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Arjun K. Gupta, Damodar N. Shanbhag, Truc T. Nguyen, J. T. Chen
August 19, 2009

### Abstract

In this expository article, we provide a unified version of the literature on certain aspects of cumulants of infinitely divisible distributions. In view of the spectral representations corresponding to the distributions referred to, with appropriate modifications to standard arguments, it follows that the cumulants of these distributions possess some nice features. Giving several illustrative examples, we attempt to explain as to what these mean. Although the emphasis of the paper is on univariate distributions, we briefly touch upon the multivariate case to verify whether this latter case has something new to offer to us on the subject.

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Hossein Hassani
August 19, 2009

### Abstract

In this paper, the sum of the sample autocorrelation function, found in many standard time series textbooks and software, at lag h ≥ 1 is considered. It is shown that this sum is always for any stationary time series with arbitrary length T ≥ 2. As an application of this quantity, it is shown that the sample spectral density of a stationary process fluctuates violently about the theoretical spectral density.

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Peng Zuoxiang, Tong Bin, Saralees Nadarajah
August 19, 2009

### Abstract

Motivated by Finner et al. (Commun. Stat., Theory Methods 37: 175–179, 2008), we investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of the Laplace and logistic distributions. In particular, we consider the asymptotic behavior of the ratio of the pdfs (cdfs) of the Laplace and Student's t -distributions (likewise for the logistic and Student's t -distributions) as the degrees of freedom parameter approaches infinity in an appropriate way. As by products, we obtain Mills' ratios for the Laplace and logistic distributions.

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Olga V. Aryasova, Andrey Yu. Pilipenko
August 19, 2009

### Abstract

We consider a Brownian motion on the plane with semipermeable membranes on n rays that have a common endpoint in the origin. We obtain the necessary and sufficient conditions for the process to reach the origin and we show that the probability of hitting the origin is equal to zero or one.

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Shrabani Banerjeer, Binayak S. Choudhury
August 19, 2009

### Abstract

In the present work, we discuss a random multi-step fixed point iteration with errors for a finite family of random asymptotically nonexpansive operators in the intermediate sense. The work is in line with the works on constructions of iterations for finding random fixed points of nonlinear random operators. The present work also generalizes some recently established results in Banach spaces.

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Petre Babilua, Elizbar Nadaraya, Albert Shatashvili, Grigol Sokhadze
August 19, 2009

### Abstract

For the Wiener integral, one property of inversion is established. This property is used for the construction of nonparametric statistical estimation of the unknown logarithmic derivative for the distribution of random processes, which is observed in Wiener noise.

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Anatoly A. Pogorui
August 19, 2009

### Abstract

In this paper, we study an asymptotic expansion for the distribution of a random motion of a particle driven by a Markov process in diffusion approximation. We show that the singularly perturbed equation of a Markovian random motion can be reduced to the regularly perturbed equation for the distribution of the random motion.

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Alexander S. Tkeshelashvili, Albert D. Shatashvili
August 19, 2009

### Abstract

The formula of integration by parts with respect to a random measure is established.