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.

Asymptotic behavior of the records of multivariate random sequences in a norm sense

2020 ◽  
Vol 70 (6) ◽  
pp. 1457-1468
Author(s):  
Haroon M. Barakat ◽  
M. H. Harpy
Keyword(s):  
Asymptotic Behavior ◽  
Weak Convergence ◽  
Sufficient Conditions ◽  
Record Values ◽  
Necessary And Sufficient Conditions ◽  
Random Sequences ◽  
Necessary And Sufficient

AbstractIn this paper, we investigate the asymptotic behavior of the multivariate record values by using the Reduced Ordering Principle (R-ordering). Necessary and sufficient conditions for weak convergence of the multivariate record values based on sup-norm are determined. Some illustrative examples are given.

Weak limits of sample range

1974 ◽  
Vol 11 (04) ◽  
pp. 836-841 ◽  
Author(s):  
Laurens De Haan
Keyword(s):  
Weak Convergence ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Weak Limits ◽  
Sample Range ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

scholarly journals On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit

2009 ◽  
Vol 46 (03) ◽  
pp. 732-755 ◽  
Author(s):  
Shai Covo
Keyword(s):  
Weak Convergence ◽  
Detailed Analysis ◽  
Sufficient Conditions ◽  
Interesting Case ◽  
Necessary And Sufficient Conditions ◽  
Limit Behavior ◽  
Gamma Process ◽  
Lévy Measure ◽  
Necessary And Sufficient ◽  
Significant Class

Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let X ε be the sum of jumps not exceeding ε, and let µ(ε)=E[X ε(1)]. We study the question of weak convergence of X ε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of X ε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

scholarly journals Hypersurface Family with a Common Isoasymptotic Curve

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Ergin Bayram ◽  
Emin Kasap
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Frenet Frame ◽  
Necessary And Sufficient ◽  
The Given

We handle the problem of finding a hypersurface family from a given asymptotic curve in R4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.

scholarly journals Weak convergence of first-rare-event times for semi-Markov processes

2020 ◽  
Author(s):  
AISDL
Keyword(s):  
Weak Convergence ◽  
Markov Processes ◽  
Sufficient Conditions ◽  
Rare Event ◽  
Necessary And Sufficient Conditions ◽  
Finite Set ◽  
In Series ◽  
Event Times ◽  
Semi Markov Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions for weak convergence of first-rareevent times for semi-Markov processes with finite set of states in series of schemes are obtained.

Weak limits of sample range

1974 ◽  
Vol 11 (4) ◽  
pp. 836-841 ◽  
Author(s):  
Laurens De Haan
Keyword(s):  
Weak Convergence ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Weak Limits ◽  
Sample Range ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

Some Relationships between Filters*

1967 ◽  
Vol 10 (2) ◽  
pp. 257-260
Author(s):  
Ivan Baggs
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Concept ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
The Given

A filter is a set theoretical concept and as such, its structure is independent of any topology which can be put on the given space. However, an O-filter, whose counterpart in the theory of nets is the O-nets of Robertson and Franklin [2], is defined with respect to the topology on the given space. The purpose of this paper is to give necessary and sufficient conditions for every O-filter to be an ultrafilter and for every Cauchy filter to be an O-filter.

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.

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