DOI 10.1007/s10986-019-09464-7Lithuanian Mathematical Journal, Vol. 59, No. 4, October, 2019, pp. 425–436

Fifty years in the field of probability:A conversation with professor Vygantas Paulauskas

Mindaugas Bloznelis and Alfredas Rackauskas

Faculty of Mathematic and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania(e-mail: mindaugas.bloznelis@mif.vu.lt; alfredas.rackauskas@mif.vu.lt)

Received October 25, 2019

Abstract. This year professor Vygantas Paulauskas is celebrating his 75 birthday. An outstanding probabilist he is wellknown for his research in the field of probability limit theorems and mathematical statistics. He was the moving spirit be-hind the organization of the Vilnius conferences in Probability and Statistics 2008–2018.The conversation gives a glimpseinto the mathematical life of our country in the 1970s, 1980s, and 1990s.

Professor Vygantas Paulauskas, we want to congratulate you on the occasion of your 75 birthday. We wouldlike to start our conversation by asking you a question about your path to mathematics. In 1961, when Lithua-nian Mathematical Journal started, you were 15 years old. Have you thought about becoming professionalmathematician being teenager? What was your path to mathematics?

I could not say that I planned to become a professional mathematician from school time. At school I was in-terested in robotics, I read specialized literature, and naturally thought about engineering. There was anotherreason for this. For the younger generation who have not lived in a country called the Soviet Union (abbreviatedas the USSR), it should be made clear that at that time there were military departments in the higher educa-tion institutions, whose graduates received a military lieutenant degree and thus avoided two or even threeyears of military service in the Soviet army. But in the sixth decade the military departments were repealedfrom many universities, and in Lithuania military department was left only in Kaunas Polytechnic Institute(KPI). To avoid military service, many of my peers, including myself, have chosen KPI. So, in September1962, with my classmate Audrius Kopustinskas (now he is Professor Emeritus of Kaunas University of Tech-nology), I joined KPI, a speciality in automation and telemechanics. Meanwhile, faculties of exact sciences(mathematics, physics) at Vilnius University (VU) lacked talented young people. Then, in the heads of VUmathematicians (first of all, VU Rector professor Jonas Kubilius), the idea to outsmart the Soviet governmentbegan to mature. It was decided to form two groups of KPI students that would formally be considered asKPI students (attending military department classes and in four year obtaining lieutenant degree), but wouldactually study mathematics and physics. This idea was endorsed by KPI rector professor K. Baršauskas. Theproject was commissioned by VU associate professor H. Jasiunas. In October 1962, two groups were formed,one in mathematics and the other in physics. In Vilnius University, they were called KPI groups. And later,associate professor Jasiunas had to take care of KPI groups. Formally we were students of the KPI branch inVilnius, we listened to engineering courses and attended lectures at the military department of KPI, but welived in VU students’ dormitory and we were getting VU scholarship. We have passed the entire VU mathe-matics program, and we have attended many special courses that were taught by the most famous Lithuanian

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mathematicians by special programs. In the third course, when professors J. Kubilius, V. Statulevicius, andB. Grigelionis started to work individually with us, we realized that mathematicians would come out of us. AsI mentioned, from school times I was interested in robotics, so I had chosen to go deeper into mathematicallogic. However, associate professor Jasiunas categorically stated that the main direction in Lithuania is thetheory of probability and assigned me to V. Statulevicius. So far I am very grateful to him for this. Even now,when writing my CV, I am hesitating how to correctly present my education—I entered KPI, four years I wasformally a student of this university, and only the last year of my studies I became a student of VU. But, aftersome hesitation, I am writing that I graduated from VU with BSc degree in mathematics.

What was life of a professional mathematician in Vilnius in the seventies and eighties?

