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In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for ‘moments’ of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical discrete distributions, occupancy problems in some generalized allocation schemes and two models related to negative multinomial distribution.
The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.
Matsumoto and Yor have recently discovered an interesting invariance property
of a product of the generalized inverse Gaussian and gamma distributions. In this
paper we obtain: (1) a complete regression version of its converse; (2) a converse
to the matrix variate Matsumoto–Yor property which extends an earlier result. Of
independent interest is a functional equation for matrix valued functions, which has
been solved in the course of investigation of the second problem.
The classical martingale characterizations of the Poisson process were obtained for point process or purely discontinuous martingale i.e. under additional assumptions on properties of trajectories. Here our aim is to search for related characterizations without relying on properties of trajectories. Except for a new martingale characterization, results based on conditional moments jointly involving the past and the nearest future are presented.
In a photoluminescence and surface photovoltage study of porous silicon films with crystallite dimensions assessed with the Atomic Force Microscope, we have found cases when the blue shifts of the luminescence spectrum and the optical absorption edge take place upon increasing crystallite dimensions, which is contrary to quantum size effects. Fourier transform infrared spectroscopy analysis of these samples shows significant differences in hydrogen and oxygen bonding, which imply that the origin of the luminescence is of chemical nature. Our results show that porous silicon luminescence is not a consequence of one mechanism, but rather results from several mechanisms with contributions depending on the chemistry and structure of porous silicon.