Conditional PASTA

Let Y be a stochastic process representing the state of a system and N a doubly stochastic Poisson process whose intensity varies with the state of a random environment represented by a stochastic process X. In this context a generalization of “PASTA” (Poisson Arrivals See Time Averages) is shown to be valid. Various applications of the result are given

Testing conditional independence using maximal nonlinear conditional correlation

In this paper, the maximal nonlinear conditional correlation of two random vectors $X$ and $Y$ given another random vector $Z$, denoted by $\rho_1(X,Y|Z)$, is defined as a measure of conditional association, which satisfies certain desirable properties. When $Z$ is continuous, a test for testing the conditional independence of $X$ and $Y$ given $Z$ is constructed based on the estimator of a weighted average of the form $\sum_{k=1}^{n_Z}f_Z(z_k)\rho^2_1(X,Y|Z=z_k)$, where $f_Z$ is the probability density function of $Z$ and the $z_k$'s are some points in the range of $Z$. Under some conditions, it is shown that the test statistic is asymptotically normal under conditional independence, and the test is consistent.Comment: Published in at http://dx.doi.org/10.1214/09-AOS770 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Conditional observability

For a quantum Hamiltonian H =H(p) the observability of the energies E may be robust (whenever all E are real at all p) or, otherwise, conditional. Using a pseudo-Hermitian family of N-state chain models H we discuss some generic properties of conditionally observable spectra.Comment: 21 pp incl. 10 fi