## Partial exchangeability and related topics.

##### Abstract

Partial exchangeability is the fundamental building block in the subjective
approach to the probability of multi-type sequences which replaces the independence
concept of the objective theory.
The aim of this thesis is to present some theory for partially exchangeable
sequences of random variables based on well-known results for exchangeable
sequences.
The reader is introduced to the concepts of partially exchangeable events,
partially exchangeable sequences of random variables and partially exchangeable
o-fields, followed by some properties of partially exchangeable
sequences of random variables.
Extending de Finetti's representation theorem for exchangeable random
variables to hold for multi-type sequences, we obtain the following result
to be used throughout the thesis:
There exists a o-field, conditional upon which, an infinite partially exchangeable
sequence of random variables behaves like an independent sequence
of random variables, identically distributed within types.
Posing (i) a stronger requirement (spherical symmetry) and (ii) a weaker
requirement (the selection property) than partial exchangeability on the
infinite multi-type sequence of random variables, we obtain results related
to de Finetti's representation theorem for partially exchangeable sequences
of random variables.
Regarding partially exchangeable sequences as mixtures of independent and
identically distributed (within types) sequences, we (i) give three possible
expressions for the directed random measures of the partially exchangeable
sequence and (ii) look at three possible expressions for the o-field mentioned
in de Finetti's representation theorem.
By manipulating random measures and using de Finetti's representation
theorem, we point out some concrete ways of constructing partially exchangeable
sequences.
The main result of this thesis follows by extending de Finetti's represen.
tation theorem in conjunction with the Chatterji principle to obtain the
following result:
Given any a.s. limit theorem for multi-type sequences of independent random
variables, identically distributed within types, there exists an analogous
theorem satisfied by all partially exchangeable sequences and by all
sub-subsequences of some subsequence of an arbitrary dependent infinite
multi-type sequence of random variables, tightly distributed within types.
We finally give some limit theorems for partially exchangeable sequences of
random variables, some of which follow from the above mentioned result.