IV International Conference
Limit Theorems in Probability Theory and Their Applications

Abstracts
 
 

 

Invariance principle for partial sum processes of moving averages

N. S. Arkashov, I. S. Borisov

Sobolev Institute of Mathematics, Novosibirsk, Russia

Let be i.i.d. random variables satisfying the conditions and . Let be a sequence of real numbers. Define the stationary sequence of random variables and the corresponding partial sum process:

   

Denote

Under the notation above,

Introduce the sequence of normalized partial sum processes of moving averages:

Theorem. Let the sequence be summable and

Moreover, let one of the following two conditions be valid:

(i)
   
(ii)
   

Then the sequence -converges in to a standard Wiener process.

Note that this Theorem improves the corresponding results in [1-3].

[1]
  Hall, P. and Heyde, C.C. Martingale limit theory and its application. -- New York: Academic Press, 1980.
[2]
 Arkashov, N. S. and Borisov, I. S. Gaussian approximation for partial sum processes of moving averages. - Siberian Math. J., 2004, V. 45, No. 6, pp. 1221-1255.
[3]
 Arkashov, N. S. Gaussian approximation and large deviation principle for partial sum processes of moving averages. Ph.D. Thesis, Sobolev Inst. of Math., 2005.

 

On the distribution of the number of crossing of a strip
for stochastic processes with independent increments

A.A. Atakhuzhaev, V.R. Khodzhibaev

Namangan Engineering-Pedagogical Institute, Namangan, Uzbekistan

Let be a stochastic process with independent stationary increments, , . Given arbitrary and , we define the random variables and respectively equal to the number of crossings of a strip on the coordinate plane of points from below to above and from above to below by the sample paths of the random walk with continuous time over the time interval from 0 to .

We study asymptotic behavior of the distribution of random variables and as ,, and . The final result consists in complete asymptotic expansions of the probabilities , , under various restrictions on , ,and compatible with the condition . This problem was solved in [1] for the discrete time case. Here we use the methods and technique of [1]. The asymptotics of the distribution of the random variables in the cases when they are finite was studied in [2].

We suppose here that and the Cramer condition holds on the analyticity of the function in a strip containing the imaginary axis. Moreover, we impose some conditions of [3] which provide some necessary properties of components of infinitely divisible factorization of the function .

[1]
 V. I. Lotov and N. G. Orlova (2004) Asymptotic Expansions for the Distribution of the Crossing Number of a Strip by Sample Paths of a Random Walk. Siberian Math. J. 45, 4, 680- 698.
[2]
 Lotov V.I., Khodjibayev V.R. (1993) On the number of crossings of a strip for stochastic processes with independent increments. Siberian Advances in Mathematics 3, 2, 145-152.
[3]
 Rogozin B.A. (1969) Distribution of the maximum of a process with independent increments Siberian Math. J. 10, 6, 989-1010.

 

Limit theorems for additive functionals of order statistic
based on dependent random variables

Evgeny Baklanov

Novosibirsk State University, Novosibirsk, Russia

The asymptotic behavior of wide class of additive functionals of order statistics is investigated. In particular, strong law of large numbers for linear combinations of functions of order statistics (-statistics) based on weakly dependent random variables is proven. As an auxiliary result we establish the Glivenko-Cantelli theorem for -mixing sequences of identically distributed random variables. Also, asymptotic normality of a class of spacings statistics based on independent and identically distributed random variables is obtained.

 

Navigation on a Poisson point process

Charles Bordenave

DI - Ecole Normale Supérieure, Paris, France

On a locally finite point set, a navigation defines a path through the point set from a point to an other. The set of paths leading to a given point defines a tree, the navigation tree. In this talk, we will analyze the properties of the navigation tree when the point set is a Poisson point process on . We examine the distribution of stable functionals, the local weak convergence of the navigation tree, the asymptotic average of a functional along a path, the shape of the navigation tree and its topological ends. We illustrate our work in the small world graphs, and new results are established.

 

Approximation of canonical von Mises statistics
based on dependent observations

I. S. Borisov, N. V. Borodikhina and A. A. Bystrov

Sobolev Institute of Mathematics, Novosibirsk, Russia

Let be a stationary sequence of -uniformly distributed r.v.'s, .Introduce the class of normalized -variate von Mises statistics:

where the kernels satisfy the degeneracy condition: for all and . Such kernels and the corresponding von Mises statistics are called canonical.

Theorem. Let and, moreover, let be square-integrable on all diagonal subspaces of the unit cube w.r.t. the corresponding induced Lebesgue measures. If the sequence satisfies -mixing condition with following restrictions on the corresponding coefficient: then

as , where is a centered Gaussian process with the covariance function

and the multiple stochastic integral in (1) is defined in [1].

In the talk, under less restrictive mixing conditions of the observations (but for more narrow subclass of the kernels), we study another representation of the limit distribution for these statistics as infinite polynomials of dependent Gaussian random variables. In the i.i.d. case this duality representation of the limit distribution is well known.

