scholarly journals On Statistics of Permutations Chosen From the Ewens Distribution

2014 ◽  
Vol 23 (6) ◽  
pp. 889-913
Author(s):  
TATJANA BAKSHAJEVA ◽  
EUGENIJUS MANSTAVIČIUS
Keyword(s):  
Probability Measure ◽  
Sufficient Conditions ◽  
Random Permutation ◽  
Asymptotic Distributions ◽  
Additive Functions ◽  
Factorial Moments ◽  
Permutation Matrices ◽  
Poisson Limit ◽  
Number Of Cycles ◽  
Necessary And Sufficient

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial moments, we establish necessary and sufficient conditions for the weak convergence of distributions to discrete laws. More attention is paid to the Poisson limit distribution. The particular case of the number-of-cycles function is analysed in more detail. The results can be applied to statistics defined on random permutation matrices.

scholarly journals Asymptotic distributions of the number of restricted cycles in a random permutation

2009 ◽  
Vol 50 ◽  
Author(s):  
Tatjana Kargina
Keyword(s):  
Symmetric Group ◽  
Sufficient Conditions ◽  
Random Permutation ◽  
Value Distribution ◽  
Asymptotic Distributions ◽  
Necessary And Sufficient Conditions ◽  
Additive Functions ◽  
Number Of Cycles ◽  
Necessary And Sufficient

The value distribution of additive functions defined on the symmetric group with respect to the Ewens probability is examined. For the number of cycles with restricted lengths, we establish necessary and sufficient conditions under which the distributions converge weakly to a limit law.

scholarly journals The factorial moments of additive functions with rational argument

2006 ◽  
Vol 81 (3) ◽  
pp. 425-440
Author(s):  
J. Šiaulys ◽  
G. Stepanauskas
Keyword(s):  
Weak Convergence ◽  
Limit Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Additive Functions ◽  
Factorial Moments ◽  
Rational Argument ◽  
Necessary And Sufficient ◽  
The Given

AbstractWe consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that f(p) and f(1/p) ∈ {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and Halász's and Ruzsa's inequalities are used. We present a few examples of application of the given results to some sets of fractions.

scholarly journals Transposable and symmetrizable matrices

1980 ◽  
Vol 29 (4) ◽  
pp. 469-474 ◽  
Author(s):  
David McCarthy ◽  
Brendan D. McKay
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Permutation Matrix ◽  
Permutation Matrices ◽  
Symmetrizable Matrices ◽  
Necessary And Sufficient ◽  
Square Matrix

AbstractA square matrix A is transposable if P(RA) = (RA)T for some permutation matrices p and R, and symmetrizable if (SA)T = SA for some permutation matrix S. In this paper we find necessary and sufficient conditions on a permutation matrix P so that A is always symmetrizable if P(RA) = (RA)T for some permutation matrix R.

The Supremum of a Family of Additive Functions

1952 ◽  
Vol 4 ◽  
pp. 463-479 ◽  
Author(s):  
Israel Halperin
Keyword(s):  
Vector Space ◽  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Boolean Ring ◽  
Additive Functions ◽  
Semi Group ◽  
Additive Functionals ◽  
Banach Theorem ◽  
Necessary And Sufficient ◽  
Natural Way

Any system S in which an addition is defined for some, but not necessarily all, pairs of elements can be imbedded in a natural way in a commutative semi-group G, although different elements in S need not always determine different elements in G (see §2). Theorem 2.1 gives necessary and sufficient conditions in order that a functional p(x) on S can be represented as the su prémuni of some family of additive functionals on S, and one such set of conditions is in terms of possible extensions of p(x) to G. This generalizes the case with 5 a Boolean ring treated by Lorentz [4], Lorentz imbeds the Boolean ring in a vector space and this could be done for the general S; but we prefer to imbed S in a commutative semi-group and to give a proof (see § 1) generalizing the classical Hahn-Banach theorem to the case of an arbitrary commutative semigroup.

