Showing papers in "Lithuanian Mathematical Journal in 1995"

A generalized fractionally differencing approach in long-memory modeling

Abstract: We extend the class of fractional ARIMA models to the class of fractional ARUMA models, which describe long-memory time series with long-range periodical behavior at a finite number of spectrum frequencies. The exact asymptotics of the covariance function and the spectrum at the points of peaks and zeros are given. To obtain asymptotic expansions, Gegenbauer polynomials are used. Consistent parameter estimation is discussed using Whittle's estimate.

On the universality of the Riemann zeta-function

Abstract: Article history: Received 2 March 2010 Revised 28 April 2010 Communicated by David Goss MSC: 11M06

On the law of the iterated logarithm in multiphase queueing systems

Abstract: Queueing systems with a single device are well developed (see, for exam­ ple, Borovkov, 1972; 1980). But there are only several works in the theory of multiphase queueing systems in heavy traffic (see Iglehart, Whitt, 1970b) and no proof of laws of the iterated logarithm for the probabilistic characteristics of multiphase queuing systems in heavy traffic. The law of the iterated logarithm for the waiting time of a customer is proved in the first part of the paper (see Minkevi~ius, 1995). In this work, theorems on laws of the iterated logarithm for the other main characteristics of multiphase queu­ ing systems in heavy traffic (a summary queue length of customers, a queue length of customers, a waiting time of a customer) are proved.

On algebraic numbers of small measure

Abstract: We give a lower bound for |α−1|, where α is an algebraic number, and also an upper bound for the number of real zeros of a polynomial. A lower bound for the maximal modulus of conjugates of a totally real algebraic integer is also obtained.

On the conditional covariance condition in the martingale CLT

Abstract: The paper develops a method allowing one to figure out how a convergence rate in the martingale central limit theorem depends on the conditional covariance structure of the martingale. The method is based on constructing “stopping projections” that control the behavior of the conditional covariances of martingale differences. A discrete time martingale taking values in either finite or infinite dimensional Hilbert space is considered.

Daubechies compactly supported wavelets with minimal heisenberg boxes

Abstract: The centers and radii of orthonormal scaling functions and wavelets are found in time and frequency domains using a two-scale relation. All compactly supported orthogonal wavelets with support on the interval [0, 3] fail to have radii in the frequency domain. On the other hand, a Daubechies wavelet with support on the interval [0, 3] has optimal resolution in the frequency domain.

On the global central limit theorem for π-mixing random variables

Abstract: The estimate of the remainder term is obtained in the global central limit theorem for π-mixing r.v.s. As a consequence of Theorem 1 the convergence rate of absolute moments for sums of π-mixing r.v.s. to corresponding absolute moments of the normal r.v. is found.

The convergence of distribution of integer-valued additive functions to the poisson law

Abstract: Letfx(m), x≥1 be a set of additive functions, and $$v_x (f_x (m)< u) = \frac{1}{{[x]}}\# \{ m \leqslant x, f_x (m)< u\} .$$ Some results are obtained about the weak convergence of the distribution functionsvx(fx(m)

Stability of one characterization by the properties of the renewal process

Abstract: (we assume that So = 0). Even in the case of random processes in continuous time we can often find one or more sequences of instants of time for which (1) holds. In such cases one succeeds in getting surprisingly exact results by elementary methods. Analytically we simply have to deal with sums of independent positive random variables, and the only justification for introduction of the term "renewal process" is its frequent use in connection with other processes as well as the fact that when applying this term we tacitly mean the use of a powerful tool, namely, the renewal equation.

The provability of certain sequents in additive arithmetic

Abstract: Shoenfield has presented the complete system for free variable additive arithmetic and noticed that also such complete systems can be given by adding RDIO to the axioms A1-A5 (see below). Shepherdson in [2, 3], investigating the replaceability of RIO in the various arithmetical systems (and using, also as Shoenfield, the models theory) has shown, that when the restricted difference, with suitable axioms, is added, the rule of double induction becomes derivable from RIO. In the additive and multiplicative arithmetics without the restricted difference this rule of double induction is definitely stronger than RIO. Shepherdson also posed the question how strong RDIO is. The answer to this question for the additive arithmetic is the purpose of our two papers (see also [5]). When investigating a question related to the replaceabitity and the provability of the double induction axiom, there arises a necessity to ascertain a possibility to prove sequents of the form

On the average difference between two conjugates of an algebraic number

Abstract: We prove that for an algebraic integer α of degreen with conjugates α = α1, α2, ...., α n $$\frac{2}{{n(n - 1)}}\sum\limits_{i \sqrt e - \varepsilon $$ where e>0,n>n 0(e).

Selberg's inequality in arithmetic progressions. II

Abstract: We obtain an elementary derivation for Selberg's formula related to the function ψ r, k * (x; q, l) for allr, k positive integers.

On convergence to the poisson law for distributions of the sum of random variables connected into a chain

Abstract: This paper deals with the triangular array of random variables where every sequence is connected into a stationary Markov chain. For the uniformly strongly mixing chain we find conditions under which the distributions of the sum converge to the Poisson law. A more general case of theL p-regular Markov chain is also considered.

On the convergence in the law of sums and maxima of independent random variables with random parameters

Abstract: The paper deals with maxima and sums of independent random variables. These random variables are the values of independent identically distributed stochastic processes at a random point in time. We obtain conditions for their weak convergence, at almost all points in time to the same infinitely divisible distribution and describe the limit distribution for these sums. Some applications of these results to statistics are considered.

The riemann boundary problem with a plus-infinite index of the logarithmic order for a complicated contour

Abstract: m1. Suppose a contour L = Uj=I Lj is given on the complex plane, consisting of infinite rays Lj = {z: argz = flj }, 0 = /~1 \"< /~2 < \"'\" < fin < 2rr with the beginning point z = 0. We will denote the angle flj < arg z < flj+l by (flj, flj+l), flm+l = 27r. A piecewise analytic function r is to be found, boundary values of which on L \\ (0, c~) satisfy the linear relation �9 +(t) = G(t)~-(t) + g(t). (1) The functions G(t) and g(t) are given so that (j = 1, m) tn IG(t)l, g(t) 6 \"Dq(Lj); g(0) = g(cx:)) = 0;