arXiv subjects

Eugenijus Manstavičius

Publications and source records attributed to Eugenijus Manstavičius.

4 recordsLinked to original sources

arXiv2025

A stochastic process defined via the random permutation divisors

The normalised partial sums of values of a nonnegative multiplicative function over divisors with appropriately restricted sizes of a random permutation from the symmetric group define trajectories of a stochastic process. We prove a functional limit theorem in the Skorokhod space when the permutations are drawn uniformly at random. Furthermore, we show that the paths of the limit process almost surely belong to the space of continuous functions on the unit interval and, exploiting the results from number-theoretical papers, we obtain rather complex formulas for the limits of joint power moments of the process.

math.PR↗

arXiv2014

Local probabilities for random permutations without long cycles

We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the saddle point method combined with ideas originated in analytic number theory. Theorem 1 and its detailed proof are included to rectify formulas for small r which have been announced by a few other authors.

math.CO↗

arXiv2013

On statistics of permutations chosen from the Ewens distribution

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 with restricted lengths function is analyzed in more detail. The results can be applied to statistics defined on random permutation matrices.

math.CO↗

arXiv2009

Strong Convergence on Weakly Logarithmic Combinatorial Assemblies

We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the author's analytic approach, we generalize the so-called Fundamental Lemma giving independent process approximation in the total variation distance of the component structure of an assembly. To evaluate the influence of strongly dependent large components, we obtain estimates of the appropriate conditional probabilities by unconditioned ones. These estimates are applied to examine additive functions defined on such a class of structures. Some analogs of Major's and Feller's theorems which concern almost sure behavior of sums of independent random variables are proved.

math.PR↗