Eugenijus Manstavičius

Mappings on Decomposable Combinatorial Structures: Analytic Approach

On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.

Limit Processes with Independent Increments for the Ewens Sampling Formula

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Techniques from probabilistic number theory are used to establish necessary and sufficient conditions for weak convergence of the associated dependent process to a process with independent increments. Not many results on the necessity part are known in the literature.

On random permutations without cycles of some lenghts

We deal with random permutations of the symmetric group SNendowed with the Haar probability measure. The main purpose of the remark is to obtain uniform lower estimates for the probability of a permutation without cycles having lengths in some J ⊂ {1, . . . . N} . ThesetJcan itself depend on N. The only information used is a bound for the sum of reciprocals of elements in J.

The Berry–Esseen Bound in the Theory of Random Permutations

The convergence rate in the central limit theorem for linear combinations of the cycle lengths of a random permutation is examined. It is shown that, in contrast to the Berry-Esseen theorem, the optimal estimate in terms of the sum of the third absolute moments has the exponent 2/3.

An analytic method in probabilistic combinatorics

We deal with the value distribution problem for the linear co mbinations of multiplicities of the cycle lengths of a random permutation . To examine the characteristic functions, we derive asymptotic formulas f or ratios of the Taylor coefficients of the relevant generating series. The propose d version of analytic method does not require any analytic continuation of these s eries outside the convergence disk.

Value Concentration of Additive Functions on Random Permutations

We prove an analog of the Kolmogorov–Rogozin inequality for the value concentration of completely additive functions defined on random permutations.

Permutations without long or short cycles

Abstract We explore probabilities that a permutation sampled from a finite symmetric group uniformly at random has only short or long cycles. Asymptotic formulas, as the order of the group increases, valid in specified regions are obtained using the saddle point method. As an application, we establish a formula with remainder term estimate for the total variation distance between the count process of the multiplicities of cycle lengths in the random permutation and a relevant process defined via independent Poisson random variables.

Strong convergence on weakly logarithmic combinatorial assemblies

We deal with the random combinatorial structures called assemblies. Instead of the traditional logarithmic condition which assures asymptotic regularity of the number of components of a given order, we assume only lower and upper bounds of this number. Using the authors analytic approach, we generalize the 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. The estimates are applied to examine additive functions defined on a new class of structures, called weakly logarithmic. Some analogs of Majors and Fellers theorems which concern almost sure behavior of sums of independent random variables are proved.

Iterated Logarithm Laws and the Cycle Lengths of a Random Permutation

We areconcerned with the iterated logarithm laws for mappings defined on thesymmetric group. For the sequences of the cycle lengthsand the different cyclelengths appearing in the decomposition of a random permutation,such laws provide asymptotical formulas valid uniformly in a wide region for the sequence parameter. The main results are analogues to Feller’s and Strassen’s theorems proved for partial sums of independent random variables.

Functional limit theorems for digital expansions

The main purpose of this paper is to discuss the asymptotic behaviour of the difference sq,k(P(n)) - k(q-1)/2 where sq,k (n) denotes the sum of the first k digits in the q-ary digital expansion of n and P(x) is an integer polynomial. We prove that this difference can be approximated by a Brownian motion and obtain under special assumptions on P, a Strassen type version of the law of the iterated logarithm. Furthermore, we extend these results to the joint distribution of q1-ary and q2-ary digital expansions where q1 and q2 are coprime.