My work as a professional mathematician began in 1969. In December 1967, our KPI Group received univer-sity diplomas, and we were all assigned to future jobs. I was appointed to Vilnius University and immediately,as of January 2, 1968, became a doctoral student. However, my post-graduate studies were short. In April1969, I requested Rector J. Kubilius to terminate my postgraduate studies and to give me a teaching position.There were two arguments for such a request: first, the opinion of my postgraduate supervisor V. Statuleviciusthat the obtained results were sufficient for the dissertation, and second, my family situation. In the summerof 1968, I married Irena Kazlauskaite, a student at the Faculty of Medicine of Vilnius University. The doc-toral scholarship was a too little source to support family. The Rector did not oppose these arguments andonly said that I was making a mistake. Nonetheless, the request was accepted “over the full measure”, andI was appointed as a senior lecturer. This means that I had jumped over two positions—that of the assistantand the lecturer. I realized later that the Rector was right about the mistake—I lost the opportunity to deepenmy mathematical knowledge for 1.5 years. On June 10, 1969, I defended the so-called Candidate Dissertation(my twins were born on May 24) and started my work in the Department of Probability Theory and NumberTheory (PTNT), headed by Rector J. Kubilius. I was assigned quite a lot new courses. I gave lectures at theFaculty of Mathematics, Natural Sciences and of Economics. The Department of Economic Cybernetics, ledby Professor Raimundas Leonas Rajeckas at the Faculty of Economics, was responsible for specialities ofEconomic Cybernetics and Mechanized Information Processing, where the program provided a strong math-ematical background. For a number of years, for students of these specialities, I gave, besides the generalcourse of Mathematics, a course of Information Theory and Coding. At the Faculty of Natural Sciences, I gavea Mathematics course for Geology students from Lithuania, Latvia, and Estonia (for each republic separately, itwas too difficult to prepare a small number of specialists in the field—it was a good example of cooperation be-tween the Baltic countries, which were all occupied at that time). And now, during the meetings of LithuanianAcademy of Sciences (AS), I talk to former student from this group, Estonian Robert Mokrik, now the aca-demician of the Lithuanian AS. For students of Mathematical Department, I provided courses on Pontriagin’smaximum principle, linear programming, and weak convergence of measures. I was even angry that so muchof my time was devoted to new courses, instead of scientific work. But later I had to admit that these coursesgreatly expanded my mathematical horizons. Another important fact related to my work in the Department ofPTNT was that my group of students began to form at that time. In 1972, I became an associate professor,which gave me the right to supervise postgraduate students. I enjoyed working with students from the secondor third year of their studies and was organizing seminars for stronger students. My group of postgraduatestudents gathered in a short time. Maybe I will talk about the pupils later; I will try to answer the questionnow. Mathematical research at Vilnius University is undoubtedly bound up with the name of Jonas Kubilius,who in 1951 returned to Vilnius University from postgraduate studies at Leningrad (now St. Petersburg) withthe famous probabilist Yurii Linnik. At once, he began to actively organize scientific work in Mathematicsin Lithuania. A lot of work has been done over the decade—the organization of mathematical Olympiads forschoolchildren, the organization of research seminars. At Vilnius University and the Academy of Sciences,groups of research mathematicians were formed, and the Lithuanian Mathematical Conferences were orga-nized since 1958. In 1961, the publication of the mathematical journal “Lietuvos matematikos rinkinys” hadstarted. The journal was published in Russian (hence the Russian title “Litovskii matematicheskii sbornik”),with short summaries of each paper in Lithuanian and English. In 1973, the US publishing house Plenumstarted to translate the journal into English as “Lithuanian Mathematical Journal”, now published by Springer.

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Mindaugas Bloznelis, Alfredas Rackauskas, and Vygantas Paulauskas.

In the 1980s, when I started working, mathematical life in Vilnius was already active; overall, the eighth andninth decades were the years of strongest growth. The main mathematical branches under development inLithuania were theory of probability and probabilistic number theory, but there were small groups of mathe-maticians working in the fields of geometry and topology, logic, differential equations, and numerical analysis.Research in mathematics was concentrated at the Faculty of Mathematics and Mechanics (now Mathematicsand Informatics) at VU and at the Institute of Mathematics and Cybernetics at Lithuanian AS. The mainmathematical library was in the institute; there was a lot of literature in the Russian language. Many math-ematical books published abroad were translated into Russian. The situation regarding journals from abroadwas relatively poor; there were very few original mathematical journals in Vilnius, mainly those obtained byVU and AS libraries by way of exchange programs between libraries abroad. But the Soviet Union, perhapsbreaking all international agreements, bought several copies of the journals and multiplied them with copy-ing machines, then bounded these copies, and was sending them to libraries of scientific institutions for acertain price. So we had the opportunity to read the main journals in our field, such as Annals of Statistics,Annals of Probability, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. Traveling abroadwas limited, especially for young scientists. As a rule, at first, scientists were allowed to travel to scientificconferences or internships into so-called socialist camp countries, and only later on to capitalistic countries.The so-called scientific tourism trips were practiced too, that is, when a person was going to a conferenceand paying most of the required amount from his funds. Such was my first trip abroad to the first EuropeanMeeting of Statisticians in Budapest in 1972.

