This paper introduces vector (multivariate, or multiple) χ2and Rayleigh random functions or random fields on a spatial, temporal, or spatio-temporal index domain, and explores their basic properties. Formulated as a sum of squares of independent Gaussian random functions, a vector χ2random function has an interesting feature that its finite-dimensional Laplace transforms are not determined by its own covariance matrix, but by that of the underlying Gaussian one. With the conditionally negative definite matrix as an important building block, this paper constructs a class of vector χ2random functions, from whose covariance matrices one can easily identify those of the underlying Gaussian one so that the resulting vector χ2random function can be easily simulated and analyzed.

This correspondence propose rational-type covariance functions for nonstationary Gaussian stochastic processes or random fields, which are rational functions of given variograms, typically have long range dependence, and include some stationary covariance functions as special cases.

This paper introduces three spatio-temporal stationary random fields with simple correlation functions and rational spectral densities. The correlation functions have similar forms, but are defined on different space-time domains or related to different spatial norms. Positive and negative correlations are allowed, and the domain of certain model parameters may rely on the spatial dimensional parameter so that the model may be available in a limited space or in all dimensions. The distinctions are discussed among the three basic correlation models. Moreover, families of spatio-temporal stationary random fields are derived by randomizing the spatial and temporal coordinates of the base random fields. The resulting correlation functions are semiparametric, and include separable spatio-temporal models as special cases

MOTIVATION: Most phylogenetic methods assume that the sequences of nucleotides or amino acids have evolved under stationary, reversible and homogeneous conditions. When these assumptions are violated by the data, there is an increased probability of errors in the phylogenetic estimates. Methods to examine aligned sequences for these violations are available, but they are rarely used, possibly because they are not widely known or because they are poorly understood. RESULTS: We describe and compare the available tests for symmetry of k-dimensional contingency tables from homologous sequences, and develop two new tests to evaluate different aspects of the evolutionary processes. For any pair of sequences, we consider a partition of the test for symmetry into a test for marginal symmetry and a test for internal symmetry. The proposed tests can be used to identify appropriate models for estimation of evolutionary relationships under a Markovian model. Simulations under more or less complex evolutionary conditions were done to display the performance of the tests. Finally, the tests were applied to an alignment of small-subunit ribosomal RNA sequences of five species of bacteria to outline the evolutionary processes under which they evolved. AVAILABILITY: Programs written in R to do the tests on nucleotides are available from http://www.maths.usyd.edu.au/u/johnr/testsym/