Showing papers in "Theory of Probability and Mathematical Statistics in 2005"

Motions with finite velocity analyzed with order statistics and differential equations

Abstract: The aim of this paper is to derive the explicit distribution of the position of randomly moving particles on the line and in the plane (with different velocities taken cyclically) by means of order statistics and by studying suitable problems of differential equations. The two approaches are compared when both are applicable (case of the telegraph process). In some specific cases (alternating motions with skipping) it is possible to use the order statistics approach also to solve the equations governing the distribution. Finally, the approach based on order statistics is also applied in order to obtain the distribution of the position in the case of planar motion with three velocities conditioned on the number of changes of directions recorded.

Wavelet estimators of a density constructed from observations of a mixture

Abstract: We construct projective estimators of a density by using a wavelet basis for the data being a sample from a mixture of several components whose concentrations vary with observations. We construct linear and adaptive estimators and prove that they converge in the mean square norm. We also prove that the linear estimator converges in the uniform norm.

Minimum ₁-norm estimation for fractional Ornstein–Uhlenbeck type process

Abstract: We investigate the asymptotic properties of the minimum L1-norm estimator of the drift parameter for fractional Ornstein-Uhlenbeck type process satisfying a linear stochastic differential equation driven by a fractional Brownian motion.

A precise upper bound for the error of interpolation of stochastic processes

Abstract: We obtain a precise upper bound for the truncation error of interpolation of functions of the Paley–Wiener class with the help of finite Whittaker–Kotelnikov– Shannon sums. We construct an example of an extremal function for which the upper bound is achieved. We study the error of interpolation and the rate of the mean square convergence for stochastic processes of the weak Cramér class. The paper contains an extensive list of references concerning the upper bounds for errors of interpolation for both deterministic and stochastic cases. The final part of the paper contains a discussion of new directions in this field.

Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space

Abstract: In this paper we consider an additive functional of a Markov process with locally independent increments switched by a Markov process. For this functional, we obtain nonhomogeneous diffusion approximation results without balance condition on the drift parameter. A more general diffusion approximation result is obtained in the case of an asymptotic split phase space of the switching Markov process.

On the joint distribution of the supremum, infimum, and the value of a semicontinuous process with independent increments

Abstract: The joint distribution of the supremum, infimum, and the value of a homogeneous lower semicontinuous process with independent increments is found in this paper. The weak convergence of the boundary distribution to the corresponding distribution of the Wiener process is proved in the case of Eξ(1) = 0 and Eξ2(1) < ∞. Exact and asymptotic relations are obtained for this distribution. Let ξ(t) ∈ R, t ≥ 0, be a homogeneous lower semicontinuous process with independent increments [1] and let k(p) be its cumulant: ξ(0) = 0, E [ e−pξ(t) ] = e, Re p = 0. The aim of this paper is to determine the joint distribution (1) Q(−y, α, β, x) = P [ −y ≤ inf u≤t ξ(u), ξ(t) ∈ (α, β), sup u≤t ξ(u) ≤ x ] where x, y > 0, −y ≤ α < β ≤ x. This problem is solved in [2] for homogeneous processes with independent increments. The problem for semicontinuous processes with independent increments can be solved in the closed form in terms of the resolvent (2) R(x) = 1 2πi ∫ γ+i∞ γ−i∞ e 1 k(p) − s dp, γ > c(s) (see [2]–[6]) where c(s) > 0 for s > 0 is a unique in the half-plane Re p > 0 positive root of the equation k(p) − s = 0 (see [2]). Now we state the main results of the paper. Theorem 1. Let ξ(t), t ≥ 0, be a homogeneous lower semicontinuous process, νs an exponential random variable with parameter s > 0, and let Q̃(−y, α, β, x) = ∫ ∞ 0 e−stP [ −y ≤ inf u≤t ξ(u), ξ(t) ∈ (α, β), sup u≤t ξ(u) ≤ x ] dt,