After PhD you chose a hot topic of Probability in Infinite-Dimensional Spaces. What motivated such a choice?Were your plans realized?

The title of my first dissertation (then it was called candidate of sciences dissertation) was “The estimationof the remainder term in the multivariate central limit theorem”. This topic was popular in the Soviet Union,

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and it suffices to mention three monographs by A.N. Kolmogorov and B.V. Gnedenko, I.A. Ibragimov andYu.V. Linnik, and by V.V. Petrov, which served as the Bible for probabilists in the world, working in the areaof limit theorems. My supervisor V. Statulevicius, during his post-graduate studies at the Leningrad Univer-sity, also worked in the field of limit theorems, so it was natural that his students were involved in this topic.I obtained a good result very quickly. By using the so-called pseudomoments, I was able strengthen the classicBerry–Esseen estimate in the central limit theorem (CLT). After defending my thesis in 1969, I worked on thesame topic for some time, estimating the rate of convergence to stable laws, both in one-dimensional and mul-tidimensional cases. During this period, I had also investigated the functions of concentration, the asymptoticproperties of sums of random number of random summands, and the distribution of the maximum of partialsums. The most important results from 1970–1976 are the estimates of the rate of the convergence of distribu-tions of sums of independent random variables to stable laws, which remain unimproved till now, especiallyin the case of nonidentically distributed summands. I want to mention the paper “Some remarks on multivari-ate stable distributions”, published in 1976 in “Journal of Multivariate Analysis”, since it was my first paperpublished abroad. Although in 1975 I had 28 publications, all papers were written in Russian and published in“Litovskii matematicheskii sbornik”, which was mentioned before, and in “Doklady Akademii Nauk SSSR”.At that time, to send officially a scientific paper to a foreign journal, it was necessary to get a special permis-sion from the so-called “first unit” at the university, which was connected with the KGB. On the other hand, weknew that most of the articles written in Russian in USSRwere translated into English by publishers in USA. Inaddition, foreign publishers paid a certain amount of dollars to the authors. These dollars were converted intocertain checks in Moscow, and the checks were used to pay in special stores with foreign goods. In a sense, mypaper in Journal of Multivariate Analysis appeared accidentally due to some happy circumstances. In 1975,Editor-in-Chief and founder of the Journal of Multivariate Analysis, Professor P.R. Krishnaiah was visiting Vil-nius, and my former supervisor Statulevicius asked Krishnaiah if it would be possible for me simply to handthe manuscript to him instead of using formal submission procedure by post. The answer was “no problem”. Soduring several days, I prepared my paper in English, gave the manuscript, and asked to use my personal homeaddress for the correspondence. And in 1976 the paper appeared [5]. This paper is one of the most cited of mypapers, but together with other papers, printed up to 1990; it is attributed to V.J. Paulauskas (not to VygantasPaulauskas). In the Soviet Union, the Russians used the patronymic name besides the first name; my father’sname was Jonas, so the Russians referred to me as Vygantas Jonovich. Of course, for Lithuanians, it wasunusual and not very acceptable, but in my case—at that time—it seemed even necessary. At our department(later we were even in one chair), there was a much older colleague, Vytautas Paulauskas, so when on a letterfrom abroad it was written “Professor V. Paulauskas”, the letter usually was presented to elder Paulauskas, hewould open the envelope, wrote “I’m sorry, I opened”, and the letter was passed unto to me. There were alsomore serious misunderstandings when VU Accounting Department transferred elder Paulauskas’ money to myaccount. An especially painful story for my senior colleague was when my first work was published in the pro-ceedings of the Young Scientists Conference; in a solid reference issue in Moscow, the article was attributedto Vytautas Paulauskas, who was at that time over 60 and had not written articles for a long time. So it seemedto me that the patronymic would solve all the problems—the names of our parents were different! Thus, inSoviet time, most of my papers written in Russian or translated from Russian into English were written underthe name V.J. Paulauskas, later, when we restored independence of Lithuania, I stopped using my father’s namebefore the family name. In this article of 1976, among other things, I introduced a measure of dependence be-tween coordinates of stable symmetric vector, and this was one of the first measure of dependence for randomvariables with infinite variance (in the case of finite variance there are well-known measures of dependency—covariance and correlation). Despite the fact that during 1976–1977 academic year, spent at the GothenburgUniversity, I generalized the measure of dependence for random stable elements with values in infinite dimen-sional Banach spaces, the measure I proposed had not attracted the attention of other probabilists (although inencyclopedic G. Samorodnicky and M.S. Taqqu monograph in 1994, this measure was mentioned). I myselfbecame interested in other problems, and only in 2013, French mathematician Bernard Garel wrote a letterto me asking several questions about this measure of dependence introduced in 1976. This letter revived myinterest to this measure of dependence, and I involved my last pupil, Julius Damarackas, who was studyingfor Master’s degree at the time, and after graduation he entered doctoral studies. We both have written fivearticles with him on this subject. Of course, starting with the answer on how I changed the topic of research,

