On the transition from a Markov chain to a continuous time process

Starting from a real-valued Markov chain X₀,X₁,…,X_n with stationary transition probabilities, a random element {Y(t);t[0, 1]} of the function space D[0, 1] is constructed by letting Y(k/n)=X_k, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t[0,1]};_n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes.     

A dimension gap for continued fractions with independent digits

Kinney and Pitcher (1966) determined the dimension of measures on [0, 1] which make the digits in the continued fraction expansion i.i.d. variables. From their formula it is not clear that these dimensions are less than 1, but this follows from the thermodynamic formalism for the Gauss map developed by Walters (1978). We prove that, in fact, these dimensions are bounded by 1−10⁻⁷. More generally, we considerf-expansions with a corresponding absolutely continuous measureμ under which the digits form a stationary process. Denote byE _δ the set of reals where the asymptotic frequency of some digit in thef-expansion differs by at leastδ from the frequency prescribed byμ. ThenE _δ has Hausdorff dimension less than 1 for anyδ>0.     

Series of independent,mean zero random variables in rearrangement-invariant spaces having the Kruglov property

This paper compares sequences of independent, mean zero random variables in a rearrangement-invariant space X on [0, 1] with sequences of disjoint copies of individual terms in the corresponding rearrangement-invariant space Z _X ² on [0, ∞). The principal results of the paper show that these sequences are equivalent in X and Z _X ² , respectively, if and only if X possesses the (so-called) Kruglov property. We also apply our technique to complement well-known results concerning the isomorphism between rearrangement-invariant spaces on [0, 1] and [0, ∞). Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 25–50.     

Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces

We define stochastic integrals of Banach valued random functions w.r.t. compensated Poisson random measures. Different notions of stochastic integrals are introduced and sufficient conditions for their existence are established. These generalize, for the case where integration is performed w.r.t. compensated Poisson random measures, the notion of stochastic integrals of real valued random functions introduced in Ikeda and Watanabe (1989) [Stochastic Differential Equations and Diffusion Processes (second edition), North-Holland Mathematical Library, Vol. 24, North Holland Publishing Company, Amsterdam/Oxford/New York.], (in a different way) in Bensoussan and Lions (1982) [Contróle impulsionnel et inquations quasi variationnelles. (French) [Impulse control and quasivariational inequalities] Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science], Vol. 11. (Gauthier-Villars, Paris), and Skorohod, A.V. (1965) [Studies in the theory of random processes (Addison-Wesley Publishing Company, Inc, Reading, MA), Translated from the Russian by Scripta Technica, Inc. ], to the case of Banach valued random functions. The relation between these two different notions of stochastic integrals is also discussed here.     

Dual representations of hulls for functions satisfying f (0) = inf f (X\{0}) *

Let Xbe a setW set of extended-real valued functions on Xand 0 an element of Xsuch that w(0) = 0 for all win W. We give a general theory of dual representations, without any extra parameters, of various hulls for extended-real valued functions on X satisfying f(0)= inf f(X\{0})We use the unifying framework of quasi-convex functions with respect to families of subsets of Xand a condition generalizing the bipolar theorem. Our results contain, as particular cases, some recent results of Thach (1991, 1993) and Rubinov and Glover (1996) on W-pseudo-affine and W-quasi-coaffine hulls.We also give some

results in the converse direction. These yield, in particular, that the bipolar theorem is equivalent to a certain property of the lower semi-continuous quasi-convex hulls of functions on locally convex spaces     

Superfractality of the set of numbers having no frequency ofn-adic digits,and fractal probability distributions

We study the fractal properties (we find the Hausdorff-Bezikovich dimension and Hausdorff measure) of the spectrum of a random variable with independentn-adic (n2,n N digits, the infinite set of which is fixed. We prove that the set of numbers of the segment [0, 1] that have no frequency of at least onen-adic digit is superfractal.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 47, No. 7, pp. 971–975, July, 1995.     

The maximal and minimal 2-correlation of a class of 1-dependent 0–1 valued processes

We compute the maximal and minimal value ofP[X _N =X _N+1=1] for fixedP[X _N =1], where (X _N ) _N∈Z is a 0–1 valued 1-dependent process obtained by a coding of an i.i.d.-sequence of uniformly [0,1] distributed random variables with a subset of the unit square. This research was supported by the Netherlands Foundation for Mathematics (S.M.C.) with financial aid from the Netherlands Organization for the Advancement of Pure Research (ZWO).     

On almost sure max-limit theorems of complete and incomplete samples from stationary sequences

Let M _n denote the partial maximum of a strictly stationary sequence (X _n ). Suppose that some of the random variables of (X _n ) can be observed and let [(M)\tilde]_n\tilde M_n stand for the maximum of observed random variables from the set {X ₁, ..., X _n }. In this paper, the almost sure limit theorems related to random vector ([(M)\tilde]_n\tilde M_n , M _n ) are considered in terms of i.i.d. case. The related results are also extended to weakly dependent stationary Gaussian sequence as its covariance function satisfies some regular conditions.     

Renewal theorem for a class of stationary sequences

Summary A renewal theorem is obtained for stationary sequences of the form _n=(...,X _n-1,X _n,X _n+1...), whereX _n, , are i.i.d. r.v.s. valued in a Polish space. This class of processes is sufficiently broad to encompass functionals of recurrent Markov chains, functionals of stationary Gaussian processes, and functionals of one-dimensional Gibbs states. The theorem is proved by a new coupling construction.Research supported by the National Science Foundation     

Transformations preserving the Hausdorff-Besicovitch dimension

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R ¹ resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.      

