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In the last twenty years extensive research has been devoted to a better understanding of the stable and other closely related infinitely divisible mod­ els. Stamatis Cambanis, a distinguished educator and researcher, played a special leadership role in the development of these research efforts, particu­ larly related to stable processes from the early seventies until his untimely death in April '95. This commemorative volume consists of a collection of research articles devoted to reviewing the state of the art of this and other rapidly developing research and to explore new directions of research in these fields. The volume is a tribute to the Life and Work of Stamatis by his students, friends, and colleagues whose personal and professional lives he has deeply touched through his generous insights and dedication to his profession. Before the idea of this volume was conceived, two conferences were held in the memory of Stamatis. The first was organized by the University of Athens and the Athens University of Economics and was held in Athens during December 18-19, 1995. The second was a significant part of a Spe­ cial IMS meeting held at the campus of the University of North Carolina at Chapel Hill during October 17-19, 1996. It is the selfless effort of sev­ eral people that brought about these conferences. We believe that this is an appropriate place to acknowledge their effort; and on behalf of all the participants, we extend sincere thanks to all these persons.



Spectral Representation and Structure of Stable Self-Similar Processes

In this paper we establish a spectral representation of any symmetric stable self-similar process in terms of multiplicative flows and cocycles. A structure of this class of self-similar processes is studied. Applying the Lamperti transformation, we obtain a unique decomposition of a symmetric stable self-similar process into three independent parts: mixed fractional motion, harmonizable and evanescent. This decomposition is illustrated by graphical presentation of corresponding kernels of their spectral representations.
K. Burnecki, J. Rosiński, A. Weron

Three Elementary Proofs of the Central Limit Theorem with Applications to Random Sums

Three simple proofs of the classical CLT are presented. The proofs are based on some basic properties of covariance kernels or ω-functions in conjunction with bounds for the total variation distance. Applications to random sum CLT’s are also given.
T. Cacoullos, N. Papadatos, V. Papathanasiou

Almost Everywhere Convergence and SLLN Under Rearrangements

The almost everywhere (a.e.) convergence of trigonometric Fourier series for L 2(0, 1) functions was conjectured by Luzin (1922) and was partially solved by Kolmogorov and Silvestrov in (1925). The full solution was given by Carleson (1966). In the work of Garsia (1964), (see Garsia (1970)) almost everywhere convergence of a rearrangement of series of orthogonal functions was initiated using the so-called Garsia Inequality (GI). His convergence result was generalized by Nikishin (1967) who removed the assumption of orthogonality. We prove an inequality which generalizes GI using the technique introduced in Chobanyan (1990) (for other references see Chobanyan (1994)). As a consequence of this inequality we derive GI and a generalization of Nikishin’s result to a series of Banach space valued random variables.
Sergei Chobanyan, V. Mandrekar

Sufficient Conditions for the Existence of Conditional Moments of Stable Random Variables

Conditional moments E[X 2|p|X 1 = x] of an α-stable random vector (X 1,X 2) may exist even if p ⩾ α. The precise conditions are stated in Cioczek-Georges and Taqqu [4]_and Samorodnitsky and Taqqu [12]. This paper provides the proof for the most delicate cases, namely 1 < α< 2 and p < 2α+ 1, which is the maximal range of possible p’s when the vector (X 1, X 2) is nondegenerate.
Renata Cioczek-Georges, Murad S. Taqqu

How Heavy are the Tails of a Stationary HARCH(k) Process? A Study of the Moments

Probabilistic properties of HARCH(k) processes as special stochastic volatility models are investigated. We present necessary and sufficient conditions for the existence of a strongly stationary version of a HARCH(k) process with finite (2m)th moments, m ⩾ 1. Our approach is based on the general Markov chain techniques of (Meyn and Tweedie, 1993). The conditions are explicit in the case of second moments, and also in the case of 4th moments of the HARCH(2) process. We also deduce explicit necessary and explicit sufficient conditions for higher order moments of general HARCH(k) models. We start by studying the HARCH(2) process (in which case our results are the most explicit) and then generalize the results to a general HARCH(k) process.
Paul Embrechts, Gennady Samorodnitsky, Michel M. Dacorogna, Ulrich A. Müller

Use of Stochastic Comparisons in Communication Networks

Many problems of optimization in the field of communication networks are difficult to track. However, use of sample-path comparison methods can occasionally be of help. In this paper we review a few instances where such comparisons have led to helpful results.
A. Ephremides

On the Conditional Variance-Covariance of Stable Random Vectors, II

Under the assumption that X = (X 1, X 2) a (n 1 + n 2)-dimensional vector is strictly α-stable distributed, the conditional variance-covariance of X 2 given X 1 is expressed in terms of the spectral measure T. Moreover, if some additional assumptions on the vector X 1 are imposed such that the coordinates are statistically independent, then an additive expression for the conditional variance-covariance is found. A trigonometric unified method is presented for establishing these expressions.
Stergios B. Fotopoulos