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I deviated a lot from the subject, now I will try to come back to this question. As I mentioned, about five yearsafter defending my dissertation, the main focus of my research was on the limit theorems in one-dimensionaland multidimensional cases. Several times after my reports at the seminar, Professor Kubilius asked why sucha significant number of probabilists in Vilnius work in the limit theorems in finite-dimensional spaces and noone is trying to move to infinite-dimensional spaces. And while that new area was tempting, the determinationto move into a new area was not easy. The US Mathematician, James Kuelbs’ visit to Vilnius (about 1974) wasa major impetus for this step. During the visit, he gave a talk at the seminar about his new paper on the conver-gence rate in CLT in Hilbert’s space, written with T. Kurtz and published in the prestigious journal “Annals ofProbability”. J. Kuelbs was a little older than me (defending his PhD in 1965), communicating was easy, myEnglish was already good enough, although there were curious situations. One day we were sitting in a cafewhile having a lunch. We were joking, and James said “you are pulling my leg”. I automatically looked underthe table, thinking maybe I accidentally kicked his leg. It was a laughable laugh when he said that in Englishthis expression means “you’re joking” (unfortunately, Google translator does not know this expression, aftertyping these words, it translates literary (in Lithuanian) “you pull my foot”; for artificial intelligence, it will notbe easy to laugh with people!) Since I was well aware of the works of Swedish mathematician H. Bergströmand Russian mathematician V.V. Sazonov, at that time, I realized that it would not be difficult for me to getinvolved in this topic. And in fact, already in 1975, my first articles on a new topic appeared, and in 1976 myarticle was published in the prestigious USSR journal “Theory of Probability and its Applications”, foundedin 1956 by A.N. Kolmogorov, and in which such leading figures in probability theory as A.N. Kolmogorov,B.V. Gnedenko, Yu.V. Prokhorov, A.N. Skorokhod, Yu.V. Linnik, I.A. Ibragimov, and others published theirworks. In this article, an estimate of the rate of convergence in the CLT in certain Banach spaces of order n−1/6,where n is the number of summands under the consideration, was obtained. This estimate not only general-ized, but also improved the above-mentioned result of Kuelbs and Kurtz, and most importantly, after 10 years,Vidmantas Bentkus showed that exponent 1/6 cannot be improved in the general situation. Thinking about thefuture, in 1974, I organized a seminar with senior students, in which we started to study functional analysis,because without knowing functional analysis it would be difficult to go into the questions of probability theoryin Banach spaces. Soon after my first postgraduate students appeared, the student seminar became a normalweekly scientific seminar on “Distributions in infinite-dimensional spaces”, which worked for about 20 years.The number of my postgraduates grew rapidly, and in 1979–1990, ten my students defended dissertations, allof them on the theory of probability in Banach spaces. I myself, in 1979, defended my second doctoral thesis(in the current terms, the habilitation doctoral thesis). Main results of our group obtained during 1974–1987are surveyed in the monograph [7] written together with Alfredas Rackauskas. All in all, the last three decadesof the last century were a year of prosperity of the theory of probability in infinite-dimensional spaces in theworld. There were also a number of research centers in the Soviet Union, where probabilists were workingon the same subject. We maintained quite close ties with Moscow scientists V.V. Sazonov, V.M. Zolotariov,V.V. Ulyanov, scientists from Leningrad Yu.A. Davydov, M.A. Lifshits, Georgia (now Sakartvelo), proba-bilists N.N. Vakhanija, V. Tarieladze, S. Chobanian, and V.V. Buldygin, V. Kolchinskii from Kijev, V.V. Yurin-ski from Novosibirsk. We had good relations with a number of foreign mathematicians, despite the fact thattravel from and to the USSR was constrained, especially when relations between the Soviet Union and theUnited States escalated. First of all, I want to mention the Americans R.M. Dudley, J. Kuelbs, M.Marcus,A. Acosta, W. Philipp, J. Zinn, E. Gine, M. Taqqu, G. Samorodnitsky, French mathematicians X. Fernique,B. Heinkel, M. Ledoux, Polish probabilists S Kwapien, W. Woyczynski, A.Weron, Z. Jurek, Germans F. GötzeandW. Linde; it is impossible to list all. It has to be said that there was a rather large and active group of proba-bilists working in probability theory in infinite-dimensional spaces regularly organizing conferences, the mostfamous of which were working on probability in Banach spaces. The first conference was held in 1975, inOberwolfach, Germany. Until 1988, five more conferences were organized, but I did not succeed to get toany of them. For foreigners, it was difficult to understand this fact. These conferences were organized quiteexpeditiously; invitations were sent less than a year before the conference, and when I filled out the requiredpapers, and they were sent by the Ministry in Vilnius to Moscow (all scientific trips abroad were regulated inMoscow), the answer was negative with the explanation that the documents arrived too late and that the tripshould have been included in the plans a year ago. And only when I wrote a letter to Anatole Beck (USA,the chief organizer of the first conferences), explaining the situation, I received an invitation from him say-ing “hold this invitation to this conference and to the next one.” And in 1988 I went to the 7th conference,

