Ionescu Tulcea Research Articles

A Probabilistic View on Predictive Constructions for Bayesian Learning

Given a sequence X=(X1,X2,…) of random observations, a Bayesian forecaster aims to predict Xn+1 based on (X1,…,Xn) for each n≥0. To this end, in principle, she only needs to select a collection σ=(σ0,σ1,…), called “strategy” in what follows, where σ0(·)=P(X1∈·) is the marginal distribution of X1 and σn(·)=P(Xn+1∈·|X1,…,Xn) the nth predictive distribution. Because of the Ionescu–Tulcea theorem, σ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence X is determined. The strategies concern generalized Pólya urns, random change points, covariates and stationary sequences.

A former student remembering Professor C. T. Ionescu Tulcea and the period when he taught mathematics in the Faculty of Mathematics and Physics in Bucharest

I was one of the students of Professor Cassius Tocqueville Ionescu Tulcea in his last years of teaching at the University of Bucharest, during the first semester of my fifth year (1956-1957). The course was on Orthogonal Series, in fact it was more linked to his monograph on Hilbert Spaces [6], published in 1956, with a Note by I. Barbalat. On my copy there is a handwritten note “May 27, 1956. The IVth Congress of the Romanian Mathematicians”, a congress of a historical importance, mainly due to the contribution of Professor Simion Stoilow. I spent my years as a student during a very difficult period of time. After Stalin’s death in March 1953, the ground began to shift and in October of 1956 we had a revolution in Hungary which, in the absence of an attitude from the West, ended in a brutal repression. We were all the more fortunate to have then excellent teachers. One of those remarkable teachers whose memory is so dear to me and to whom I am so grateful was professor C.T. Ionescu Tulcea. I also recall with feelings of gratitude my other professors, Miron Nicolescu (1903 1975), Solomon Marcus, Gr. C. Moisil (19061973), Dan Barbilian (1895-1961), Mihail Benado (1920-?). I had the privilege to have very fine colleagues such as Sergiu Rudeanu, Alexandra Bagdasar Bellow, Kostache Teleman, Ion Cuculescu, presently a member of the Romanian Academy, Aldo Lazar, Ion Zamfirescu, Paul Voiculescu who graduated with a thesis supervised by Professor C. T. Ionescu Tulcea, Horia Banea, Mariuca Halalau, Laura Constantinescu who all graduated with me in 1957. Also Valentin Poenaru, Gheorghe Gussi, Ciprian Foias, Nicu Boboc who graduated one year earlier than me. During those years scientific criteria were consistently observed. Our mathematical education was of high quality and it was not elitist. For those years one should mention an increase of interest for partial ordering, e.g., the doctoral thesis of Professor Solomon Marcus “Monotone functions for two variables”in 1956, and the paper of Mihail Benado on multilatices (multistructures) in 1953. I mention below a few courses taught during those years. They are somewhat

From Euler to Bourbaki, by way of C. T. Ionescu Tulcea. An episode from the history of university mathematics in Romania

At the end of August 2013, the Department of Mathematics and Informatics of the University of Bucharest marked, through a scientific meeting, the anniversary of 150 years since the founding of the Faculty of Sciences in Bucharest, which included Mathematics and Natural Sciences. Being invited to present in the inaugural plenary session the history of the university mathematics in Bucharest, I examined documents dating from the 19th century until today. I found out that, taking over the relay from the Royal Academies (in Romanian: Academiile Domnesti) of Iasi and Bucharest, and from the Saint Sava School initiated by Gheorghe Lazar at the beginning of the 19th century, the university mathematics in Bucharest has evolved, throughout decades, very slowly, frozen in programs that expressed the state of mathematics in the 18th century. This situation lasted until the fourth decade of past century. Of course, due to mathematicians like David Emmanuel, Traian Lalescu, Simion Stoilow, Alexandru Myller, Gheorghe Titeica, Dimitrie Pompeiu, Gheorghe Vranceanu, Miron Nicolescu, Grigore C. Moisil, and a few others, there were some isolated attempts of renewal, of incorporating, in university courses some achievements of mathematics in the 19th century and at the beginning of 20th century, such as the epsilon-delta analysis, crystallized by Cauchy, Riemann and Weierstrass, the Galois theory, the Cantor set theory, the theory of integral equations, modern logic, etc. But the dominant spirit of the Romanian university mathematics remained that based on calculations and less on ideas, a typical situation in the 18th century. To a large extent, that situation didn’t exist only in Romania. It is surprising to find out how difficult it was for the epsilon-delta analysis to get into the education programs in France. Goursat’s treatise, in the spirit preceding the epsilon-delta rigor, dominated the teaching of Mathematical Analysis in French universities until the fifth decade of past century. But the moment of change arrived. When? At the time when we would have the least expected it: during the Second World War. Even at the beginning of the forties, but especially in 1942, several Romanian mathematicians, primarily Stoilow,