[1]
 Borisov, I. S. and Bystrov, A. A. Constructing stochastic integral of a nonrandom function without orthogonality of a noise, Theory Probab. Appl., 50, 52-80 (2005).

 

Nonorthogonal noises and stochastic integrals

I. S. Borisov,  A. A. Bystrov

Sobolev Institute of Mathematics, Novosibirsk, Russia

Let be a nonempty set and be a semiring of its subsets with identity. Let be a random process defined on a probability space and satisfying the condition for all The process is called an elementary stochastic measure or a noise if

for all subsets satisfying the conditions and . We study -construction of stochastic integrals of the form

where is a nonrandom -measurable kernel function and is the minimal -field generated by in the case when the elementary stochastic measure does not satisfy the classical orthogonality condition , where is a measure on . As examples, we consider several classes of random processes (not necessarily Gaussian!) with nonorthogonal increments on the real line generating the corresponding elementary stochastic measures, for which the stochastic integral exists under minimal restrictions on the kernel functions.

As an application of the above-mentioned general scheme, we also study multiple stochastic integrals of the form

and compare this construction with the corresponding results in [1] and [2].

In the second part of the talk we discuss some applications of this construction to asymptotic analysis of normalized canonical von Mises statistics and U-statistics based on samples from a stationary sequence of observations under some dependency conditions.

[1]
 Cambanis, S. and Huang, S. T. Stochastic and multiple Wiener integrals for Gaussian processes -- Ann. Probability, 1978, v.6, p. 585-614.
[2]
 Dasgupta, A. and Kallianpur, G. Multiple fractional integrals -- Probability Theory and Related Fields, 1999, v. 115, No. 4, p. 505-526.

 

On large deviations for sums of random variables

A. A. Borovkov

Sobolev Institute of Mathematics, Novosibirsk, Russia

A huge number of papers are devoted to the study of large deviation (l.d.) problems for sums of i.i.d. random variables. In spite of this until very recent times the systematical unified approach to the solution of these problems was absent and some problems remained to be open.

Let be i.i.d. random variables,

If the right hand side Cramer condition is met:

then we can define parameters

If , , then asymptotics of is studied rather well. But if , , then the situation differs and the nature of asymptotics of is getting completely different. General enough results in this direction (especially in multidimensional case) were missing. The same can be said on superlarge deviations  in case .

The talk exposes the systematical unified approach to the solution of all principal l.d. problems in all areas of l.d. The approach is based on the following elements:

In the last case as well as in case , when Cramer condition is met, one needs with necessity additional conditions concerning regularity of as .

 

Change-point problem for large samples
and incomplete information on distributions

A. A. Borovkov,  Yu. Yu. Linke

Sobolev Institute of Mathematics, Novosibirsk, Russia

Suppose we are given a sample consisting of independent observations in an arbitrary measurable space such that the first observations have a common distribution , and the remaining a distribution . The distributions and are unknown, and both and are assumed large. Under certain assumptions on and concerning either their moments or the behavior of their tails we construct estimates for the change point with a ``proper'' error, i.e., such that converges to zero with growing (at an exponential or polynomial rate). In the problem of sequential estimation of we construct a stopping time with explicit bounds for , for which asymptotics of as is found within (or within a constant if the distributions and are unknown).

 

Integro-local and integral theorems for sums of random variables
with semiexponential distribution

A. A. Borovkov, A. A. Mogulskii

Sobolev Institute of Mathematics, Novosibirsk, Russia


Let be i.i.d. random variables,

The distribution of is semiexponential, i.e. it has the form

where is smooth enough slowly varying function.

Integro-local and integral limit theorems for are obtained. They describe for any fixed the exact asymptotics of

as in all the areas of large deviations: in Cramer's zone, in so-called intermediate zone and in "extreme" zone, where the distribution of is close to the distribution of . Boundaries between these zones are included.

 

On small deviations of series of weighted i.i.d. random variables

A. A. Borovkov,  P. S. Ruzankin

Sobolev Institute of Mathematics, Novosibirsk, Russia

Let be positive i.i.d. random variables,

where coefficients are such that . We obtain an explicit form of the asymptotics of , , for the following three cases:

(i)   sequence is regularly varying with exponent , and as for some ,

(ii)   is regularly varying with exponent as , and as for some ,

(iii)  and are regularly varying with positive exponents (as and , respectively).

We show, in particular, that the asymptotics of as can be expressed as the product of known constant and function , where depends on , , and coefficients , whereas is determined only by the asymptotics of:

The asymptotics of , , is refined provided that is the geometric progression and as , , .

 

Asymptotics for first passage times of Levy processes
and random walks

Denis Denisov

EURANDOM, Eindhoven, The Netherlands

Joint work with Vsevolod Shneer (Novosibirsk, Russia & Edinburgh, UK)

We present exact asymptotics for the distribution of the first time a Levy process becomes negative, starting from positive level. We apply these results to find asymptotics for the distribution of the busy period in an M/GI/1 queue.

