TIGHTNESS CRITERIA FOR RANDOM MEASURES WITH APPLICATIONTO THE PRINCIPLE OF CONDITIONING IN HILBERT SPACES

Adam Jakubowski

Abstract: Suppose that is a sequence of random probability measures on a real and
separable Hilbert space such that, for each is a pointwisely convergent
convolution of some sequence of random measures. The sequence is
said to be if one can find random vectors such that the ”centered”
sequence is tight.

It is proved that for a shift-tight sequence there exists a ”progressively
measurable” centering which changes into a tight sequence.

As an application, Principle of Conditioning and Martingale Central Limit Theorem in a
Hilbert space are proved.