Common Contracts

Library Declaration and Deposit Agreement

February 5th, 2007

• Filed
  February 5th, 2007

Let P have density with respect to the Lebesgue measure and take a sequence of measures Pn defined on a common space X (where we do not assume that Pn is the empirical measure). Then (Pn, fn) → (P, f ) in TLp is equivalent to weak convergence of measures (due to the first term) and ǁµ(n) ◦ Tn − µǁLp(X;P ) → 0 (due to the second term), see [69, Proposition 3.6]. Since we we will always work with the empirical measure then with probability one Pn converges weakly to P . Hence the first term plays no role in this thesis and so is not included.