 

Lower deviation probabilities for supercritical
Galton-Watson processes

Klaus Fleischmann, Vitali Wachtel

Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

Let denote a Galton-Watson process with .We restrict attention to the supercritical case with . It is well-known that there exist constants such that

By "lower deviation probabilities" we refer to with . We give a detailed picture of the asymptotic behavior of such lower deviation probabilities.

 

Non-classical limit theorems in terms of characteristic functions

Sh. K. Formanov, L. D. Sharypova

Romanovskii Institute of Mathematics, Tashkent, Uzbekistan

It is well known that the classical theory of summation of independent random variables is based on the condition of uniform infinite smallness (UN-condition). Under this condition the class of limiting distributions for sums of independent random variables coincides with the class of infinitely divisible distributions. Nevertheless, P. Levy noticed long ago that UN-condition is inessential if we consider the convergence to a fixed limiting distribution.

Limit theorems for distributions of sums of independent random variables proved without UN-condition are called non-classical. This terminology was introduced by V. M. Zolotarev who proved a non-classical version of central limit theorem that generalized the classical Lindeberg-Feller theorem.

Our talk contains non-classical versions of the central limit theorem and the theorem on the convergence to the Poisson distribution in terms of characteristic functions of summands. The proofs of our theorems use modified versions of the well-known Stein-Tikhomirov and Stein-Chen methods. Moreover, the modifications of the above-indicated methods are connected with solutions of characterization problems for the Gaussian and Poisson distributions in terms of characteristic functions.

The theorems presented in the talk can be generalized for sequences of independent random variables taking values in separable Hilbert spaces. In addition, we prove some non-classical theorems on the convergence of the distributions of sums of independent random variables that provide the convergence of the moments of an arbitrary order.

 

On the lower bound for convolution tails of heavy-tailed distributions

S. G. Foss

Sobolev Institute of Mathematics, Novosibirsk, Russia & Heriot-Watt University, Edinburgh, UK

We present a short new proof of the following result.

Let be a distribution on the positive half-line and . Assume that, for any ,

Then

We also provide various comments or relative problems.

Based on joint work with D. Denisov and D. Korshunov.

 

On minimal conditions of weak dependence
in Local Limit Theorem

A. G. Grin'

Omsk State University, Omsk, Russia

Let be a strictly stationary sequence, and . Denote , . If random variables , , and have density functions , and , then the symbol  denotes that

,

and we write , if

as .

Let stand for the standard normal random variable. We say that the local limit theorem is applicable to the sequence , if for each random variables have densities and .

The sequence is called regularly varying with exponent , if is a regularly varying function with exponent  (here is the greatest integer not greater ). We denote by independent random variables with the same distributions as . The symbol means that and is an arbitrary sequence of positive integers.

Theorem 1. The local limit theorem is applicable to and is regularly varying with exponent  if and only if

(i)
 the following relation hold

(ii)
 the sequence is uniformly integrable,

(iii)
 for any positive integer ,

Theorem 1 can be interpreted as follows. (RL) is the minimal condition of weak dependence for the sequence providing applicability the local limit theorem with regularly varying with the exponent .

Replacing by stochastic equivalence in (RL), we get the minimal condition for central limit theorem (see [1]).

[1]
 Grin' A. G. On minimal condition of weak dependence in central limit theorem for stationary sequences.- TVP, 2002, v. 47, No. 3, p. 554-558.

 

Estimation of analytical functions

I. A. Ibragimov

St. Petersburg Department of Steklov Mathematical Institute,
Saint-Petersburg, Russia

In this talk we consider a few problems of statistical estimation of analytic functions (analytic function observed in an additive noise, estimation of analytic distribution density function, estimation of analytic spectral density function). The problems concerne the unicity theorem for analytic functions. An analytic function is fully determined by its values on an interval. We study how stable is the theorem with respect to the abovementioned random perturbations.

 

On boundary crossing problem for a random walk with switching

Dmitry Kim

Al-Farabi Kazakh National University, Almaty, Kazakhstan

Let , , be two independent sequences of independent random variables identically distributed within each sequence. Introduce a random walk with one level of switching as follows: and given , put

It is assumed that jumps from positions that are below point have a negative mean (then the random walk may drift to minus infinity with a positive probability) and that there are exponential moments for the distributions of jumps , (the so-called light-tailed case).

Let , be arbitrary numbers and . I study tail asymptotics for the distribution of the supremum , as and .

