Financial models with long-tailed distributions and volatility clustering

2013-10-16T12:45:33Z

195.37.142.72: /* Infinitely divisible distributions */ missing power

In [[linear algebra]], a '''Moore matrix''', introduced by {{harvs|txt|authorlink=E. H. Moore|first=E. H.| last=Moore|year=1896}}, is a [[matrix (math)|matrix]] defined over a [[finite field]]. When it is a square matrix its [[determinant]] is called a '''Moore determinant''' (this is unrelated to the [[Moore determinant of a quaternionic Hermitian matrix]]). The Moore matrix has successive powers of the [[Frobenius automorphism]] applied to the first column, so it is an ''m'' × ''n'' matrix

:<math>M=\begin{bmatrix}
\alpha_1 & \alpha_1^q & \dots & \alpha_1^{q^{n-1}}\\
\alpha_2 & \alpha_2^q & \dots & \alpha_2^{q^{n-1}}\\
\alpha_3 & \alpha_3^q & \dots & \alpha_3^{q^{n-1}}\\
\vdots & \vdots & \ddots &\vdots \\
\alpha_m & \alpha_m^q & \dots & \alpha_m^{q^{n-1}}\\
\end{bmatrix}</math>
or
:<math>M_{i,j} = \alpha_i^{q^{j-1}}</math>
for all indices ''i'' and ''j''. (Some authors use the [[transpose]] of the above matrix.)

The Moore determinant of a square Moore matrix (so ''m'' = ''n'') can be expressed as:

:<math>\det(V) = \prod_{\mathbf{c}} \left( c_1\alpha_1 + \cdots + c_n\alpha_n \right), </math>

where '''c''' runs over a complete set of direction vectors, made specific by having the last non-zero entry equal to 1, i.e.

:<math>\det(V) = \prod_{1 \le i \le n} \prod_{c_1, \dots, c_{i-1}} \left( c_1\alpha_1 + \cdots + c_{i-1}\alpha_{i-1} + \alpha_i \right). </math>

In particular the Moore determinant vanishes if and only if the elements in the left hand column are [[linearly dependent]] over the finite field of order ''q''. So it is analogous to the [[Wronskian]] of several functions.

Dickson used the Moore determinant in finding the [[modular invariant of a group|modular invariants]] of the [[general linear group]] over a finite field.

==See also==
* [[Alternant matrix]]
* [[Vandermonde determinant]]
* [[List of matrices]]

==References==

*{{Citation | last1=Dickson | first1=Leonard Eugene | author1-link=Leonard Eugene Dickson | editor1-last=Magnus | editor1-first=Wilhelm | editor1-link=Wilhelm Magnus | title=Linear groups: With an exposition of the Galois field theory | origyear=1901 | url=http://www.archive.org/details/lineargroupswith00dickuoft | publisher=[[Dover Publications]] | location=New York | series=Dover Phoenix editions | isbn=978-0-486-49548-4 | mr=0104735 | year=1958}}
* {{cite book | authorlink=David Goss | author=David Goss | title=Basic Structures of Function Field Arithmetic | year=1996 | publisher=[[Springer Verlag]] | isbn=3-540-63541-6}} Chapter 1.
*{{Citation | last1=Moore | first1=E. H. | author1-link=E. H. Moore | title=A two-fold generalization of Fermat's theorem. | doi=10.1090/S0002-9904-1896-00337-2 | jfm=27.0139.05 | year=1896 | journal=American M. S. Bull. | volume=2 | pages=189–199}}

[[Category:Matrices]]
[[Category:Determinants]]

{{Linear-algebra-stub}}

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Financial models with long-tailed distributions and volatility clustering