The stability of almost homogeneous in time Markov semigroups of operators

Abstract: A homogeneous in time semigroup of Markov operators defined by its infinitesimal operator with a dense domain is considered. The operator is perturbed by another bounded operator that depends on time, and this results in a nonhomogeneous semigroup. Under certain assumptions, we prove that the perturbed semigroup is a unique solution of a weak integral equation determined by the initial semigroup and an operator perturbation function; this equation is an integral analog of the perturbed Kolmogorov equation. We find explicit estimates for the stability of the perturbed semigroup in the case where the perturbation operator is uniformly small. The stability of perturbed homogeneous semigroups of operators is studied in the author monograph [1] for discrete time. The general questions of the perturbation theory of operators are discussed in the Kato monograph [2]. Problems concerning the stability of nonhomogeneous semigroups with continuous time become more important in view of the growing number of models in risk theory, insurance, and finance mathematics. These models are nonhomogeneous in time (in view of the season phenomena, say) and are not yet studied in detail. 1. Setting of the problem 1. Let (E, Ξ) be a measurable space. By fΞ and mΞ we denote the classes of measurable functions and finite measures that may attain negative values on (E, Ξ). Let א ⊂ mΞ be a Banach subspace of mΞ equipped with the norm ‖ · ‖ and such that (M) Var(μ) ≤ c‖μ‖, ‖μ‖ ≤ ‖μ + ν‖ for all μ, ν ∈ א, ν ≥ 0, and some constant c. Consider the dual space ⊂ fΞ of א that consists of functions equipped with the norm ‖f‖ = sup(|μf |, ‖μ‖ ≤ 1, μ ∈ א) where the dual linear form μ is such that μf = ∫ E f(x) μ(dx), μ ∈ א, f ∈ , and (1) ‖μ‖ = sup(|μf |, ‖f‖ ≤ 1, f ∈ ). The space contains all measurable bounded functions if condition (M) holds. Some examples of such spaces and their dual counterparts are given in [1, Chapter 1]. 2000 Mathematics Subject Classification. Primary 60J45; Secondary 60A05.

Improved estimators for moments constructed from observations of a mixture

Abstract: Procedures for improving weighted empirical distribution functions constructed from mixtures with varying concentrations are considered. The procedures are such that the estimators of moments of the mixture components constructed from weighted empirical distribution functions have specified properties (say, estimators of the variance must not be negative). We prove that the moment estimators constructed from improved weighted empirical distribution functions have the same asymptotic behavior as those constructed from the original weighted empirical distribution functions.

On the product of a random and a real measure

Abstract: The product of a random measure X and a real measure Y is defined as a random measure on X × Y . We obtain conditions under which the integral of a real function with respect to the product measure equals the iterated integrals of this function. Let (X,BX) and (Y,BY ) be measurable spaces, Z = X × Y , and BZ = BX ⊗ BY . By L0 = L0 (Ω,F ,P) we denote the set of all random variables defined on the probability space (Ω,F ,P) (to be more specific, L0 is the set of classes of equivalent random variables). The convergence in L0 is the convergence in probability. Definition 1. Any σ-additive mapping μ : BX → L0 is called a random measure on BX . Note that we do not assume that μ is nonnegative and we do not pose any moment condition. Here are some examples. If X(t), 0 ≤ t ≤ T , is a continuous square-integrable martingale, then μ(A) = ∫ T 0 IA(t) dX(t) is a random measure on Borel sets of [0, T ]. A fractional Brownian motion B(t) for H > 12 defines a random measure in a similar way (this follows from inequality (3.11) in [1]). Other examples as well as conditions for increments of a stochastic process to generate a random measure can be found in Chapters 7 and 8 of [2]. Further let μ be a random measure on BX , and m a finite nonnegative measure on BY . A set A ∈ BX is called μ-negligible if μ(B) = 0 a.s. for all B ∈ BX such that B ⊂ A. Let ξ be a random variable and put ‖ξ‖ = sup{δ : P{|ξ| > δ} > δ}. The integral ∫ A f dμ is defined and studied in [3] where f : X → R is a real measurable function and A ∈ BX . When constructing this integral one starts with simple functions and proceeds similarly to [2, Chapter 7] (see also [4]). In particular, any measurable bounded function f is integrable with respect to any measure μ. In this paper, we define the product of a random and a real measure and prove analogs of Fubini’s theorem for integrals of real functions. Theorem 1. There exists a unique random measure η on BZ such that η(A1 ×A2) = μ(A1)m(A2) 2000 Mathematics Subject Classification. Primary 60G57.