The stable case where jumps from positions that are below point have a positive mean and jumps from positions that are above point have a negative mean has been studied by A.A.Borovkov and D.A.Korshunov. In this situation a stationary distribution always exists. A.A.Borovkov (1980) has found the Laplace-Stieltjes transform for this stationary distribution. A.A.Borovkov and D.A.Korshunov (1996, 2001, 2002) considered a more general class of "asymptotically homogenous" Markov chains and derived a number of large deviations results for their stationary distributions. D.V.Gusak, O.I.Eleiko (1981) and N.S.Bratiychuk, D.V.Gusak, O.I.Eleiko (1984) have found the Laplace-Stieltjes transforms for the supremum of a random walk with one level of switching model for some special types of walks. I am unaware of any other results on (asymptotic) behavior of the supremum.

 

Mean fixation time estimates in populations of constant size

S. A. Klokov, V. A. Topchii

Omsk Branch of Sobolev's Institute of Mathematics SB RAS, Omsk, Russia

We consider a population of a fixed size where each particle has a type ascribed to it. In integer time moments each particle generates a random number of offspring of the same type in such a way that the population keeps its size constant and joint distributions of offspring numbers are exchangeable.

Let be the number of offspring produced by one particle, , and .

Several upper estimates for expected fixation time , i.e. the random time when all the particles have one type, are obtained for an arbitrary initial configuration of particles.

Theorem. Let an initial population have particles of j-th type, , , and . There exists a sequence , , such that and

where . Particularly, if , , then , .


Supported by grants: RFBR-NWO 047.016.013, RFBR 06-01-00127, Russian Scientific School 4129.2006.1

 

An aggregation of a normal sample for a minimization
of an asymptotic variance of a sign change

Artyom Kovalevskii

Novosibirsk State Technical University, Novosibirsk, Russia

A Sign Method of correlation function estimation of a zero-mean Gaussian stationary series is based on calculation of a frequence of a sign change of series' elements.

If the stationary series is a Fractional Gaussian Noise (FGN), that is, a variance of its particular sums grows as , then a probability of a sign change does not depend on an analysing object to be an initial series or series of sums of a fixed number of elements: if is a FGN then for any fixed

Thus one can made an estimator of better by an aggregation of the random series, that is, by fixing a set of divisions of the series on blocks of an equal size.

We propose an algorithm of search of an aggregation giving an asymptotically minimal variance in the case . In this case the random sequence is a sample from a normal distribution with a zero mean.

This algorithm is a solution of a problem of quadratic programming with Boolean variables. We give examples of the algorithm's realizations under different restrictions on a maximal block size.

 

Towards a theoretical foundation for wireless and sensor networks

P. R. Kumar

University of Illinois, Urbana, USA

What guidance can information theory provide to the emerging technologies of wireless and sensor networks that may be at the cusp of a possible takeoff? This talk explores the themes of how much information wireless networks can transport, what should be their architecture, what protocols are appropriate for their operation, and a possible Maxwellian model of computation for sensor networks.

 

First exit time of Brownian motion
from a parabolic domain

Mikhail Lifshits, Zhan Shi

St.Petersburg State University, Russia & Université Paris VI, France

Consider a planar Brownian motion starting from an interior point of the parabolic domain , and let denote the first time the Brownian motion exits from . The tail behaviour [or equivalently, the integrability property] of is somewhat unusual since it arises from an interference of large deviation and small deviation events. Our main result implies that the limit of [as ] exists and equals , thus improving previous estimates by Bañuelos et al. (2001) and W. Li (2003). The existence of the limit is proved by applying the classical Schilder large deviation theorem. The identification of the limit leads to a variational problem, which is solved by exploiting a theorem of Biane and Yor (1987) relating different additive functionals of Bessel processes. Our result actually applies to more general parabolic domains in any finite dimension , namely

with . For this case, we give an explicit formula for the non-degenerated limit of as .

 

Factorization method in a sojourn time problem

V. I. Lotov

Sobolev Institute of Mathematics, Novosibirsk, Russia

Let be a sequence of i.i.d. random variables, , . For and we consider the sojourn time

where if and otherwise.

The problem consists of studying the joint distribution .

Denote and

We prove that, for and ,

This equation contains two unknown functions. Using a Wiener-Hopf type method we solve it and find in terms of certain operators containing double transforms of joint distributions of ladder heights and ladder epochs. The representations for $Q_i$ obtained in this way look rather complicated so we give asymptotic formulas for them under Cramer condition, as $a\to\infty$ and $b\to\infty$. The main parts of these asymptotic formulas can be used for direct inversion or for asymptotic analysis of probabilities.

 

On functionals on the Markov chain transitions

V. S. Lugavov

Kurgan University, Kurgan, Russia

Let be a homogeneous Markov chain with state space and transition matrix .For any subset consider the random set

and functionals, determined by this set

where $Card (A)$ is the number of elements in the set A.

With help of these functionals, by varying set $B$ we get a wide class of functionals for chain $\kappa$. Some of these functionals, especially in the case , are researched in much detail (see [1]-[3] and references there).

