We show that the weak infinitesimal generator of a class of Markov processes acts on bounded continuous functions with bounded continuous second derivative as a singular integral with respect to the orthogonality measure of the explicit family of polynomials.

Free quadratic harness is a Markov process from the class of quadratic harnesses, i.e. processes with linear regressions and quadratic conditional variances. The process has recently been constructed for a restricted range of parameters in Bryc et al. (2010) [7] using Askey-Wilson...

We give an elementary construction of a time-invertible Markov process which is discrete except at one instance. The process is one of the quadratic harnesses studied in Bryc and Wesolowski [2005. Conditional moments of q-Meixner processes. Probab. Theory Related Fields 131, 415-441 <arxiv.org/abs/math.PR/0403016>], Bryc et...</arxiv.org/abs/math.pr/0403016>

In Bryc (Ann. Probab., (1998), to appear), we determined one-dimensional distributions of a stationary field with linear regressions (1) and quadratic conditional variances (2) under a linear constraint (7) on the coefficients of the quadratic expression (3). In this paper, we show that for...

We consider pairs of random variables X,Y with the property that regressions E(XnY), E(YnX) are polynomial. We show that the leading terms of these polynomials and the marginal distributions determine uniquely the bivariate distribution.

We show that the large deviation principle with respect to the weak topology holds for the empirical measure of any stationary continuous-time Gaussian process with continuous vanishing at infinity spectral density. We also point out that large deviation principle might fail in both continuous...

We point out that under a suitable regularity condition the central limit theorem can be obtained as a consequence of the large deviation principle.

For the sum of squares of an autoregressive system, the large deviation principle with the explicit rate function is established.

The behavior of the conditional expectation Es{;Xvb;Ys}; under a small perturbation Z of the conditioning random variable Y is analyzed. We show that if Y and Z are independent then Es{;Xvb;Y + [var epsilon]Zs}; converges to Es{;Xvb;Ys}; in mean as [var epsilon] -- 0 for all integrable X, provided...

We prove the large deviation principle for the arithmetic means of a uniform strong mixing stationary sequence which has either fast enough [phi]-mixing rate or is [psi]-mixing.