A new (probabilistic) proof of the Diaz–Metcalf and Pólya–Szegő inequalities and some applications

Abstract: The Diaz–Metcalf and Polya–Szegő inequalities are proved in the probabilistic setting. These results generalize the classical case for both sums and integrals. Using these results we obtain some other well-known inequalities in the probabilistic setting, namely the Kantorovich, Rennie, and Schweitzer inequalities.

A structural approach to solving the 6th Hilbert problem

Abstract: The paper deals with an approach to solving the 6th Hilbert problem based on interpreting the field of random events as a partially ordered set endowed with a natural order of random events obtained by formalization and modification of the frequency definition of probability. It is shown that the field of events forms an atomic generated, complete, and completely distributive Boolean algebra. The probability distribution of the field of events generated by random variables is studied. It is proved that the probability distribution generated by random variables is not a measure but only a finitely additive function of events in the case of continuous random variables (both rationaland real-valued).

Properties of distributions of random variables with independent differences of consecutive elements of the Ostrogradskiĭ series

Abstract: Several metric relations for representations of real numbers by the Ostrogradskĭı type 1 series are obtained. These relations are used to prove that a random variable with independent differences of consecutive elements of the Ostrogradskĭı type 1 series has a pure distribution, that is, its distribution is either purely discrete, or purely singular, or purely absolutely continuous. The form of the distribution function and that of its derivative are found. A criterion for discreteness and sufficient conditions for the distribution spectrum to have zero Lebesgue measure are established. Introduction By the Ostrogradskĭı algorithms, any real number x ∈ [0; 1] can be represented as follows: 1 q1 − 1 q1q2 + · · · + (−1) n−1 q1q2 · · · qn + · · · where qn are positive integers such that qn+1 > qn for any n ∈ N, or 1 q1 − 1 q2 + · · · + (−1) n−1 qn + · · · where qn are again positive integers such that qn+1 ≥ qn(qn + 1) for any n ∈ N. The problems related to algorithms of sign alternating series expansion of a number were investigated by M. V. Ostrogradskĭı not very long before his death and were not published. His short notes on the problem were discovered in 1951 in the manuscript section of the Academy of Sciences of Ukraine and further deciphered by E. Ya. Remez in [1]. In that paper, the author points out at a certain analogy between the Ostrogradskĭı series and a continued fraction and devotes much attention to the application of the Ostrogradskĭı series for finding approximate solutions of algebraic equations. Sierpiński [2] studied similar problems independently of Ostrogradskĭı (in the paper [2], there are several algorithms for series expansion of a real number; two of these algorithms give the Ostrogradskĭı series expansion). Pierce probably also worked on the problem (the book [4, p. 10] mentions, referring to [3], the Pierce algorithm for sign alternating series expansion of a real number; the result is the Ostrogradskĭı type 1 series). Gnedenko mentions two algorithms due to Ostrogradskĭı in editor’s remarks to the book [5] noting that there had been no detailed studies of these series by the time of writing (1961). There exist other papers dealing with applications of the Ostrogradskĭı series. Let us mention some of them. The paper [6, pp. 91–96] establishes a link of the Ostrogradskĭı 2000 Mathematics Subject Classification. Primary 60E05, 26A30; Secondary 11A67, 11K55. c ©2005 American Mathematical Society 147 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 148 M. V. PRATS’OVYTYĬ AND O. M. BARANOVS’KĬI algorithms with the algorithm of continued fraction expansion of a number. Some generalizations of these algorithms related to branching continued fractions are also obtained there. In [7], p-adic analogs of the Euclid algorithm and the Ostrogradskĭı algorithm are used for constructing p-adic continued fractions, and unimprovable rates of convergence of the corresponding convergents to the real number are obtained. The same author “combines” in [8] the Engel algorithm and the Ostrogradskĭı algorithm for constructing an algorithm of representation of real numbers by series whose convergence rate is higher than that of the Engel series and that of the Ostrogradskĭı series. The paper [9] should perhaps be considered the first to contain a metric theory of numbers represented by the Ostrogradskĭı series. In this paper, the first Ostrogradskĭı algorithm is studied and estimates for the error of the nth approximation are found. A generalization of the Ostrogradskĭı algorithm for approximations in Banach spaces is also proposed in [9]. In the paper [10], an algorithm for sign alternating series expansion of a number is introduced leading, under a certain choice of the parameters, to the Lüroth series, Engel series, and Ostrogradskĭı series (though the latter series are not studied in that paper). We also note that the algorithm of the Q̃∞-representation (see [11] or [12]) can give, under a certain choice of the set Q̃∞, the sign alternating series expansions of the Lüroth type. In this paper, we study a random variable such that the consecutive terms in the expansion of this variable in the Ostrogradskĭı type 1 series have differences