For the matrix and the set put

In the sequel we shall assume that all essential states of are aperiodic states. Under this assumption we denote . The following statement holds:

Theorem 1. For , ,

Expressions for factorial moments of the functional are given in report. The next assertions are proved too.

Theorem 2. For

Theorem 3. Let be an arbitrary essential state of or be the nonessential state of such that

Then for the sequence

converges in probability to zero.

[1]
 Borovkov A.A. Theory of probability [in Russian], Nauka, Moscow (1986).
[2]
 Kemeny J.G., Snell J.L. Finite Markov chains [in Russian], Nauka, Moscow (1970).
[3]
 Chung K.L. Homogeneous Markov chain [in Russian], Mir, Moscow (1964).

 

Extended Ito formula and Ito integrals

F. S. Nasyrov

Ufa State Aviation Technical University, Ufa, Russia

Consider a Wiener process with the jointly continuous local time , , ,put , , . Let be an arbitrary function, denote

Generalized Ito formula is proved for left-continuous predictable functions and for a Wiener process .

Theorem 1. Let , $s\in [0,1]$, be a standard Wiener process. Then

where $g(p)$ be arbitrary left continuous predictable functions, such that Ito and Lebesgue-Stieltjes integrals in (1) are finite. It is proved that the limits (in probability) in right-side part of formula (1) always exist.

We have to notice that right-side part of the formula (1) is valid for unpredictable left-continuous functions $g(p)$ and arbitrary continuous functions $X(s)$, which admit the jointly continuous local time $\alpha(t,x)$, $t\in [0,1]$, $x\in R$. Hence the formula (1) allows to define Ito type integrals for unpredictable integrands and continuous functions $X(s)$.

[1]
 Follmer H., Protter P., Shiryayev A. Quadratic covariation and an extension of Ito's formula. Bernoulli, 1995, v.1, p.149-169.
[2]
 Nasyrov F.S. Extended Ito formula and path-wise Ito integrals. Vestnik UGATU. v.6, n.1(12), p.33-40 (in Russian).
[3]
 Nasyrov F.S. Extended Ito formula and Ito integrals. Theor. Probab. Appl. to appear.

${}^a$ Supported by RFFI grants 04-01-00286-a, 05-01-97909.

 

Dominating points in fluid queues with priority classes

Ilkka Norros

VTT Technical Research Centre of Finland

Joint work with Petteri Mannersalo and Michel Mandjes

Consider a fluid queue with two priority classes. We show that the event that the low priority queue exceeds a level and the event that the delay of a fluid molecule exceeds a level can both be written in the form of a union of intersections of elementary univariate events related to the cumulative input processes. In "many sources" -type large deviation limits of these events, the respective dominating points (most probable paths) minimizing the rate function are characterized as those of infinite intersections of simple events. More exactly, the basic problem turns out to be the identification of the most probable path that lies above a straight line on a certain interval. In the case that the input processes are Gaussian, the dominating points can be computed more explicitly, since the problem is then a minimum norm problem in the reproducing kernel Hilbert space. A general technique to solve this problem is presented.

 

On the optimal dividend problem for
a spectrally negative Lévy process


Zbigniew Palmowski

Wroclaw University, Wroclaw, Poland

In this talk we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative Lévy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative Lévy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal amongst all admissible ones takes the form of a barrier strategy.

 

Estimation of the constant in the Burkholder moment inequality
for supermartingales and martingales

Ernst Presman

Central Economics & Mathematics Institute of RAS, Moscow, Russia

Let a sequence of the random variables $S_k,\ k\geq 0,$ forms a supermartingale defined on a filtered probability space with $S_0=0$, , i.e. .

Let , , .

Theorem. The following inequality holds: ,

where ,   

Corollary. Let be a martingale, . Then

The last inequality was obtained by Burkholder in [1] without explicit expression for .

Let . For the function decreases slowly (as ) from to , so that , ,, , , .

The proofs are based on the modification of the proofs and results from [2] (see also [3]).

[1]
 Burkholder D.L. Distribution function inequalities for martingales. - Ann. Probab. 1973, 1, 19-42.
[2]
 Nagaev S.V. On Probability and Moment Inequalities for Supermartingales and Martingales. - Acta Appl. Math., 2003, 79, 35-46.
[3]
 Nagaev S.V. On Probability and Moment Inequalities for Supermartingales and Martingales. - The Theory of Probability and Its Application, 2006, 51, i. 2

 

The large deviation principle for queueing networks

A. A. Puhalskii

University of Colorado, Denver, USA and IITP, Moscow, Russia

We concern ourselves with studying the large deviation principle (LDP) for queueing networks with homogeneous customer populations. The processes of the exogenous arrivals, service and routing are assumed to be general and are only required to obey LDPs in the associated function spaces with action functionals of integral form. We show that under additional conditions this implies an LDP for the queue length processes and find the action functional explicitly. In particular, an LDP for a broad class of generalised open Jackson networks is obtained. Our approach is based on the large deviation relative compactness property of exponentially tight sequences of probability measures. The action functional is identified in terms of a weak solution to a system of idempotent equations obtained as large deviation limits of the stochastic equations governing the network.

