<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field> <Text-field style="Normal" layout="Normal">Abstract: The algorithm constructs natural statistics of a spectral sample, by using convolutions on the symmetric group and the Weingarten function. These statistics are unbiased estimators of cumulants of traces. </Text-field> <Text-field style="Normal" layout="Normal"> Application Areas/Subject: Computational statistics</Text-field> <Text-field style="Normal" layout="Normal">Keyword: Random matrix, cumulant of traces, polykays</Text-field>

This Maple worksheet accompanies the papers:

(2013) Di Nardo E., McCullagh P., Senato D. Natural statistics for spectral samples. Annals of Statistics. 41(2), 982-1004. http://arxiv.org/abs/1302.5892

Spectral k-statistics

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E. Di Nardo*

elvira.dinardo@unibas.it

http://www.unibas.it/utenti/dinardo/home.html;

Tel: +39 0971205890, Fax: +39 0971205896

G. Guarino**

giuseppe.guarino@rete.basilicata.it

* Dipartimento di Matematica e Informatica, Universit\303\240 degli Studi della Basilicata,Viale dell'Ateneo Lucano n.10, 85100 Potenza, Italy

**Medical Scool, Universit\303\240 del Sacro Cuore (Rome branch), Largo Agostino Gemelli n.8, 00168 Roma, Italy

<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field> <Text-field style="Normal" layout="Normal">Abstract: The algorithm constructs natural statistics of a spectral sample, by using convolutions on the symmetric group and the Weingarten function. These statistics are unbiased estimators of cumulants of traces. </Text-field> <Text-field style="Normal" layout="Normal"> Application Areas/Subject: Computational statistics</Text-field> <Text-field style="Normal" layout="Normal">Keyword: Random matrix, cumulant of traces, polykays</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Initialization</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JFEocmVzdGFydEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvRjBRJ25vcm1hbEYnLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic=

QygtSSV3aXRoRzYiNiZJKWNvbWJpbmF0R0YlSSRDaGlHRiVJKnBhcnRpdGlvbkdGJUkocGVybXV0ZUdGJSIiIi1GJDYlSSZncm91cEdGJUkoaW52cGVybUdGJUkpbXVscGVybXNHRiVGKy1GJDYkSSpMaXN0VG9vbHNHRiVJKEZsYXR0ZW5HRiVGKw==

NyVJJENoaUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IkkqcGFydGl0aW9uR0YnSShwZXJtdXRlR0Yn

NyRJKGludnBlcm1HNiJJKW11bHBlcm1zR0Yk

NyNJKEZsYXR0ZW5HNiI=

<Text-field style="Heading 1" layout="Heading 1">Background</Text-field>

<Text-field style="Heading 2" layout="Heading 2"> Consruction of Schur function</Text-field>

The procedure Sch takes in input an integer partition and returns the Schur polynomial in N indeterminates all evaluated in 1.

QyQ+SSRTY2hHNiJmKjYkSSdsYW1iZGFHRiVJIk5HRiU2I0kjbXVHRiVGJUYlQyU+OCQtJkkpY29tYmluYXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kpY29uanBhcnRHRiU2IzkkLUknTWF0cml4R0YlNiUtSSVub3BzR0YzNiNGLkY8Zio2JEkiaUdGJUkiakdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjaWZHRjM2JTIsKCZUJUY3IiIiRjhGTTklISIiIiIhRlAtSSliaW5vbWlhbEdGJTYkVCRGSkYlRiU2JkYpRk5GK0YuLUknZXhwYW5kR0YzNiMtJkknbGluYWxnR0YlNiNJJGRldEdGJTYjSSIlR0YlRiVGJUYlRk8=

Example: for the partition (1,2,3) of the integer 6

LUkkU2NoRzYiNiQ3JSIiIiIiIyIiJEkiTkdGJA==

LCgqJEkiTkc2IiIiJyMiIiIiI1gqJEYkIiIlIyEiIiIiKiokRiQiIiMjRitGKQ==

for the partition (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiQtSSVtc3VwR0YkNiUtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUY3NiRRIjJGJ0Y6LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y6RitGOkYrRjo=,2) of the integer 4

LUkkU2NoRzYiNiQ3JSIiIkYnIiIjSSJOR0Yk

LCoqJEkiTkc2IiIiJSMiIiIiIikqJEYkIiIkIyEiIkYmKiRGJCIiIyNGLUYpRiQjRihGJg==

<Text-field style="Heading 2" layout="Heading 2"> Weingarten function<Equation executable="false" style="2D Comment" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>

The procedure Wg takes in input an integer partition and returns the Weingarten function as a rational function in N.

The algorithm makes use of Schur polynomials and the character of the symmetric group.