Interacting Particle Approximation for Fractal Burgers Equation

The paper reports on the existence of McKean’s nonlinear processes and the related propagation of chaos results for a class of one-dimensional (1-D) generalized Burgers-type equations with a fractional power of the Laplacian in the principal part and a quadratic nonlinearity. Such equations naturally appear in continuum mechanics.
T. Funaki, W. A. Woyczyński

Optimal Transformations for Prediction in Continuous-Time Stochastic Processes

In the classical Wiener-Kolmogorov prediction problem, one fixes a functional of the “future” and seeks its best predictor (in the L 2-sense). In this paper we treat a variant of this problem, whereby we seek the “most predictable” non-trivial functional of the future and its best predictor. In contrast to the Wiener-Kolmogorov problem, our problem may not have solutions, and if solutions exist, they might not be unique. We prove the existence of solutions for linear functionals under appropriate conditions on the spectral function of weakly stationary, continuous-time processes.
B. Gidas, A. Murua

Algebraic Methods Toward Higher-Order Probability Inequalities

In work motivated by problems in the analysis of variance, Kimball [15]_proved that if Vi = Qi/Q0, i = 1,…,n, where the Q i are mutually independent positive random variables, then V 1,...,V n are positively upper orthant dependent (PUOD),
$$P(\mathop \cap \limits_{i = 1}^n \{ {V_i} \geqslant {v_i}\} ) \geqslant \prod\limits_{i = 1}^n {P({V_i} \geqslant {v_i}),} {v_1}, \ldots ,{v_n} \geqslant 0,$$
and also are positively lower orthant dependent (PLOD),
$$P(\mathop \cap \limits_{i = 1}^n \{ {V_i} \leqslant {v_i}\} ) \geqslant \prod\limits_{i = 1}^n {P({V_i} \leqslant {v_i}),} {v_1}, \ldots ,{v_n} \geqslant 0.$$
These results motivated much research on other inequalities; cf. [4], [10], [20], [24].
Kenneth I. Gross, Donald St. P. Richards

Comparison and Deviation from a Representation Formula

Let X ~ ID(b, ΣE, v), i.e., let X be a d-dimensional infinitely divisible random vector with characteristic function
$$\varphi (t) = \exp \left\{ {i\langle t,b\rangle - \frac{1}{2}\langle \sum t ,t\rangle + \int_{{\mathbb{R}^d}} {({e^{i\langle t,u\rangle }} - 1 - i\langle t,u\rangle 1(\left| u \right|} < 1))\nu (du)} \right\},$$
where t, b ∈ ℝ d , Σ is a positive semidefinite d × d matrix and v (the Lévy measure) is a positive measure on B(ℝ d ), the Borel σ-algebra of ℝ d , without atom at the origin and such that \(\int_{\mathbb{R}^d } {(\left| u \right|} ^2 \wedge 1)\nu (du) < + \infty (\langle \cdot , \cdot \rangle {e_{1}}, \ldots ,{e_{n}},e_{i}^{2} = - 1, \) and ∣ · ∣ are respectively the Euclidean inner product and norm in ℝ d ).
Christian Houdré

Components of the Strong Markov Property

The strong Markov property of a process X at an optional time π < ∞ may be thought of as a combination of the conditional independence XT+hM-xrFT with the homogeneity for a suitable set of probability kernels. In an earlier paper, a stronger version of the latter condition was shown to imply the former property. Our present aim is to examine to what extent the two properties are in fact equivalent
Olav Kallenberg

The Russian Options

In the European and American options of option pricing theory, the time period between the time the option is purchased and the time at or before which the option has to be exercised is fixed and known. If the purchase time is taken to be t = 0 and the exercise time t = T, then the European option pricing theory requires the option to be exercised at t = T (the date of maturity) while under the American option, you can exercise it at any time up to T and moreover, the exercise time can be random.
G. Kallianpur

Cycle Representations of Markov Processes: An Application to Rotational Partitions