 

Large deviation theorems for certain self-normalized sums

L. V. Rozovsky

St.Petersburg Chemical-Pharmaceutical Academy, St.Petersburg, Russia

Consider a sequence of independent identically distributed random variables , , , and denote , , .

Theorem. For

This result is one of the Chernoff type theorems for self-normalized sums of i.i.d. random variables (see also Qi-Man Shao "Self-normalized large deviations", The Annals of Probability 1997, Vol.25, No.1, 285-328).

Note that if then (1) is identical to the classical Chernoff theorem as in Bahadur R.R. "Some limits theorems in statistics", Regional Conference Series in Applied Mathematics 4. SIAM, Philadelphia.


The investigations were supported by Grant of Scientific School No. 4222.2006.1 and RFFI Grant No. 06-01-00179-a.

 

Poisson hypothesis for information networks
(a study in non-linear Markov processes)

A. N. Rybko

Institute for Information Transmission Problems, Moscow, Russia

We study the Poisson Hypothesis, which is device to analize approximately the behavior of large queuing networks. We prove it in some simple limiting cases. We show in particular that the corresponding dynamical system, defined by the non-linear Markov process, has a lane of fixed points which are global attractors. To do this we derive the corresponding non-linear equation and we explore its self-averaging properties. We also argue that in cases of heavy-tail service times the PH can be violated.

 

Estimates in the Invariance Principle
in terms of truncated moments

Alexander I. Sakhanenko

Ugra State University, Khanty-Mansiysk, Russia

Let be an infinite sequence of independent random variables with and .Consider a random function such that

and suppose that is monotone on every interval of the form

Introduce now the main notations

Theorem. For any fixed numbers , and there exists a Wiener process such that

where is an absolute constant.

In the partial case when and a weaker estimate with instead of truncated moments was obtained earlier in [1].

Theorem sharpens also several known results from [2].

Note, that a simple case when for is also possible in Theorem.

[1]
 A. I. Sakhanenko, On the accuracy of normal approximation in the invariance principle. Proceedings of the Institute of Mathematics Novosibirsk, 1989, v.13, p.40-66. (English translation in: Siberian Advances in Mathematics, 1991, V.1, No. 4, p.58-91.)
[2]
 J. Komlos, P. Major, G. Tusnady, An approximation of partial sums of independent RV's and sample DF. II, Z. Wahrscheinlichkeitstheorie verw. Geb. (1976) B. 34. H. 1, pp. 33-58.

 

Pricing of financial assets in exponential Levy models

A. T. Semenov

Novosibirsk State University of Economics and Management, Novosibirsk, Russia

The model of the financial market consisting of a riskless asset (bond) and risk asset (stock) is considered. Their respective prices follow the equations:

The constant is the riskless interest rate, is a Lévy process, that is, a process with stationary independent increments.

It is well known (see for example [1], [2]) that under standard assumptions (no arbitrage, completeness of financial market etc.) the pricing of financial assets (securities) can be reduced to calculating expectations. The main problem is that if we are not the Black-Scholes setting this calculation becomes a serious technical hurdle.

We consider the use of Fourier-analytic methods for pricing a wide class of securities and give explicit formulae for security prices by means of the Fourier transform. Implications for option pricing are discussed.

[1]
Melnikov A.V., Volkov S.N., Nechaev M.L. (2001). Mathematics of financial commitments. - M.: SU HSE.
[2]
Shiryaev A.N. (1998). Essentials of Stochastic Finance. V.1, 2. - M.: FAZIS.

 

On mixing conditions for sequences of moving averages

D. I. Sidorov

Sobolev Institute of Mathematics, Novosibirsk, Russia

Let be i.i.d. random variables and be real numbers such that and Consider a stationary sequence of random variables having a structure of the so-called moving averages: . We study possible types of mixing for the sequence . We say that the sequence satisfies -mixing if

and it satisfies -mixing if

where and are -fields generated by the corresponding random variables.

In the talk, we formulate conditions on and the distribution of providing (or not!) - or -mixing of the sequence It is clear that if the set of integers is finite then, for some natural , the sequence consists of -dependent random variables (i.e., satisfies the mixing conditions above). In the opposite case we obtained the following results.

Proposition 1. Let be a bounded random variable having a continuous density, and . Then the sequence satisfies -mixing.

Proposition 2. Let be a discrete random variable with a finite number of atoms , where . Moreover, let one of the following two conditions be fulfilled:

  for some
  if and if , where

 

Then the sequence does not satisfy -mixing.

Theorem 1. Let be a bounded nondegenerated random variable and the set be infinite. Then the sequence does not satisfy -mixing.