On the effect of perturbation of conditional probabilities in total variation

A celebrated result by A. Ionescu Tulcea provides a construction of a probability measure on a product space given a sequence of regular conditional probabilities. We study how the perturbations of the latter in the total variation metric affect the resulting product probability measure.

Cassius Tocqueville Ionescu Tulcea in occasion of his 90-th Birthday

C. T. Ionescu Tulcea is celebrating this year his 90th Birthday. He is joined by the Romanian Mathematical Diaspora and a great number of colleagues from Romania and United States of America, who are sending him their best wishes for a continuous research activity and long life. Professor Ionescu Tulcea was born in Bucharest, Romania, on October 14, 1923. He received his elementary and secondary education in distinguished schools from Bucharest and in 1942 became a student at his Alma Matter, the University of Bucharest, the Faculty of Sciences, Mathematical Section. He graduated from this school in 1946, obtaining his Diploma in Mathematics. The University of Bucharest counted among its Professors some distinguished mathematicians, like Octav Onicescu (Probability Theory), Miron Nicolescu (Mathematical Analysis), Gheorghe Vranceanu (Geometry), Nicolae Theodorescu (Differential Equations), Simion Stoilow (Complex Variables), all of them former students of great mathematicians from France or Italy. During his four years of study at the University of Bucharest, Ionescu Tulcea has acquired a solid training as a research mathematician, not only in classical mathematics, as represented in Europe before the Second World War, but also in the relatively new branches of Topology or Functional Analysis. Following his graduation in 1946, Ionescu Tulcea has been appointed on a position of Assistant at the Chair of Probability Theory and remained with the University of Bucharest for the next 11 years. He has been promoted from the Assistant position to Associate Professor (1951) and to Full Professor (1957), at the Chair of Mathematical Analysis. As a faculty member, Ionescu Tulcea has developed a teaching activity that qualified him as the main promoter of modern mathematics at the Bucharest University, in the field of Modern Analysis. To be more specific, he gave his lectures using Bourbaki’s books and other up to date treatises, which found their place in the programs only after World War II. Moreover, as a research fellow of the Mathematical Institute of the Romanian Academy (1949-1957), Ionescu Tulcea organized seminars on Operator Theory, substantially contributing to a generation of Romanian specialists in this field, among

General theories unifying ergodic averages and martingales

The curious fact that the behavior of ergodic averages is analogous to the behavior of (reversed) martingales has long been known. For the last 60 years, at least five different approaches have been proposed to the construction of a unifying theory: by M. Jerison (1955), G.-C. Rota (1961), A. and C. Ionescu Tulcea (1963), A.M. Vershik (1960s), and A.G. Kachurovskii (1998). The aim of the present survey is to carry out a comparative analysis of all the approaches, with special focus being placed on the fifth approach, which is due to the present author and has not yet been published in full detail. Moreover, the comparative analysis carried out in the survey has led to a new, the sixth, approach.