The power digraphs associated with generalized dihedral groups

2015 ◽  
Vol 07 (04) ◽  
pp. 1550057 ◽  
Author(s):  
Uzma Ahmad
Keyword(s):  
Dihedral Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dihedral Groups ◽  
Degree Structure ◽  
Number Of Cycles ◽  
Necessary And Sufficient ◽  
Power Mapping

We study the digraphs based on dihedral group [Formula: see text] by using the power mapping, i.e., the set of vertices of these digraphs is [Formula: see text] and the set of edges is [Formula: see text]. These are called the power digraphs and denoted by [Formula: see text]. The cycle and in-degree structure of these digraphs are completely examined. This investigation leads to the derivation of various formulae regarding the number of cycle vertices, the length of the cycles, the number of cycles of certain lengths and the in-degrees of all vertices. We also establish necessary and sufficient conditions for a vertex to be a cycle vertex. The analysis of distance between vertices culminates at different expressions in terms of [Formula: see text] and [Formula: see text] to determine the heights of vertices, components and the power digraph itself. Moreover, all regular and semi-regular power digraphs [Formula: see text] are completely classified.

scholarly journals Conditions for equality between Lyapunov and Morse decompositions

2015 ◽  
Vol 36 (4) ◽  
pp. 1007-1036 ◽  
Author(s):  
LUCIANA A. ALVES ◽  
LUIZ A. B. SAN MARTIN
Keyword(s):  
Probability Measure ◽  
Ergodic Theorem ◽  
Principal Bundle ◽  
Sufficient Conditions ◽  
Base Space ◽  
Morse Decomposition ◽  
Lyapunov Spectra ◽  
Multiplicative Ergodic Theorem ◽  
Morse Decompositions ◽  
Necessary And Sufficient

Let$Q\rightarrow X$be a continuous principal bundle whose group$G$is reductive. A flow${\it\phi}$of automorphisms of$Q$endowed with an ergodic probability measure on the compact base space$X$induces two decompositions of the flag bundles associated to$Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

scholarly journals The necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution

2019 ◽  
Vol 22 (1) ◽  
pp. 143-146
Author(s):  
Tran Loc Hung ◽  
Phan Tri Kien
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Sufficient Conditions ◽  
Laplace Distribution ◽  
Asymptotic Distributions ◽  
Necessary And Sufficient Conditions ◽  
Characteristic Functions ◽  
Necessary And Sufficient ◽  
Geometric Sums ◽  
Risk Processes

The geometric sums have been arisen from the necessity to resolve practical problems in ruin prob- ability, risk processes, queueing theory and reliability models, etc. Up to the present, the results related to geometric sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned domains of geometric attraction are still limitative. The main purpose of this article is to introduce concepts on the domain of geometric attraction of standard Laplace distribution. Using method of characteristic functions, the necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution are shown. In special case, obtained result is a weak limit theorem for geometric sums of independent and identically distributed random variables which has been well-known as the second central limit theorem. Furthermore, based on the obtained results of this paper, the analogous results for the domains of geometric attraction of exponential distribution and Linnik distribution can be established. More generally, we may extend results to the domain of geometric attraction of geometrically strictly stable distributions.      

scholarly journals On bounds for convergence rates in combinatorial strong limit theorems and its applications

2020 ◽  
Vol 65 (4) ◽  
pp. 688-698
Author(s):  
Andrei N. Frolov ◽  
Keyword(s):  
Limit Theorems ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Permutation ◽  
Independent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Set Of Permutations ◽  
Combinatorial Sums ◽  
Necessary And Sufficient

We find necessary and sufficient conditions for convergences of series of weighted probabilities of large deviations for combinatorial sums i Xniπn(i), where Xnij is a matrix of order n of independent random variables and (πn(1), πn(2), . . . , πn(n)) is a random permutation with the uniform distribution on the set of permutations of numbers 1, 2, . . . , n, independent with Xnij. We obtain combinatorial variants of results on convergence rates in the strong law of large numbers and the law of the iterated logarithm under conditions closed to optimal ones. We discuss applications to rank statistics.

A general form of the covering principle and relative differentiation of additive functions. II

1946 ◽  
Vol 42 (1) ◽  
pp. 1-10 ◽  
Author(s):  
A. S. Besicovitch
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Additive Functions ◽  
General Measure ◽  
General Sense ◽  
Measure Function ◽  
Concentric Circles ◽  
Almost All ◽  
Necessary And Sufficient ◽  
Convergent Sequences

In my paper under the same title, to which I shall refer in future as I‡, I generalized the Vitali covering principle from the case of Lebesgue measure to the case of any non-negative additive function. This allowed me to establish the relative differentiation of additive functions. The convergent sequences of sets in this generalized form of the covering principle were restricted to sequences of concentric circles, and therefore the differentiation arrived at was that in the symmetrical sense. In the present paper, I extend the principle to the case of any regular convergent sequences of covering sets; and then establish the relative differentiation of additive functions in the general sense, and in particular the differentiation of indefinite integrals with respect to any measure function. This problem has a complete solution. It is established that indefinite integrals are differentiable at almost all points. In the case of the general measure function, it is not true that the derivative is equal to the integrand at almost all points, but necessary and sufficient conditions are given under which this is true.

Sign in / Sign up

Export Citation Format

Share Document