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also in Oberwolfach, and later attended the 8th (1991, Brunswick, Main, USA) and the 9th (1993, Sandjberg,Denmark) conferences. The relationship among scientists working in the field of probability theory in infinite-dimensional spaces was so strong that, after the last, ninth, conference around 2000, when interest in thisarea began to decline (which is a frequent phenomenon after many decades of development, when most of thetopical issues have been resolved, or there are left only minor generalizations or extremely difficult tasks), theprobability in Banach spaces conferences were superseded by high-dimensional probability conferences, nineof them have already taken place. Maybe it is interesting to note that my seminar “Distributions on infinite-dimensional spaces” stopped its activity in 1995, about the same time as the conferences on “Probability inBanach spaces”.

Speaking about the work of the seminar and my group of students, I would like to tell the story how onebright scientist, Vidmantas Bentkus, who this year would celebrate 70th birthday (unfortunately, suddenly andvery unexpectedly passed away in 2010) became interested in probability in Banach spaces. In the eighties,Lithuanian school of probability, due to the efforts of Professors J. Kubilius, V. Statulevicius, and B. Grigelio-nis was gaining strength, but research in other branches of mathematics was at a rather low level. Therefore,by initiative of Vilnius University Rector, J. Kubilius, the strongest students from VU were sent to the bestuniversities of Soviet Union (Moscow, Leningrad (now St. Petersburg), Kijev) to study Topology, Differen-tial Equations, Functional Analysis. Vidmantas Bentkus was among such students, he had chosen FunctionalAnalysis, the area which was most needed in Lithuania: at this time there was no professional mathemati-cian working in functional analysis. In 1973, he graduated from Moscow State University (MSU) with cumlaude, and during the same year, Vidmantas entered graduate studies at MSU under supervision of ProfessorO.G. Smolyanov. In 1977, he defended his PhD thesis, in which he considered problems of differentiation ofmeasures and infinite-dimensional differential equations. After returning to Vilnius in 1977 and spending someyears in Vilnius, Vidmantas suddenly and very decisively changed his research area. Since I and my studentswere working on limit theorems in Banach spaces, and differentiation of functions and measures, among otherproblems, were actual in this area, I asked Vidmantas to give several talks on his own work on this problem.After attending several more seminars, Vidmantas found that he liked probability in Banach spaces very much.As sometimes it is said, it was love from the first glance. After his first paper on the convergence rate in CLT inBanach space in 1981, only in 2003–2004 Vidmantas returned to functional analysis. This was when in 2002I returned from three years stay as Visiting Professor at Georgia Institute of Technology (Atlanta) and showedmy paper, accepted in Journal of Functional Analysis, in which ideas from the CLT in probability were used inapproximation theory of operators. Vidmantas became interested, we wrote one joint paper [1] on this topic,and Vidmantas alone wrote another one. Thus, it seemed for me that the beginning was very promising, butunfortunately, very soon, Vidmantas’ interest in operator theory diminished. During last ten years of his life,he was completely absorbed with inequalities for sums of random variables. Thus it is possible to say that atthe end of the 1980s probability theory in Lithuania had gained very much with Vidmantas joining this area,but at the same time functional analysis had lost even more, since this area in Lithuania remains still in infantstage.

