Description

Classicalexamples of moreand more oscillatingreal-valued functions on a domain N ?of R are the functions u (x)=sin(nx)with x=(x, ..., x ) or the so-called n 1 1 n n+1 Rademacherfunctionson]0,1[, u (x)=r (x) = sgn(sin(2 x))(seelater3.1.4). n n They may appear as the gradients?v of minimizing sequences (v ) in some n n n?N variationalproblems. Intheseexamples, thefunctionu convergesinsomesenseto n ameasure µ on ? ×R, called Young measure. In Functional Analysis formulation, this is the narrow convergence to µ of the image of the Lebesgue measure on ? by ? ? (?, u (?)). In the disintegrated form (µ ), the parametrized measure µ n ? ? captures the possible scattering of the u around ?. n Curiously if (X ) is a sequence of random variables deriving from indep- n n?N dent ones, the n-th one may appear more and more far from the k ?rst ones as 2 if it was oscillating (think of orthonormal vectors in L which converge weakly to 0). More precisely when the laws L(X ) narrowly converge to some probability n measure, it often happens that for any k and any A in the algebra generated by X, ..., X, the conditional law L(XA) still converges to (see Chapter 9) 1 k n which means 1 C (R) ?(X (?))dP(?) d b n P(A) A R or equivalently, ? denoting the image of P by ? ? (?, X (?)), n X n (1l )d? (1l )d[P? ].