that are independent random variables. The main question in the study of this variable is to determine the structure of its distribution; if it is singular, then the structure of this singular distribution must be established. By the Lebesgue theorem, any distribution function admits a unique representation in the form (1) F (x) = α1Fd(x) + α2Fac(x) + α3Fs(x) where Fd is a discrete distribution function, Fac is an absolutely continuous distribution function, Fs is a singular distribution function, αk ≥ 0, and α1 + α2 + α3 = 1. Representation (1) is called the structure of the distribution function (or the structure of the distribution). Any singular distribution function can be represented as follows: (2) F (x) = γ1F(x) + γ2F(x) + γ3F(x) where F , F , and F are an S-type, a C-type, and a K-type distribution function, respectively, γk ≥ 0, and γ1 + γ2 + γ3 = 1. Representation (2) is called the structure of the singular distribution function (or the structure of the singular distribution) [11, p. 74]. Solving the problem of the structure of a distribution (or that of the structure of a singular distribution) consists in determining the numbers α1, α2, α3, and the functions Fd, Fac, Fs (γ1, γ2, γ3, and F , F , F , respectively). Recall also that random variables having Q-, Q∞-, Q̃∞-representations have already been studied, as well as those represented by a continued fraction or by an Ostrogradskĭı type 2 series whose elements are either independent random variables or form a Markov chain (see [11]). Besides the problem of the structure of the distribution, fractal properties of these random variables have been studied, that is, fractal properties of the distribution spectrum and distribution support. This paper contains four sections. In Section 1, an Ostrogradskĭı type 1 series and its approximant numbers are defined and some lemmas describing certain properties of the approximant numbers are given, as well as a theorem stating that a real number can be represented by an Ostrogradskĭı type 1 series and that this representation is unique. An Ō1-representation of a number is introduced together with the cylindric sets corresponding to the Ō1-representation of a number. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use PROPERTIES OF DISTRIBUTIONS OF RANDOM VARIABLES 149 In Section 2, the set C[Ō1, {Vk}] is introduced containing all numbers x having Ō1representations whose elements take values in the sets V1, V2, . . . , Vk, . . . , respectively. In particular, some sufficient conditions for this set to have zero Lebesgue measure are obtained. In Section 3, we consider a random variable whose elements of the Ō1-representation are independent. The forms of the distribution function of this variable and its derivative are obtained, a criterion for the distribution to be discrete is proved, and sufficient conditions for being Cantor-type singular are established. Section 4 contains a proof of the fact that the Ō1-elements of a uniformly distributed random variable cannot be independent and cannot form a homogeneous Markov chain. 1. Representation of numbers by the Ostrogradskĭı type 1 series Definition 1. An expression of the form (3) q0 + 1 q1 − 1 q1q2 + · · · + (−1) n−1 q1q2 · · · qn + · · · is called an Ostrogradskĭı type 1 series, which is written for brevity as O1(q0; q1, q2, . . . , qn, . . . ) where q0 is an integer, q1, q2, q3, . . . are positive integers and qk+1 > qk for any k ∈ N. The numbers qk are called the elements of the Ostrogradskĭı type 1 series. It is clear that any finite partial sum of series (3) is a rational number as the result of a finite number of rational operations with rational numbers. An infinite series of the form (3) is absolutely convergent under the above assumptions imposed on qk (which can be readily checked by the d’Alembert criterion of convergence of positive series) and is therefore a finite real number (which is irrational by Theorem 1). Definition 2. A number having the form Ak Bk = O1(q0; q1, q2, . . . , qk) = q0 + 1 q1 − 1 q1q2 + · · · + (−1) k−1 q1q2 · · · qk is called an approximant number of order k of the Ostrogradskĭı type 1 series. The following results can readily be proved [13]. Lemma 1 (The law of creation of the approximant numbers). For any positive integer k, we have Ak = Ak−1qk + (−1)k−1, Bk = Bk−1qk = q1q2 · · · qk (assuming that A0 = q0, B0 = 1). Lemma 2. The approximant numbers of even orders form an increasing sequence, while the approximant numbers of odd orders form a decreasing sequence. Moreover, each approximant number of an odd order is greater than any approximant number of an even order. Theorem 1 (M. V. Ostrogradskĭı). Each real number x can be represented by Ostrogradskĭı series (3). Moreover, if x is irrational, this representation is unique and expression (3) contains infinitely many terms; if x is rational, it can be represented in the form (3) with a finite number of terms in two different ways: O1(q0; q1, q2, . . . , qn), O1(q0; q1, q2, . . . , qn − 1, qn). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 150 M. V. PRATS’OVYTYĬ AND O. M. BARANOVS’KĬI The elements q1, q2, q3, . . . of the Ostrogradskĭı series expansion of a number x ∈ (0; 1) can be calculated applying the first Ostrogradskĭı algorithm to the number x: 1 = q1x + α1, 0 ≤ α1 < x, 1 = q2α1 + α2, 0 ≤ α2 < α1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 = qnαn−1 + αn, 0 ≤ αn < αn−1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The representation of a number x in the form (3) is also called the O1-representatio