ON STRONG LIFTINGS ON PROJECTIVE LIMITS

The main problem investigated in this paper is the following. Assume that we are given a convergent projective system of topological measure spaces ordered by ordinals. When does there exist a consistent system of liftings (densities, linear liftings) on the projective system converging to a lifting (density, linear lifting) on the limit space. We look mainly for strong or strong completion Baire liftings. We reduce the problem to the question about the existence of strong liftings being inverse images of other strong liftings under measure preserving mappings (Proposition 2.4) and then we adapt a condition applied earlier by A. and C. Ionescu Tulcea [14] to get a strong lifting for an arbitrary measure on a product space (Theorem 2.7). In this way we get some results (see Theorems 2.7, 5.3, 5.7, 6.4 and 6.5) extending the well known achievements of A. and C. Ionescu Tulcea [14] and Fremlin [9].The application of projective limits allows us to carry over results obtained earlier only for product spaces (see e.g. [23], [18], [19], [20], [21]) to more general classes of topological probability spaces. In particular, we can extend the class of spaces for which there is a positive answer to a problem of J. Kupka [17] concerning the permanence of the strong lifting property under the formation of products (see Theorem 6.5).

On strong liftings for projective limits

The main problem investigated in this paper is the following. Assume that we are given a convergent projective system of topological measure spaces ordered by ordinals. When does there exist a consistent system of liftings (densities, linear liftings) on the projective system converging to a lifting (density, linear lifting) on the limit space. We look mainly for strong or strong completion Baire liftings. We reduce the problem to the question about the existence of strong liftings being inverse images of other strong liftings under measure preserving mappings (Proposition 2.4) and then we adapt a condition applied earlier by A. and C. Ionescu Tulcea (14 )t o get as trong lifting for an arbitrary measure on a product space (Theorem 2.7). In this way we get some results (see Theorems 2.7, 5.3, 5.7, 6.4 and 6.5) extending the well known achievements of A. and C. Ionescu Tulcea (14 )a nd Fremlin (9). The application of projective limits allows us to carry over results obtained earlier only for product spaces (see e.g. (23), (18), (19), (20), (21)) to more general classes of topological probability spaces. In particular, we can extend the class of spaces for which there is a positive answer to a problem of J. Kupka (17 )c oncerning the permanence of the strong lifting property under the formation of products (see Theorem 6.5).

Sur un théorème spectral et son application aux noyaux lipchitziens

The aim of this note is to point out that the Ionescu Tulcea and Marinescu theorem can be reinforced, using a Nussbaum formula for the essential spectral radius of an operator. In this stronger version, this theorem is suitable for the spectral analysis of lipschitzian, positive, not necessarily markovian kernels, Ruelle theorem follows. As an application to Probability Theory, a large deviation theorem is proved.

The Lattice Property of Uniform Amarts

In this note it is shown that the $L^1$-bounded uniform amarts in $l^1$ form a vector lattice. This extends a result of Austin, Edgar and Ionescu Tulcea on real-valued $L^1$-bounded amarts and parallels a result of Ghoussoub on $L^1$-bounded order amarts.

A Comparison of Methods for Constructing Probability Measures on Infinite Product Spaces

AbstractThe construction, from a consistent family of finite dimensional probability measures, of a probability measure on a product space when the marginal measures are perfect is shown to follow from a classical theorem due to Ionescu Tulcea and known results on the existence of regular conditional probability functions.

On the existence of markov processes with given transition functions

It is known that a Markov transition function defines the corresponding Markov process (measure) if phase spaces are “good” or, for arbitrary phase spaces, if the parameter set is just the set of natural numbers with the natural order, by a theorem of Ionescu Tulcea. An example of a Markov transition function is given to which no corresponding Markov process exists. In this example, the time is not well-ordered and phase spaces are bad. Conditions in terms of three-dimensional distributions are obtained which incorporate both the known positive cases.

Boolean algebras of projections

Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a σ-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a σ-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of σ-completeness to weak relative compactness; indeed, a Boolean algebra of projections has σ-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see (9)), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).