QyQ+SSNXZ0c2ImYqNiRJI211R0YlSSJOR0YlNiRJInFHRiVJJHVub0dGJUYlRiVDJT44JC1JJGFkZEclKnByb3RlY3RlZEc2JEkieEdGJS9GNDkkPjglNyMtSSIkR0YyNiQiIiJGLy1JJ2ZhY3RvckdGJTYjKiYtRjE2JCooLUkkQ2hpRzYkRjJJKF9zeXNsaWJHRiU2JEknbGFtYmRhR0YlRjgiIiMtRkY2JEZKRjZGPS1JJFNjaEdGJTYkRko5JSEiIi9GSi1JKnBhcnRpdGlvbkdGJTYjRi9GPS1JKmZhY3RvcmlhbEdGMkZWISIjRiVGJUYlRlI=

Example: for the partition (1,2,3) of the integer 6

LUkjV2dHNiI2JDclIiIiIiIjIiIkSSJOR0Yk

LCQqNiwmKiRJIk5HNiIiIiNGKCIjOCIiIkYqRiYhIiIsJkYmRioiIiZGKkYrLCZGJkYqIiIlRipGKywmRiZGKkYoRipGKywmRiZGKkYqRiohIiMsJkYmRipGK0YqRjIsJkYmRipGMkYqRissJkYmRiohIiVGKkYrLCZGJkYqISImRipGK0Yr

Example: for the partition (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiQtSSVtc3VwR0YkNiUtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUY3NiRRIjJGJ0Y6LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y6RitGOkYrRjo=,2) of the integer 4

LUkjV2dHNiI2JDclIiIiRiciIiNJIk5HRiQ=

LCQqLCwmSSJORzYiIiIiIiIkRichIiIsJkYlRidGJ0YnRiksJkYlRidGKUYnRiksJkYlRichIiRGJ0YpRiVGKUYp

<Text-field style="Heading 1" layout="Heading 1">Spectral k-statistics</Text-field>

<Text-field style="Heading 2" layout="Heading 2">The Maple routines</Text-field>

<Text-field style="Text" size="14" bold="true" family="Serif" layout="Heading 3"> Some details on secondary Maple routines</Text-field>

The procedure compldisjcyc takes as input a permutation and returns its decomposition in disjoint cycles.

In the output there are also the fixed points;

QyQ+SS1jb21wbGRpc2pjeWNHNiJmKjYjSSJhR0YlNiRJInZHRiVJIlNHRiVGJUYlQyU+OCQtSShjb252ZXJ0RyUqcHJvdGVjdGVkRzYkOSRJKGRpc2pjeWNHRiU+OCUtSSZtaW51c0dGMTYkPCMtSSNvcEdGMTYjRjM8Iy1JJHNlcUdGMTYkLUY8NiNJImNHRiUvRkRGLjckLUZANiQ3I0kiaUdGJS9GSkY2LUY8NiNGLkYlRiVGJSEiIg==

Example: for the permutation which fix 1 and 4 and switch 2 and 3

LUktY29tcGxkaXNqY3ljRzYiNiM3JiIiIiIiJCIiIyIiJQ==

NyU3IyIiIjcjIiIlNyQiIiMiIiQ=

Example: for the permutation which sends 1 in 2, 2 in 3, 3 in 4 and 4 in 1

LUktY29tcGxkaXNqY3ljRzYiNiM3JiIiIyIiJCIiJSIiIg==

NyM3JiIiIiIiIyIiJCIiJQ==

Example: for the identity permutations

LUktY29tcGxkaXNqY3ljRzYiNiM3JiIiIiIiIyIiJCIiJQ==

NyY3IyIiIjcjIiIjNyMiIiQ3IyIiJQ==

JSFH

The procedure tipo takes as input a permutation in disjoint cycles and returns its cycle type, that is how many cycles of each length are present in the cycle decomposition of the permutation.

QyQ+SSV0aXBvRzYiZipGJTYkSSJuR0YlSSJ2R0YlRiVGJUMlQCUvOSMiIiI+OCQtSSRtYXhHJSpwcm90ZWN0ZWRHNiMtSSRzZXFHRjM2JC1JI29wR0YzNiNJInhHRiUvRjsmOSI2I0YuPkYwJkY+NiMiIiM+OCUtSSVzb3J0R0YzNiM3Iy1GNjYkLUklbm9wc0dGMzYjSSJjR0YlL0ZPRj03JC1JIiRHRjM2JEYuLCZGMEYuLUkkYWRkR0YzNiRGOy9GO0ZFISIiLUY5NiNGRUYlRiVGJUZa

Examples: for the permutation which fix 1 and 4 and switch 2 and 3

LUkldGlwb0c2IjYjNyU3JCIiIiIiJTcjIiIjNyMiIiQ=

NyUiIiJGIyIiIw==

Examples: for the permutation which sends 1 in 2, 2 in 3, 3 in 4 and 4 in 1

LUkldGlwb0c2IjYjNyM3JiIiIiIiIyIiJCIiJQ==

NyMiIiU=

Examples: for the identity permutation

LUkldGlwb0c2IjYkNyIiIiU=

NyYiIiJGI0YjRiM=

<Text-field style="Text" size="14" bold="true" family="Serif" layout="Heading 3"> The master function</Text-field>

The procedure CXX takes as input a permutation and returns the formula (5.6) in Theorem 5.2, see [1]. This formula corresponds to the convolution between products of traces of a spectral sample X and the inverse of a function giving the spectral sample size powered by the number of disjoint cycles.

