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Cayley-Hamilton theorem for left eigenvalues of 3 × 3 quaternionic matrices

100%
Macías-Virgós E., Pereira-Sáez M. J.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
We prove that any quaternionic matrix of order n ≤3 admits a characteristic function, whose roots are the left eigenvalues, that satisfes Cayley-Hamilton theorem.
2
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Simple fractions and linear decomposition of some convolutions of measures

100%
Misiewicz J., Cooke R.
Discussiones Mathematicae Probability and Statistics
|
2001
|
tom 21
|
nr 2
149-157
EN
Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a}(ξ) = [a/(h(ξ) + a)]$, where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain that convolutions of measures $μ_{a}$ with $μ̂_{a}(ξ) = φ_{a}(ξ)$ are linear combinations of powers of such measures. This can simplify calculations. It is interesting that this simplification uses signed measures since coefficients of linear combinations can be negative numbers. All the results of this paper except Proposition 1 remain true if we replace probability measures with complex valued measures with finite variation, and replace the characteristic function with Fourier transform.
3
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Near-exact distributions for the generalized Wilks Lambda statistic

88%
Grilo L., Coelho C.
Discussiones Mathematicae Probability and Statistics
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2010
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tom 30
|
nr 1
53-86
EN
Two near-exact distributions for the generalized Wilks Lambda statistic, used to test the independence of several sets of variables with a multivariate normal distribution, are developed for the case where two or more of these sets have an odd number of variables. Using the concept of near-exact distribution and based on a factorization of the exact characteristic function we obtain two approximations, which are very close to the exact distribution but far more manageable. These near-exact distributions equate, by construction, some of the first exact moments and correspond to cumulative distribution functions which are practical to use, allowing for an easy computation of quantiles. We also develop three asymptotic distributions which also equate some of the first exact moments. We assess the proximity of the asymptotic and near-exact distributions obtained to the exact distribution using two measures based on the Berry-Esseen bounds. In our comparative numerical study we consider different numbers of sets of variables, different numbers of variables per set and different sample sizes.
4
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Geometrically strictly semistable laws as the limit laws

75%
Malinowski M.
Discussiones Mathematicae Probability and Statistics
|
2007
|
tom 27
|
nr 1-2
79-97
EN
A random variable X is geometrically infinitely divisible iff for every p ∈ (0,1) there exists random variable $X_p$ such that $X\stackrel{d}{=} ∑_{k=1}^{T(p)}X_{p,k}$, where $X_{p,k}$'s are i.i.d. copies of $X_p$, and random variable T(p) independent of ${X_{p,1},X_{p,2},...}$ has geometric distribution with the parameter p. In the paper we give some new characterization of geometrically infinitely divisible distribution. The main results concern geometrically strictly semistable distributions which form a subset of geometrically infinitely divisible distributions. We show that they are limit laws for random and deterministic sums of independent random variables.
5
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On some limit distributions for geometric random sums

63%
Malinowski M.
Discussiones Mathematicae Probability and Statistics
|
2008
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tom 28
|
nr 2
247-266
EN
We define and give the various characterizations of a new subclass of geometrically infinitely divisible random variables. This subclass, called geometrically semistable, is given as the set of all these random variables which are the limits in distribution of geometric, weighted and shifted random sums. Introduced class is the extension of, considered until now, classes of geometrically stable [5] and geometrically strictly semistable random variables [10]. All the results can be straightforward transfered to the case of random vectors in $ℝ^d$.
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