Singular and fractal properties of distributions of random variables digits of polybasic representations of which a form homogeneous Markov chain

We study the fractal properties of distributions of random variables digits of polybasic Q-representations (a generalization of n-adic digits) of which form a homogeneous Markov chain in the case where the matrix of transition probabilities contains at least one zero.     

Characterizations and embedding properties in exchangeability

Summary Our key result is the characterization of exchangeable sequences as being strongly stationary, i.e. invariant in distribution under stopping time shifts. From this we prove homogeneity characterizations of pure and mixed Markov chains. These results carry over to continuous time processes and random sets, and on a whole, our theory provides a unified approach to exchangeability. The key result above is closely related to Dacunha-Castelle's embedding characterization of exchangeability, which is partially extended here to processes on [0, 1]. In the other direction, we prove that a previsible sample from a finite or infinite exchangeable sequence X may be embedded into a copy of X. We finally establish some uniqueness results for exponentially and uniformly killed exchangeable random sets.     

A remark on random and equidistributed sequences

The average of the values of a function f on the points of an equidistributed sequence in [0, 1] ^s converges to the integral of f as soon as f is Riemann integrable. Some known low discrepancy sequences perform faster integration than independent random sampling (cf. [1]). We show that a small random absolutely continuous perturbation of an equidistributed sequence allows to integrate bounded Borel functions, and more generally that, if the law of the random perturbation doesn't charge polar sets, such perturbed sequences allow to integrate bounded quasi-continuous functions.     

Semi‐proper forcing,remarkable cardinals,and Bounded Martin's Maximum

We show that L(?) absoluteness for semi‐proper forcings is equiconsistent with the existence of a remarkable cardinal, and hence by [6] with L(?) absoluteness for proper forcings. By [7], L(?) absoluteness for stationary set preserving forcings gives an inner model with a strong cardinal. By [3], the Bounded Semi‐Proper Forcing Axiom (BSPFA) is equiconsistent with the Bounded Proper Forcing Axiom (BPFA), which in turn is equiconsistent with a reflecting cardinal. We show that Bounded Martin's Maximum (BMM) is much stronger than BSPFA in that if BMM holds, then for every X ∈ V , X^# exists. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compactness properties of the integration map for Volterra measures in non-normable spaces

We make a detailed study of the compactness properties of the integration map associated with the classical Volterra (vector valued) measure inX=L ^p([0, 1]), forp in [1, ∞], and inX=C([0, 1]), whereX is equipped with its relevant weak or weak-star topology. The spaceX (a Frechet space) of all complex valued sequences, equipped with the pointwise convergence topology, is also considered.     

Minimal expansions in redundant number systems: Fibonacci bases and Greedy algorithms

We study digit expansions with arbitrary integer digits in base q (q integer) and the Fibonacci base such that the sum of the absolute values of the digits is minimal. For the Fibonacci case, we describe a unique minimal expansion and give a greedy algorithm to compute it. Additionally, transducers to calculate minimal expansions from other expansions are given. For the case of even integer bases q, similar results are given which complement those given in [6].     

An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables

Summary In this paper we establish an almost sure invariance principle with an error termo((t log logt)^1/2) (ast) for partial sums of stationary ergodic martingale difference sequences taking values in a real separable Banach space. As partial sums of weakly dependent random variables can often be well approximated by martingales, this result also leads to almost sure invariance principles for a wide class of stationary ergodic sequences such as ø-mixing and -mixing sequences and functionals of such sequences. Compared with previous related work for vector valued random variables (starting with an article by Kuelbs and Philipp [27]), the present approach leads to a unification of the theory (at least for stationary sequences), moment conditions required by earlier authors are relaxed (only second order weak moments are needed), and our proofs are easier in that we do not employ estimates of the rate of convergence in the central limit theorem but merely the central limit theorem itself.     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Superbranching processes and projections of random Cantor sets

We study sequences (X ₀, X ₁, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that , for m, n0, where the X _n (m, i) are distributed as X _n and have certain properties of independence. We prove that, under appropriate conditions, X _n ^1/n almost surely and in L ¹, where =sup E(X _n )^1/n . Our principal application of this result is to study the Lebesgue measure and (Hausdorff) dimension of certain projections of sets in a class of random Cantor sets, being those obtained by repeated random subdivisions of the M-adic subcubes of [0, 1] ^d . We establish a necessary and sufficient condition for the Lebesgue measure of a projection of such a random set to be non-zero, and determine the box dimension of this projection.Work done partly whilst visiting Cornell University with the aid of a Fulbright travel grant     

The parity of the Zeckendorf sum-of-digits function

Let s(n) denote the sum of digits of the Zeckendorf representation of n and . The aim of this paper is to discuss the behaviour of $S_{q,i}(N)$. First it is shown that that the values of admit a Gaussian limit law with bounded mean and variance of order log N. Conversely, for q≤1 (mostly) has a periodic fractal structure. We also prove that which is an analogue to a well-known result by Newman [14] for binary digit expansions. Received: 15 March 1999 / Revised version: 5 November 1999