Boolean valued interpretation of Hilbert space theory

Hilbert Space-valued Mixingales

A Boolean-valued probability theory

New foundations for probability theory are suggested and investigated, derived from the formalized systems theory and using also some other branches of mathematical logic. The probability measures are supposed to take their values not in the unit interval of reals, but in a special kind of Boolean algebras. Some analogies as well as differences of this approach with respect to the classical Kol...

Zakai Equations for Hilbert Space Valued Processes*

versity A state process is described by either a discrete time Hilbert space valued process, or a stochastic differential equation in Hilbert space. The state is observed through a finite dimensional process. Using a change of measure and a Fusive theorem the Zakai equation is obtained in discrete or continuous time. A risk sensitive state estimate is also defined.

Elementary Hilbert Space Theory

A Hilbert Space is an inner product space that is complete. Let H be a Hilbert space and for f, g ∈ H, let 〈f, g〉 be the inner product and ‖f‖ = √ 〈f, f〉. (1.1) We will consider Hilbert spaces over C. Most arguments go through for Hilbert spaces over R, and the arguments are simpler. We assume that the reader is familiar with some of the basic facts about the inner product. Let us prove a few t...

Rate of convergence for Hilbert space valued processes

Consider a stationary, linear Hilbert space valued process. We establish Berry-Essen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of non-linear Bernoulli-shift sequences is also considered. If the sequence is m-dependent, the optimal rate (n/m)1/2 is reached. If the sequence is weakly geome...