The cycle formula asserts that any finite-order recurrent stochastic matrix P is a linear combination of matrices J c = (J c(i, j)) associated with the cycles c of the graph of P, and defined as follows: J c (i, ii) = 1 or 0, according to whether i, j are consecutive points of c or not.
In the present paper we investigate, by using the cycle formula, the asymptotic behavior of the sequence (t, m S), t ⩾ 0, of rotational representations associated to the powers P m, m = 0, 1, 2,…, of an irreducible stochastic matrix on a finite set S = {1,…, n}, n ⩾ 2. In particular, we give a criterion on the rotational partitions m S, m = 0, 1,…, for the sequence P m m to be convergent. A pair (t, S) is a rotational representation of P = (p ij, i, j = 1,…, n) if S is a partition of [0, 1) into n sets S 1,…, S n, each of positive Lebesgue measure and consisting of a finite union of arcs, such that Pij = . Here f t is the A-preserving transformation of [0, 1) onto itself defined by f t (x) = (x + t)(mod 1), and A denotes Lebesgue measure
S. Kalpazidou

On Extreme Values in Stationary Random Fields

This paper develops distributional extremal theory for maxima M T = max(X t: 0 ⩽ t ⩽ T) of a stationary random field X t. A general form of “extremal types theorem” is proven and shown to apply to M T under very weak dependence restrictions. That is, any non-degenerate distributional limit for the normalized family a T(MT - b T) (a T > 0) must be one of the three classical types. Domain of attraction criteria are discussed.
The dependence structure used here for fields involves a potentially very weak type of strong-mixing, “Coordinatewise (Cw) mixing”) using mild individual “past-future” conditions in each coordinate direction. Together with careful control of numbers and sizes of sets involved, this avoids the over-restrictive nature of common generalizations of mixing conditions to apply to random fields. Futher, the conditions may be readily adpated to deal with other quite general problems of Centeral Limit type (cf. [6]).
M. R. Leadbetter, Holger Rootzén

Norming Operators for Operator-Self-Similar Processes

There are many similarities between the theory of operator-stable distributions and that of operator-self-similar processes as discussed by Mason [6].
Makoto Maejima

Multivariate Probability Density and Regression Functions Estimation of Continuous-Time Stationary Processes from Discrete-Time Data

Let be a real-valued continuous-time jointly stationary processes and let tj be a renewal point process on [0,00], with finite mean rate independent of (Y,X). Given the observations and a measurable function, we estimate the multivariate probability density and the regression function of given X(0) = xo, X(T) = XI, …, X(r m ) = x m for arbitrary lags m. We present consistency and asymptotic normality results for appropriate estimates of f and r.
Elias Masry

Tracing the Path of a Wright-Fisher Process with One-way Mutation in the Case of a Large Deviation

The large deviations theory developed by Wentzell for discrete time Markov chains is used to show that if the state of a Wright-Fisher process modeling the frequency of an allele in a biological population undergoes a transition from one value to another over a number of generations which is large but much smaller than the size of the population, then there is a preferred path which the process follows closely with near certainty in the intervening time. This path was identified in [8]_in the case in which there is only random drift acting on the genes. The case in which one-way mutation is added to the drift was cursorily mentioned in [8]_and is fully treated here. The preferred path in this case is shown to be an exponential, a parabola, a hyperbolic cosine or a trigonometric cosine, depending on the mutation parameter and the boundary conditions involved.
F. Papangelou

A Distribution Inequality For Martingales with Bounded Symmetric Differences

We discuss (with some generalizations) the question of optimal strategies for the game of red-and-black with a fixed goal, with fixed subfair odds, and with both the size of legal bets allowed and the playing time limited.
Loren D. Pitt

Moment Comparison of Multilinear Forms in Stable and Semistable Random Variables with Application to Semistable Multiple Integrals

Let 1 < α< 2. We provide a uniform comparison of the tail probabilities of (non-symmetric) strictly α-semistable random variables with the tail probabilities of their symmetrized counterparts as well as of their “associated” strictly & symmetric α-stable random variables. We use this to obtain a uniform comparison between the moments of the multilinear forms in (non-symmetric) strictly α-semistable random variables on the one hand and in their symmetrized counterparts as well as in their “associated” strictly, & symmetric α-stable random variables on the other. In turn, using this and following the approach of Krakowiak and Szulga in the stable case, we construct strictly and symmetric α-semistable multiple stochastic integrals of Banach space-valued integrands.
Balram S. Rajput, Kavi Rama-Murthy, Xavier R. Retnam

Global Dependency Measure for Sets of Random Elements: “The Italian Problem” and Some Consequences

We suggest a detailed analysis of the classical independence/dependence properties for finite sets of random events or variables. All possible combinations of random elements are considered as a configuration obeying a hierarchical property. We define a function called a Dependency Measure taking values in the interval [0, 1] (0 corresponds to mutually independent sets; 1 corresponds to totally dependent sets) and serving as a global measure of the amount of dependency which is contained in the whole set of random elements. This leads to “The Italian Problem” about the existence of a probability model and a set of random elements with any prescribed independence/dependence structure. Some consequences and nonstandard illustrative examples are given. Related properties such as exchangeability and association are also discussed.
Jordan Stoyanov
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