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

Examples: for the permutation which fix 1 and 4 and switch 2 and 3

LUkkQ1hYRzYiNiM3JTckIiIiIiIlNyMiIiM3IyIiJA==

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

Examples: for the permutation which sends 1 in 2, 2 in 3, 3 in 4 and 4 in 1

LUkkQ1hYRzYiNiM3IzcmIiIjIiIkIiIlIiIi

LDIqMCwmSSJORzYiIiIiIiIkRichIiIsJkYlRiciIiNGJ0YpLCZGJUYnRidGJ0YpLCZGJUYnRilGJ0YpLCZGJUYnISIjRidGKSwmRiVGJyEiJEYnRiktSSJFR0YmNiMqJi1JI1RyR0YmNiNJIlhHRiZGKy1GNzYjKiRGOUYrRidGJyIjNSoyRiRGKUYqRilGLEYpRi1GKUYuRilGMEYpRiVGKS1GMzYjKiRGOkYrRichIiYqMkYkRilGKkYpRixGKUYtRilGLkYpRjBGKUYlRiktRjM2IyomRjZGJy1GNzYjKiRGOUYoRidGJyEjPyouRiRGKUYsRilGLUYpRjBGKUYlRilGP0YnRi8qLkYkRilGLEYpRi1GKUYwRilGJUYpRkRGJyEiJSoyRiRGKUYqRilGLEYpRi1GKUYuRilGMEYpRiVGKS1GMzYjKiRGNiIiJUYnRkIqMEYkRilGKkYpRixGKUYtRilGLkYpRjBGKS1GMzYjLUY3NiMqJEY5RlJGJ0YnKjJGJEYpRipGKUYsRilGLUYpRi5GKUYwRilGJUYrRlRGJ0Yn

The procedure eTr takes as input the output of the procedure CXX and replaces traces with power sums indexed by their powers.

QyQ+SSRlVHJHNiJmKjYjSSJ5R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUmSSJTR0YlNiMtSSdkZWdyZWVHJSpwcm90ZWN0ZWRHNiM5JEYlRiVGJSEiIg==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

Example: for the permutation which sends 4 in 1, with fixed 2 and 3

LUklZUNYWEc2IjYjNyM3JiIiJSIiIyIiJCIiIg==

KjIsLCokJkkiU0c2IjYjIiIiIiIlISImKihGJSIiIyZGJjYjRi1GKUkiTkdGJ0YpIiM1KigsJiokRjBGLSEiJUY1RilGKSZGJjYjIiIkRilGJUYpRikqJiwmRjhGKUY0ISIjRilGLkYtRikqJiwmKiRGMEY4RilGMEYpRikmRiY2I0YqRilGKUYpLCZGMEYpRjhGKSEiIiwmRjBGKUYtRilGQiwmRjBGKUYpRilGQiwmRjBGKUZCRilGQiwmRjBGKUY7RilGQiwmRjBGKSEiJEYpRkJGMEZC

Example: for the permutation which sends 1 in 2, 2 in 3, 3 in 4 and 4 in 1

LUklZUNYWEc2IjYjNyM3JiIiIyIiJCIiJSIiIg==

LCQqMiwsKiQmSSJTRzYiNiMiIiIiIiUhIiYqKEYmIiIjJkYnNiNGLkYqSSJOR0YoRioiIzUqKCwmKiRGMUYuISIlRjZGKkYqJkYnNiMiIiRGKkYmRipGKiomLCZGOUYqRjUhIiNGKkYvRi5GKiomLCYqJEYxRjlGKkYxRipGKiZGJzYjRitGKkYqRiosJkYxRipGOUYqISIiLCZGMUYqRi5GKkZDLCZGMUYqRipGKkZDLCZGMUYqRkNGKkZDLCZGMUYqRjxGKkZDLCZGMUYqISIkRipGQ0YxRkMiIic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Conclusions</Text-field>

This algorithm extends the symmetric functions k-statistics and polykays to spectral sampling. Spectral samples are eigenvalues of freely randomized classical sample. The notion of freely randomized classical sample has been introduced for the first time in [1].

<Text-field style="Heading 1" layout="Heading 1">References</Text-field>

1] Di Nardo E., McCullagh P., Senato D. (2013) Natural statistics for spectral samples. Annals of Statistics. 41(2), 982-1004. http://arxiv.org/abs/1302.5892

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=