Some large and moderate deviations results for exchangeable sequences

Let (Ω,cal A,P) be a probability space and assume that the probability measure P has the following representation:

P(·) = intΘ P(θ,·)dm(θ)

where (Θ,cal M,m) is a probability space (with Θ a first countable topological space) and P(·,·) a probability transition function on Θ×cal A. aLet also, Yjspj=1infty be a sequence of random variables defined on (Ω,cal A,P) and taking values in an arbitrary measurable space (S,cal S) and assume that for each θinΘ, Yjspj=1infty is an i.i.d. sequence under P(θ,·). Let the empirical measures of Yjspj=1infty be defined by:

Ln=1over nsumspj=1nδYj

with δx the Dirac measure at the point x, and let νn=cal LP(Ln). a

Define:

Mn=nover bn(Ln-μ)

with μ=P o (Y1)-1 and nspn=1infty a positive real sequence such that:

bnover n1over2uildreln→inftyoverlongrightarrowinfty, bnover nuildreln→inftyoverlongrightarrow 0eqno(*)

and let ildeνn=cal LP(Mn).

In chapter 2 of this dissertation, we study Large Deviations for the sequence of probability measures νnspn=1infty. In chapter 3, a Moderate Deviations result with normalizing constants spn2over nspn=1infty, for the sequence ildeνspn=1infty is proved aNow, let (S,cal S) (R^d,cal Bd,dge1 and define

eqalign S0 & = 0cr Sj &= sumspi=1jYi, j=1,2,3,···

and denote by sn(t),t in [0,1] the polygonal line in R^d determined by the points (jover n, Sjover xn), j=0,1,2,···, n (trajectories of Yjspj=1infty), with xnspn=1infty a positive real sequence and let μn=cal LP=(sn(·)).

In chapter 4, we study Large Deviations for the sequence of probability measures μnspn=1infty, when xn = n. Finally, in chapter 5 we prove a Moderate Deviations result with normalizing constants xspn2over nspn=1infty, for the sequence μnspn=1infty when xn=bn and bn is as in (*)