Randomly Weighted Series of Contractions in Hilbert Spaces

G. Peskir, D. Schneider, M. Weber
1996 Mathematica Scandinavica  
Conditions are given for the convergence of randomly weighted series of contractions in Hilbert spaces. It is shown that under these conditions the series converges in operator norm outside of a (universal) null set simultaneously for all Hilbert spaces and all contractions of them. The conditions obtained are moreover shown to be as optimal as possible. The method of proof relies upon the spectral lemma for Hilbert space contractions (which allows us to imbed the initial problem into a setting
more » ... blem into a setting of Fourier analysis), the standard Gaussian randomization (which allows us further to transfer the problem into the theory of Gaussian processes), and finally an inequality due to Fernique [3] (which gives an estimate of the expectation of the supremum of the Gaussian (stationary) process over a finite interval in terms of the spectral measure associated with the process by means of the Bochner theorem). As a consequence of the main result we obtain: Given a sequence of independent and identically distributed mean zero random variables fZ k g k1 defined on (; F;P) satisfying EjZ 1 j 2 < 1 , and > 1=2 , there exists a (universal) P -null set N 3 2 F such that the series: 1 X k=1 Z k (!) k T k converges in operator norm for all ! 2 n N 3 , whenever H is a Hilbert space and T is a contraction in H . in quadratic mean, stationary, the spectral measure, the spectral representation theorem, orthogonal stochastic measure, the covariance function, the Bochner theorem, the Herglotz theorem, nonnegative definite, the dilation theorem of Sz.-Nagy, Kolmogorov's three series theorem, Szidon's theorem on lacunary trigonometrical series. © goran@imf.au.dk
doi:10.7146/math.scand.a-12606 fatcat:mtkfyhyn7ncqbjoeupbvnehgne