Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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.

Conditional moments of q-Meixner processes

We analyze certain stationary fields with linear regressions and quadratic conditional variances. This classic probabilistic problem leads somewhat unexpectedly to stationary Markov processes closely tied to non-commutative probability through the q-Hermite polynomials.

Stationary random fields 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.

Stationary Markov chains with linear regressions

Following a general theorem concerning the relationship between the first two conditional moments of some random variables and the distribution of the conditioning random vector, we derive differential equations which involve the conditional regression and covariance of the mean square derivative of the process {Wt} and the finite dimensional densities of this process We use these equations to define a subclass of ms differentiable processes having smooth conditional moments and finite dimensional densities and having the property that the first two conditional moments identify a given process completely Among these are processes having linear first moments We show that under some differentiability assumptions these processes have elliptically contoured distributions

Can the first two conditional moments identify a mean square differentiable process

In 2017, the presence of the fly Suillia gigantea (Meigen, 1830) was noted in Poland, after many years of research related to the ecology of insects associated with the fruiting bodies of hypogeous fungi. Finally, in 2020, after further studies, the distribution of the truffle fly in Poland was confirmed. Six adults were reared from larvae inhabiting the fruiting bodies of Burgundy truffle (Tuber aestivum Vittad. (1831)). The fungi were harvested in southern Poland. Morphological and genetic analyses of the insect specimens were performed. For the first time, the DNA sequence of this fly species was isolated. This is the first record of S. gigantea in Poland, although it has already been reported from neighbouring countries. The development of S. gigantea, also known as the truffle fly, is associated with hypogeous fungi, mainly belonging to the genus Tuber P. Micheli ex F.H. Wigg. (1780). The diptera larvae develop inside the fruiting bodies. This type of damage causes great losses in truffle production. Containment of these losses is of great interest to the truffle industry. Knowledge of truffle-inhabiting insects is crucial for the development of effective methods to protect truffle crops in Europe.

First molecular identification of Suillia gigantea (Meigen, 1830) (Diptera: Heleomyzidae).

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Can the first two conditional moments identify a mean square differentiable process

First molecular identification of Suillia gigantea (Meigen, 1830) (Diptera: Heleomyzidae).