The local asymptotic normality of a family of measures generated by solutions of stochastic differential equations with a small fractional Brownian motion

Abstract: A formula for the likelihood ratio of measures generated by solutions of a stochastic differential equation with a fractional Brownian motion is established in the paper. We find sufficient conditions that the family of measures generated by solutions of such an equation is locally asymptotically normal. Introduction We consider the stochastic differential equation (1) Xt = x0 + ∫ t 0 S(θ, u, Xu) du + εBt, t ∈ [0, T ], where x0 ∈ R, ε ∈ (0, 1); S(θ, t, x) : R × [0, T ]×R → R is a nonrandom function of drift; θ ∈ Θ ⊂ R is an unknown parameter of the system; Bt = B t is a fractional Brownian motion with the Hurst parameter H ∈ ( 1 2 , 1). Along with equation (1) we consider the deterministic equation (2) xt = x0 + ∫ t 0 S(θ, u, xu) du, t ∈ [0, T ], whose solution is x = x(θ). Equation (1) describes the evolution of a dynamic system with a small noise being a fractional Brownian motion. The problem of the statistical estimation is well studied for systems with a small noise being a standard Brownian process (see [1]). In particular, the consistency and asymptotic normality of the maximum likelihood estimator of the parameter θ is proved under certain assumptions for systems with Brownian noise. As shown in the monograph [2, Chapter II], several important properties of statistical estimators follow from the local asymptotic normality of a system of measures generated by the random element X θ . Thus the proof of the local asymptotic normality is a necessary step to obtain results similar to the Kutoyants results [1] in the case of a fractional Brownian motion. In this paper, we obtain some conditions under which the family of probability measures {P (ε) θ , θ ∈ Θ} generated by solutions of equation (1) that correspond to different parameters θ in the measurable space ( C[0, T ],BT ) is locally asymptotically normal as ε → 0. 2000 Mathematics Subject Classification. Primary 62F12; Secondary 60G15, 60H10.

The invariance principle for a class of dependent random fields

Abstract: Sufficient conditions for the tightness of a family of distributions of partial sum set-indexed processes constructed from symmetric random fields are obtained in this paper. We require that the moments of order s, s > 2, exist. The dependence structure of the field is described by the β1-mixing coefficients decreasing with a power rate. Assuming that a field is stationary and applying a result of D. Chen (1991) on the convergence of finite-dimensional distributions of the processes we obtain the invariance principle.

A strong law of large numbers for generalized almost sure central limit theorems

Abstract: We prove a Strong Law of Large Numbers in which the variables are assumed to be asymptotically negligible and a generalized Almost Sure Central Limit Theorem is given. As an application we obtain a result about the so-called intersective ASCLT.

