654 research outputs found
Expansions of one density via polynomials orthogonal with respect to the other
We expand the Chebyshev polynomials and some of its linear combination in
linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the
Al-Salam--Chihara polynomials and vice versa. We use these expansions to obtain
expansions of some densities, including q-Normal and some related to it, in
infinite series constructed of the products of the other density times
polynomials orthogonal to it, allowing deeper analysis and discovering new
properties. On the way we find an easy proof of expansion of the
Poisson--Mehler kernel as well as its reciprocal. We also formulate simple rule
relating one set of orthogonal polynomials to the other given the properties of
the ratio of the respective densities of measures orthogonalizing these
polynomials sets
Separation of the largest eigenvalues in eigenanalysis of genotype data from discrete subpopulations
We present a mathematical model, and the corresponding mathematical analysis,
that justifies and quantifies the use of principal component analysis of
biallelic genetic marker data for a set of individuals to detect the number of
subpopulations represented in the data. We indicate that the power of the
technique relies more on the number of individuals genotyped than on the number
of markers.Comment: Corrected typos in Section 3.1 (M=120, N=2500) and proof of Lemma
Markov processes with free Meixner laws
We study a time-non-homogeneous Markov process which arose from free
probability, and which also appeared in the study of stochastic processes with
linear regressions and quadratic conditional variances. Our main result is the
explicit expression for the generator of the (non-homogeneous) transition
operator acting on functions that extend analytically to complex domain.
The paper is self-contained and does not use free probability techniques
Stationary Markov chains with linear regressions
In a previous paper we determined one dimensional distributions of a
stationary field with linear regressions and quadratic conditional variances
under a linear constraint on the coefficients of the quadratic expression. In
this paper we show that for stationary Markov chains with linear regressions
and quadratic conditional variances the coefficients of the quadratic
expression are indeed tied by a linear constraint which can take only one of
the two alternative forms
Compound real Wishart and q-Wishart matrices
We introduce a family of matrices with non-commutative entries that
generalize the classical real Wishart matrices.
With the help of the Brauer product, we derive a non-asymptotic expression
for the moments of traces of monomials in such matrices; the expression is
quite similar to the formula derived in our previous work for independent
complex Wishart matrices. We then analyze the fluctuations about the
Marchenko-Pastur law. We show that after centering by the mean, traces of real
symmetric polynomials in q-Wishart matrices converge in distribution, and we
identify the asymptotic law as the normal law when q=1, and as the semicircle
law when q=0
Conditional moments of q-Meixner processes
We show that stochastic processes with linear conditional expectations and
quadratic conditional variances are Markov, and their transition probabilities
are related to a three-parameter family of orthogonal polynomials which
generalize the Meixner polynomials. Special cases of these processes are known
to arise from the non-commutative generalizations of the Levy processes.Comment: LaTeX, 24 pages. Corrections to published version affect formulas in
Theorem 4.
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