What are you favorite topics in probability? May we ask about the motivation behind your choice of a researchproblem: interesting/important, both?

This is a complex and multifaceted question. Often, the problem was of interest if it was clearly formu-lated and it seemed to me that I could say something new in the solution to that problem. For example,I became interested in the rate of convergence in CLT still being an undergraduate student. Well-knownprobabilist from Moscov V.M. Zolotarev gave several talks at the probability seminar in Vilnius about theuse of pseudomoments in estimating the rate of convergence in limit theorems. The problem was to find andprove a proper equivalent to the classical Berry–Esseen bound, where the third moment were replaced bya pseudomoment. Pseudomoments indicate the proximity of the distribution of summands to the normal law,and therefore they would be a better characteristics to assess the rate of convergence than absolute momentsof the summands. Zolotariov and his disciples used the characteristic function method of Swedish mathemati-cian C.-G. Esseen, and I worked with the so-called composition method, which was introduced by Swedish

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Paulauskas at the terrace of his house.

mathematician H. Bergström and revived by the Moscowmathematician V.V. Sazonov. H. Bergström also intro-duced pseudomoments. I quickly realized that the com-position method was a much more natural tool for work-ing with pseudomoments. Thus I managed to solve theproblem quite quickly [4], and it was the main result ofmy candidate dissertation. Subsequently, the pseudomo-ments become an important instrument for examining therate of convergence to stable laws, and some results ob-tained in 1974–1975 remain unimproved up to now.

Another example of a problem I was interested in wasthe CLT for a Banach space valued random elements.When I started working with my students in this field,many results have already been achieved in this area,for example, the problem was solved for the so-calledtype 2 Banach spaces (a particular case of type p Ba-nach spaces, where p = 2), but there were many Banachspaces and other function spaces where little was knownabout the validity of CLT. Together with my studentsD. Jukneviciene and M. Bloznelis and trainee from Ger-many Ch. Stieve, in 1988–1994, we published a wholeseries of articles on the CLT in the Skorokhod spaceD[0, 1]. It happened that in France the famous probabilistX. Fernique workedwith his student P.H. Bezandry on thesame problem. I remember it like a race about who willget better results. In January 1993, with M. Bloznelis, wesubmitted a paper to “Stochastic Processes and Applica-tions” [2], which gave optimal conditions for the validityof the CLT inD[0, 1] expressed in terms of process incre-ments. A tiny bit later X. Fernique got the same result bya completely different method and, perhaps acknowledg-ing our contribution, published his article in “LithuanianMathematical Journal” [3].

At this place I would like to mention my links withfunctional analysis. After graduating from university for