Theorem 2. Let the following conditions be fulfilled:

(1)
 for some , ,

(2)
 ;
(3)
 .
Then the sequence does not satisfy -mixing.

 

Individuals at the origin in the critical multidimensional
catalytic branching random walk

Valentin Topchii
Omsk Branch of Sobolev Institute of Mathematics of SB RAS, Omsk, Russia


Vladimir Vatutin
Steklov Mathematical Institute of RAS, Moscow, Russia

A continuous time branching random walk on the lattice is considered in which individuals may produce children at the origin only. Each individual spends at the origin an exponentially distributed time with parameter 1 and then either jumps to a point with probability , or dies with probability producing just before the death a random number of children . Individuals outside the origin perform a Markov random walk without reproduction, i.e., an individual spends in a state an exponential time with parameter and then jumps to a point with probability 

Let and be the numbers of individuals at the origin and outside the origin at moment respectively. Assuming that the underlying Markov random walk is homogeneous and symmetric, and the reproduction law of $\xi$ is critical, we describe the asymptotic behavior, as of the conditional distribution of the two-dimensional vector (scaled in an appropriate way), given .

The case has been investigated by the authors in

Individuals at the origin in the critical catalytic branching random walk. Discrete Mathematics Theretical Computer Science (electronic), v.6 (2003), 325-332.
http://dmtcs.loria.fr/proceedings/html/dmAC7130.abs.html

Two-dimensional limit theorem for a critical catalytic branching random walk. In Mathematics and Computer Science III, Algoritms, Trees, Combinatorics and Probabilities, Editors M.Drmota, P.Flajolet, D.Gardy, B.Gittenberger. Birkhauser, Verlag, Basel-Boston-Berlin, (2004), 387-395.

Limit Theorem for Critical Catalytic Branching Random Walks. Theory of Probability & Its Applications, V.49 (2005), no. 3. pp. 498-518.


${}^a$ Supported by grants: RFBR-NWO 047.016.013, RFBR 06-01-00127, Russian Scientific School 4129.2006.1
$^b$ Supported by grants: RFBR-NWO 047.016.013, RFBR 05-01-00035, Russian Scientific School 4129.2006.1

 

Limit distributions in queueing networks with unreliable elements

G. Sh. Tsitsiashvili, M. A. Osipova

Institute of Applied Mathematics, Far Eastern Branch of RAS, Vladivostok, Russia

In this paper new product theorems for opened queueing networks with unreliable elements: nodes, links between nodes and servers in nodes are proved. Limit distributions calculations are based on different schemes of elements recovery (independent recovery, recovery with one repair place and restoration network scheme), routing algorithms and service disciplines. Thus an introduction of special control allows to connect queueing networks theory with reliability theory. Obtained results may be spread onto closed queueing networks practically without changes.


${}^a$ Partially supported by RFBR, project 06-01-00063-à and FEB RAS, project 06-III-A-01-016

 

Berry-Esseen type bounds for transformations of chi-squared
variables and its statistical applications

Vladimir V. Ulyanov

Moscow State University, Moscow, Russia
Joint work with Gerd Christoph (Magdeburg, Germany)
and Yasunori Fujikoshi (Tokyo, Japan)

Put

where the random matrix $W$ has a Wishart distribution . The distribution of appears as the null distribution of LR statistic for testing a hypothesis that a covariance matrix is equal to a given covariance matrix.

Let be the distribution function of chi-squared random variable with degrees of freedom.

We prove that

where $q = p (p + 1)/2$ and is a computable constant, depending only on $p$ and $n$. Moreover, $C(p,n) = O(n^{-1})$ as .

The possible extensions of the result are discussed.

 

Branching processes in random environment
and the bottlenecks in evolution of populations

V. A. Vatutin

Steklov Mathematical Institute, Moscow, Russia

First models of branching processes were investigated in the 80th of 19th century by two British scientists - Galton and Watson in connection with the study of extinction of nobel families. Now, owing to the efforts of Kolmogorov, Sevastuanov, Bellman, Harris, Athreya, Ney, Dawson, Dynkin and many others the theory of branching processes becomes one of the important parts of probability theory . Many results of this theory occur to be useful in various fields of science from physics and chemistry to biology. However, the classical models of branching processes do not reflect phenomena essential to the evolution of populations one of the most important of which is oscillation of the number of individuals (and not permanent exponential growth or rapid extinction as is usual in the classical models).

In our talk we consider a model of branching processes in random environment where oscillation of the size of populations is an essential feature. In particular, we formulate mathematical results which show (at least in the framework of the model under consideration and at the theoretical level) that the evolution of a population consists of favorable and unfavorable stages. Unfavorable periods reduce the size of the population to a finite (bounded) number of individuals. These periods are followed by favorable time-intervals when the population ``recovers' and rapidly reestablishes its size. In real populations such phenomena is called the bottleneck of evolution of populations.

