On embedding set functions into covariance functions

Abstract:

We consider any continuous hermitian kernel $M(\Delta ,\Delta ’)$ on $\mathcal {P} \times \mathcal {P}$ where $\mathcal {P}$ is the prering of intervals of [0,1]. Conditions on M are given to find an interval covariance function $K(\Delta ,\Delta ’)$ so that $K(\Delta ,\Delta ’) = M(\Delta ,\Delta ’)$ for all nonoverlapping $\Delta$ and $\Delta ’$ in $\mathcal {P}$. The problem is solved by first treating finite hermitian matrices A and finding a positive definite matrix B so that ${b_{ij}} = {a_{ij}},i \ne j$, so that tr B is minimized. Using natural correspondence between interval covariance functions and stochastic processes, a decomposition theorem is derived for stochastic processes of bounded quadratic variation into an orthogonal process and a process having minimal quadratic variation.