over 10 years, I worked at the Department of PTNT led by Rector J. Kubilius. After my group became involvedin probability theory in the infinite-dimensional spaces, which required a lot of knowledge from functionalanalysis, J. Kubilius came up with the idea of consolidation of the Department of Mathematical Analysis. In1981, he transferred all my group and several members of the Department of Applied Mathematics who taughtmathematical analysis to The Department of Mathematical Analysis. I was elected Head of that Department.My main task was to strengthen scientific work and teaching in functional analysis. Things worked quite wellwith teaching, I began to teach the basic course of functional analysis, my students also joined, and togetherwith A. Rackauskas we prepared two parts textbook on functional analysis. Research was more difficult. Andalthough our group’s papers used many functional analysis concepts and results (geometry of Banach spaces,operator theory, topological concepts), however, there were no papers devoted purely to functional analysis,except for one my paper in 1981, which appeared in the proceedings of the conference “Functional Analysisand Applications” (Oberwolfach). Things changed when I went to Georgia Institute of Technology in Atlantain 1998–2001. This institute is ranked among the top five US technology universities, with a very strong andversatile school of mathematics. The mathematical life in the school was very active, often in a week there wereup to ten seminars. In addition to probability theory and statistical seminars, time from time I was attending theanalysis seminar (it included harmonic and functional analysis), sometimes I went to combinatorics seminar.One day I saw the announcement that, at the analysis seminar, V. Zagrebnov, coming from Marseille, will

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report on the Kato–Troter formula in operator theory. I was familiar with Trotter’s method that is used inCLT and decided that maybe I will understand the message. And, in fact, the approximation in the theoryof semigroups of operators has some similarity to the central limit theorem in the probability theory. Duringthe talk I realized that we have more progress in the theory of probabilities and I can immediately improvesome of the results formulated in the talk of Zagrebnov. After the seminar, I talked to Zagrebnov and, aftera few months of studying the approximation of the semigroups of operators, I prepared a short article [6] thatI sent to L. Gross, a member of the editorial board of the Journal of Functional Analysis. One might think thatsending his first article to a prestigious functional analysis journal was a bit offensive, but L. Gross was theonly specialist in functional analysis I knew: he attended one of Vilnius conferences and we later met at CornellUniversity, where I was a visiting professor in 1994. Gross wrote to me that as a rule the journal does not acceptshort articles, but the editorial board decided to make an exception to my paper because the article uses ideasof probability theory in functional analysis (usually the situation is opposite—functional analysis methodsand results are used in probability theory). As I mentioned earlier, after returning from Atlanta in 2004, wewith Vidmantas Bentkus published another paper on operator approximation in the prestigious journal “Lettersin Mathematical Physics” [1]. Then after an eight-year break, I returned to that topic and in 2012 publishedanother paper (now normal, 25 pages) in the Journal of Functional Analysis. And I still have the hope ofreturning to functional analysis to accomplish some of my unrealized ideas. These three papers from operatortheory make up a very small part of all my publications, but thanks to them, I took part in two internationalconferences (Lumini, France, and Eilat, Israel) on functional analysis, I made reports in functional analysisseminars in Warsaw, Lille, and Atlanta.

How did scientific life changed after SU collapsed: a look from your perspective.

The restoration of Lithuania’s independence and the ensuing collapse of the Soviet Union were epoch eventsthat touched people of several generations and had great significance in the history of all mankind. And therecould be a lot to talk about, but the question is what impact these events had on the scientific life and onme personally. As for science, the transition from a Soviet system of higher education to a new, often calledWestern, system was both daunting and painful, and I would say it was not always successful. As an example,in 1990–1995, with the opening of the borders and the start of “wild capitalism”, there was an opinion amongyoung people that it was not worth to seek the education, that in Gariunai (a huge market in Vilnius whereeverything was traded) you can become rich within a few years, just by transporting large quantities of goodsfrom the abroad. And we felt the moods in the classrooms, the number of students quite sharply reduced.With the opening of borders, and especially when Lithuania became a member of the EU, another problemarose—more and more of the best graduates began to study at foreign universities. A well-known problem,the so-called brain drain, has been a painful problem for developing countries, and now it is also actual forLithuania. As positive changes in science one should probably mention the ideological issues. Though themathematicians may have felt these issues the least, in many sciences it was felt. When I looked at old papers,I discovered the summary of the thesis of my father, associated professor at the Faculty of Civil Engineeringof KPI. The thesis dealt with wood resistance to splitting, but the first words of this summary (in Russian)were: “The XXII congress of the CPSU adopted the program of building communism in our country.” Itcontinues to justify by means of this program the need to study and increase the use of wood. Fortunately,mathematicians did not have to refer to declarations of communist party meetings in their research. Anotherpositive change was the opportunity to communicate and collaborate with foreign colleagues. Thanks to theVilnius conferences, Lithuanian probabilists had quite extensive contacts with many scientific centres in theworld. One of the most successful examples of such co-operation was the co-operation between Vilnius andBielefeld probabilists. During the First Bernoulli World Congress in Tashkent in 1986, Friedrich Götze toldme that he was planning a long-term big project and asked if Vilnius’s probabilists will be able to contributeto the project. Since Gorbachev’s “perestroika” has already started in the Soviet Union, we expected that itwill be possible. Until 1990, the cooperation was not very intensive, but when Lithuania became independent,the cooperation just flourished. Vidmantas Bentkus worked in Bielefeld in 1993–2001, Alfredas Rackauskasspent a couple of years, I and Mindaugas Bloznelis were visiting Bielefeld for shorter periods, one or twomonths. During this collaboration, we four have written 25 joint papers with Friedrich. In total, over 10 Vilnius