Asymptotic behavior of median estimators of multiple change points

Abstract: We consider the problem of posterior estimation of multiple change points in the case of only two distributions. We find the asymptotic distribution of the difference between the median estimator of a single change point and the true change point and show that the distribution does not change if the unknown parameter is estimated by a median of the sample. We generalize the results to the case of multiple change points.

Asymptotics of empirical Bayes risk in the classification of a mixture of two components with varying concentrations

Abstract: We consider the problem of classification for a sample from a mixture of several components. For the problem of classification of a two-component mixture with the space of characteristics = [a, b] ⊂ R and smooth distribution densities, we find the precise rate of convergence for the error LN of the empirical Bayes classifier gN to the error L ∗ of the Bayes classifier, namely we prove that N(LN − L∗) ⇒ [A + Bς] where ς is a standard normal random variable, and the empirical Bayes classifier gN is constructed from the kernel estimator of the density of a mixture with varying concentrations. We prove that the kernel estimator with the Epanechnikov kernel is optimal for the empirical Bayes classifier.

On the pricing of equity-linked life insurance contracts in Gaussian financial environment

Abstract: The paper deals with the problem of pricing an equity-linked insurance contract based on stock prices. The stock prices are supposed to follow a stochastic exponent model with respect to a given Gaussian martingale. The model gives a possibility to obtain unified formulas for “mean–variance” hedging and the corresponding premium for both natural cases: Black–Scholes and Gaussian discrete time models.

Series expansion for the probability that a random Boolean matrix is of maximal rank

Abstract: We consider a random (N × n) matrix in the field GF (2) and establish relations that allow one to find the coefficients of the expansion of the probability that a given matrix is of maximal rank into a series in powers of a small parameter. We give explicit formulas for the cases of n = 1 and n = 2, N ≥ n. 1. Setting of the problem Let A = (aij)i∈I,j∈J be a matrix with N rows and n columns, where I = {1, . . . , N} and J = {1, . . . , n}. The entries of the matrix A are independent random variables that assume values in the field GF (2) and have distribution (1) P{aij = 0} = 1 − P{aij = 1} = 2−1 (1 + εxij) where ε is a fixed small number, ε ≥ 0, and xij ∈ (−∞,∞). Denote by χ(A) the following indicator: χ(A) = ⎧⎪⎨ ⎪⎩ 1 if the matrix A contains n linearly independent (in the field GF (2)) N -dimensional columns; 0, otherwise. Using relation (1), the probability of the event {χ(A) = 1} can be represented in the following form: (2) P{χ(A) = 1} = nN ∑ s=0 εf (xij , i ∈ I, j ∈ J) where the coefficients f (xij , i ∈ I, j ∈ J), s ≥ 0, are real numbers that do not depend on ε. Let m = N −n. In the case of m = 0, a recurrence relation with respect to n is found in [1] to evaluate f (xij , i ∈ I, j ∈ J), s ≥ 0; for the case of (3) m ≥ 0 the coefficients f (xij , i ∈ I, j ∈ J), s ∈ {0, 1, 2}, are found in [2] in an explicit form by applying different approaches depending on s ∈ {0, 1, 2}. The aim of this paper is to find a relation that allows one to evaluate the coefficients f (xij , i ∈ I, j ∈ J), s ≥ 1, of the expansion of the probability that a random (N × n) matrix in the field GF (2) is of the maximal rank n into a series in terms of powers of a small parameter ε. Our 2000 Mathematics Subject Classification. Primary 60C05, 15A52, 15A03. c ©2005 American Mathematical Society 93 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

On an extension of the Lyapunov criterion of stability for quasi-linear systems via integral inequalities methods

Abstract: In this article, we concern ourselves with a new concept for comparing the stability degree of two dynamical systems. By using the integral inequality method, we give a criterion which allows us to compare the growth rate of two Itô quasi-linear differential equations. It can be viewed as an extension of the Lyapunov criterion to the stochastic case.

Limit distributions of extreme values of bounded independent random functions

Abstract: We study the limit probabilities that extreme values of a sequence of independent normal random functions belong to extending intervals.