify scaling limits. However it
is difficult to carry out step (2) rigorously, except for hierarchical
lattices (leaves of trees). I will review some modest successes within this
program for models on hierarchical and Euclidean lattices.
abstract:
Stochastic chains with memory of
variable length constitute an interesting family of stochastic chains of
infinite order on a finite alphabet. The idea is that for each past, only a
finite suffix of the past, called context, is enough to predict the next
symbol. These models were first introduced in the information theory
literature by Rissanen (1983) as a universal tool to perform data
compression. Subsequently, they have been used to model up scientific data
in areas as different as biology, linguistics and music.
In recent years chains with memory of variable length received a lot of
attention in the statistics literature, with several papers dedicated to the
study of the properties of the algorithm Context and other estimators of the
probabilistic context tree defining the chain. But not much has been done to
better understand the probabilistic structure of these chains. Not even the
basic problem of the existence of these chains when the tree of contexts is
unbounded has been properly addressed.
In my talk I will present a new way to perform a perfect simulation of a
chain with memory of variable length. This will imply the existence and
uniqueness of the stationary chain. This will also provide an upper bound
for the rate at which the chain converges to the stationary regime. The
success of the procedure is assured by a new type of condition on the rate
at which the length of the contexts grow.