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Asymptotics of infinitely divisible distributions onR
Siberian Mathematical Journal, 1990Let G be a probability distribution on \([0,\infty)\) with the properties: 1) \(G([x,\infty))>0\) for every \(x>0,\) 2) \(G^{2*}([x,\infty))/G([x,\infty))\to 2\int_{R}e^{\gamma u}dG(u)=\hat G(\gamma)\) as \(x\to \infty\), for some \(\gamma\geq 0,\) where * denotes convolution, 3) for every real number y, \[ G([x+y,\infty))/G([x,\infty))\to e^{- \gamma ...
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Infinitely Divisible Distributions
2014For every n, the normal distribution with expectation μ and variance σ 2 is the nth convolution power of a probability measure (namely of the normal distribution with expectation μ/n and variance σ 2/n). This property is called infinite divisibility and is shared by other probability distributions such as the Poisson distribution and the Gamma ...
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Infinitely Divisible Distributions
1975A distribution function F (x) and the corresponding c.f. f (t) are said to be infinitely divisible if for every positive integer n there exists a c.f. f n (t) such that $$f\left( t \right) = {\left( {{f_n}\left( t \right)} \right)^n}$$ (1.1)
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OnV-Infinitely Divisible Distributions
Theory of Probability & Its Applications, 1996\(V\)-infinite divisibility of a probability measure \(\mu\) on the real line means that, for each \(n\in\mathbb{N}\), there exist another measure \(\mu_n\) and a constant \(a_n\geq 0\) such that \(\mu\) can be written as the convolution product of the normal distribution with mean 0 and variance \(a_n\) and the \(n\)th convolution power of \(\mu_n ...
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Distribution of the Superposition of Infinitely Divisible Processes
Theory of Probability & Its Applications, 1958In this paper it is proved that for an arbitrary infinitely divisible process $\xi (t)$ and any non-negative infinitely divisible process $\eta (t)$ the distribution of their superposition $\xi (t) = \xi [\eta (t)]$ is also infinitely divisible. The corresponding spectral function $H(x)$ of that process (Levy function) is constructed. The second result
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On infinitely divisible generalized distributions in \(R_k\)
1975Artykuł w: Annales Universitatis Mariae Curie-Skłodowska. Sectio A, Mathematica. Vol. 27 (1973), s. 121-129 ; streszcz. pol., ros. ; Artykuł w: Annales Universitatis Mariae Curie-Skłodowska. Sectio A, Mathematica. Vol. 27 (1973), s. 121-129 ; streszcz. pol., ros.
Szynal, Dominik (1937- ) +1 more
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Cumulants of infinitely divisible distributions
ROSE, 2009Abstract 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 ...
Gupta, Arjun K. +3 more
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On the tails of infinitely divisible distributions
Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1974No abstract.
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On the infinite divisibility of the lognormal distribution
Scandinavian Actuarial Journal, 1977Summary In the present paper the author proves that the lognormal distribution is infinitely divisible. This is achieved by showing that the lognormal is the weak limit of a sequence of probability distributions all of which are generalized Γ-convolutions and thus infinitely divisible.
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On the unimodality of infinitely divisible distribution functions II
Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1978A great deal of work has been done during the last 45 years concerning the unimodality of one-dimensional infinitely divisible distribution functions. Recently, a few results have been obtained for multivariate infinitely divisible distribution functions.
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