Browsing by Subject "covariance"

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Access status: Open Access ,
Decomposition of Gaussian processes, and factorization of positive definite kernels
(Wydawnictwa AGH, 2019) Jørgensen, Palle E.T.; Tian, Feng
We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.
Access status: Open Access ,
Finitely additive functions in measure theory and applications
(Wydawnictwa AGH, 2024) Alpay, Daniel; Jørgensen, Palle E.T.
In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory, in particular, to the study of $\mu$-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures $\mu$ , and to adjoints of composition operators.