 

An exponential estimate for error of wavelet density estimator

V. Yurinsky

Universidade da Beira Interior, Covilhã, Portugal

This communication is based on joint work with J.M.R. Gama. It considers some properties of wavelet density estimators (WDE) that reconstruct a density from i.i.d. observations with this density. A WDE is essentially a partial sum of the multiresolution expansion based on a sequence of scales ,


(1)

with unknown coefficients , substituted by appropriate statistics of the sample.

For the so-called linear WDE , which changes coefficients to sample means, the squared -norm of the random component of its error proves asymptotically normal [1] if the number of "detail layers" retained goes to infinity with sample size: , and the distribution of
weakly converges to the standard normal law on .

A "thresholded" WDE is obtained from a linear one by suppression of less relevant terms. The efficiency of these non-linear WDE's was studied predominantly in terms of integral risks. Yet, a thresholded WDE admits an exponential inequality, resembling (at least in form) that of S.N.Bernstein,

for the discrepancy between WDE and , a linear approximation to it. (This latter is obtained by same procedure of exclusion of non-informative terms that relies on the exact coefficients of (1) instead of sample estimates.) All elements of the above inequality can be written down in a reasonably explicit form (see [2]).

[1]
 Yurinsky V., Gama dos Reis J. M. On distribution of error of linear wavelet estimator. Lith. Math. J. (2005) vol. 45, No. 2, 152-172.
[2]
 Gama J.M.R., Yurinsky V. Exponential estimate for wavelet density estimator. Theory Probab. Appl. (2006) vol. 50. (to appear)


${}^a$ Work supported by Fundação para a Ciência e a Tecnologia (Portugal) through Centro de Matem'atica  da Universidade da Beira Interior, Sub-Projecto DECONT.

 

Estimates for the strong approximation in multidimensional
Central Limit Theorem

A. Yu. Zaitsev

St. Petersburg Department of Steklov Mathematical Institute,

St.Petersburg, Russia

Let be an even positive function such that and the function is non-decreasing.

Theorem 1. Suppose that are independent random vectors with , . Let and . Write $ K_{H}=H^{-1}(e^{\lambda_H}L_{H})$, where $H^{-1}(\cdot )$ denote the inverse function for $H(\cdot )$ Assume that there exists a strictly increasing sequence of non-negative integers $m_{0}=0$, $m_{1},\dots ,m_{s}=n$ satisfying the following conditions. Let ,, ,and assume that for all , and ,

with some constant and . Then one can construct on a probability space a sequence of independent random vectors $X_{1},\dots ,X_{n}$ and a corresponding sequence of independent Gaussian random vectors so that , , , , and, for , ,

where , for , and , are positive quantities depending only on and .

Theorem 1 is a consequence of the result of the author [2] and can be considered as multidimensional generalization of a weakened version of a result of Sakhanenko [1]. The conditions of Theorem 1 are satisfied, for example, for the functions ,, with .

[1]
 A. I. Sakhanenko, Estimates in the invariance principles, In: Trudy Inst. Mat. SO AN SSSR 5, Nauka, Novosibirsk, 1985, 27-44 (in Russian).
[2]
 A. Yu. Zaitsev, Multidimensional version of the results of Sakhanenko in the invariance principle for vectors with finite exponential moments. I; II; III, Theor. Probab. Appl., 45 (2000), 718-738; 46 (2001), 535-561; 744-769.


Research partially supported by Russian Foundation of Basic Research (RFBR) Grant 05-01-00911, by RFBR-DFG Grant 04-01-04000, and by INTAS Grant 03-51-5018.

 

Strictly stable distributions on convex cones

Sergei Zuyev (Glasgow, UK & Moscow, Russia)

Statistics and Modelling Science dept., University of Strathclyde

Joint work with Yury Davydov and Ilya Molchanov

Notion of a stable distribution is one of central in the Theory of Probability. It naturally appears in the Central Limit Theorem (CLT) for random vectors (and generally, random elements in a Banach space $B$) and characterised by the parameter $0<\alpha \leq 2$. The case $\alpha=2$ corresponds to the classical Gaussian CLT, while stable distributions with arise when the second and/or the first moment of the summands does not exist. The symmetric alpha-stable random elements can be represented as a sum of points of a Poisson process known as the LePage representation. This point process is union-stable, i.e. the union of its two independent copies coincides in distribution with the rescaled original point process. This shows that the classical definition of stable random elements is closely related to the union-stability property of point processes.

These concepts make sense in any convex cone , i.e. in a semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. Examples include compact sets with union operation, measures with convolution or with superposition operation, positive numbers with harmonic mean operation etc. We establish limit theorems for normalised sums of random elements with $\alpha$-stable limit for $0<\alpha<1$ and deduce the LePage representation for strictly stable random vectors in these general cones.

By using the technique of harmonic analysis on semigroups we characterise distributions of $\alpha$-stable random elements and show how possible values of the characteristic exponent $\alpha$ relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties.




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