Convolution on locally compact spaces

In this paper a convolution similar to that defined by Ionescu TulceaSimon [3] is developed by means of an extension of the underlying convolution of functions to convolution of measures. The hypotheses under which the extension obtains are relatively weak and it is to be noted that associativity is not assumed (an elementary non-associative example is given in Section 2). Under various hypotheses additional to those assumed in Section 2 one may obtain (using other results in this paper) generalized translation operators in the sense of Levitan [4] and several Banach algebras. An interesting example where solutions to partial differential equations are used to generate the underlying convolution is given by Povzner [6]. The general thrust of this work is towards the spectral representation of certain families of operators. In particular a left-convolution (see [2]) is obtained in Section 4. This may be used in the construction of locally compact spaces on which the integrations are performed (see [3] p. 1765 and [5] p. 6). The remaining sections of this paper are as follows: Section 2. AnLl-type convolution, Section 3. Convolution of measures, Section 4. Translation of functions and measures, and Section 5. The compactness condition (iii) of Section 2. The main results are Theorems 1 and 2, and for applications, the corollaries of Theorem 2. The author gratefully acknowledges his indebtedness to C. Ionescu Tulcea for many fruitful discussions of material in this paper.

On commutative V*-algebras

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.

Uniform integrability and the pointwise ergodic theorem

Let (X, 6B, m) be a finite measure space. We shall denote by Lv(m) (1 ?p < oo) the Banach space of all real-valued 63-measurable functionsf defined on X such that if I P is m-integrable, and by Lo?(m) the Banach space of all real-valued, 63-measurable, m-essentially bounded functions defined on X; as usual, the norm in LP(m) is given by lIpIfI = {fxlf(x)IPdm}1/P, and the norm in Lcr(m) by IIg||OO -m-ess. supzEEx lg(x)l. Two functions in LP(m) or Lo((m) will be identified if they differ only on a set of m-measure zero. In this note, we shall be concerned with a positive linear operator T of L'(m) into L1(m) with III1Ti<1. We say that the pointwise ergodic theorem (the L'(m)-mean ergodic theorem, respectively) holds for such an operator T if for every f in L'(m), the sequence of averages { 1 (/n) k=0 Thf } converges m-almost everywhere (in the norm of L'(m), respectively) to a function in Ll(m). Recently, R. V. Chacon [1] constructed a class of positive linear operators in L1(m) with the norm equal to 1 for which the pointwise ergodic theorem fails to hold. Also, A. Ionescu Tulcea [5], [61 showed that in the group of all positive invertible linear isometries of L(r(m) the set of all T for which the pointwise ergodic theorem fails to hold forms a set of second category with respect to the strong operator topology. On the other hand, the ergodic theorem of HopfDunford-Schwartz [4] tells us that if, in addition, T maps L(r(m) into Lco (m) and IIT| < 1, then the pointwise ergodic theorem is valid for such T. In view of these facts, it is interesting to find out what other additional conditions on T would guarantee the validity of the pointwise ergodic theorem. In this note, we shall find a few such conditions which are weaker than the condition of the Hopf-Dunford-Schwartz theorem (though our conditions seem to work for a finite measure space only). We also obtain a result (corollary to Theorem 1, below) which generalizes a result obtained by N. Dunford and D. S. Miller in [3]. First of all, let us observe that if our operator T satisfies ITI1 <1, then the pointwise ergodic theorem is always valid. This is because, for such an operator T, nI Tnf(x) I < rnm-almost everywhere for every f in L' (m), since

On the lifting property

Such determinations of representatives have been studied by A. and C. Ionescu Tulcea [2] who call them liftings of L*(R). The theorem of von Neumann has been extended to general finite measure spaces by D. Maharam [1]. Since L*(R) is a commutative Banach algebra, it is clear that a lifting assigns to every real x a multiplicative linear functional, viz. F_(J) =f*(x). The null space of this functional is necessarily a maximal ideal, and since a neighborhood of the unit element consists exclusively of invertible elements, that ideal is not dense, and is therefore closed. Thus the multiplicative linear functional is continuous, and the lifting can be written

Spectral representations for some unbounded normal operators

Introduction. In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (U,,, = U,LUV) as well as many others. In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures. We shall consider therefore a mapping z -* UZ of a locally compact space Z into the set of unbounded normal operators on a Hilbert space. Since the operators are not necessarily bounded, the conditions imposed on the mapping z -* Uz vary somewhat from the conditions in the bounded case. The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8]. The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea. For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21]. Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A. E. Nussbaum [19]. Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]). The setting of the problem in algebras of measures is given in ??1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in ??3 and 4. The main result is stated and proved in ?6. In ?8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].