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mathematicians went to Bielefeld, using this collaboration. For his merits to Lithuanian science, F. Götze waselected as foreign member of the Lithuanian AS. Another good example of international relations was thecooperation of probabilists of Vilnius and Lille lasting about 20 years. The basis for this cooperation wasa group of 8 scientists, 4 from Vilnius (R. Leipus, V. Paulauskas, A. Rackauskas, D. Surgailis) and 4 fromLille (Y. Davydov, A. Philippe, Ch. Suquet, M.-C. Viano), although more researchers, mainly pupils of theseeight researchers, participated in joint projects. During 1996–2004 years joint work was carried out usingCNRS grants (i.e., French money), and in 2005–2012 we were working under the Lithuanian–French scientificcooperation program Gillibert. For her impact to this collaboration, M.-C. Viano was awarded a HonoraryDoctorate of Vilnius University in 2017.

Along with these positive things, it has to be pointed out that Lithuania failed to successfully reorganize thehigher education system. Reforms were made hastily. Moreover, when new political forces came to power,these reforms were alternated or hindered by interests of different groups. It becomes clear that almost 30 yearsafter the restoration of independence, the situation in science and education is of great concern. I rememberwell that in 1998, when I returned after working three years at the University of California, Santa Barbara,one newspaper printed a large interview in which I said that a rather deep decline was waiting for Lithuanianmathematicians because the profession of teacher became less and less prestigious for young people. Butthese thoughts are tied to politics and maybe not suitable for a mathematical journal. . . Personally, there havebeen a number of important changes in my life after the restoration of independence. It suffices to mentionthat between 1990 and 2001, I spent almost half of my time abroad. For the first time, I visited the UnitedStates (I went to a conference in 1991), and then I spent almost six years as a visiting professor at severalUSA universities. The situation with travel to America was strange: during the Soviet years, I was allowedto travel to various capitalist countries, but never was allowed to go to the USA. Despite that I was receivinginvitations to conferences with full covering of travel and living costs, the answer from Moscow was “no”.In 1985, I received a very good invitation from prof. Ronald Shonkwiler (with whom I met at a conferencein Poland) to work one semester as a visiting professor at the Georgia Institute of Technology in Atlanta.I was told by the Ministry of Education and Science that definitely I will succeed with such an invitationbecause Moscow is interested in such trips (some of the money earned had to be returned to Moscow), but, tomy big surprise, the answer from Moscow was “no” again, and it was written that even I am not allowed toexplain to the Americans why I cannot come. I realized that I could not write the truth to Ronald (because thenmaybe I shall not be able to travel abroad at all), so I stopped correspondence with Ronald at all. And onlyin 1992 the reasons for those denials to go to the United States came to light: it turned out that my mother’sbrother, who left for the West in 1944, when the front was approaching Lithuania, and who by all relatives inLithuania was considered dead during the bombing in Germany, was actually alive and lived with his familyin Chicago. My guess is that the KGB knew this fact, and because it tried to limit the relations between thediaspora and Lithuania as much as possible, it was natural that it was decided not to allow a professor fromLithuania to go to the USA. And when we were found by my cousin, she was sent by the uncle to Lithuaniato find relatives. After two months of hassle finally she contacted my mother. The end of this story (aboutwhich the book can be written) is as follows: after more than 10 years of silence, I wrote a letter to Ronald,explaining everything why I kept silent for a long time and asked half-laughing at the end of the letter thatmaybe I can use the 1985 invitation. And the answer was “no problem”, I worked in Atlanta not for one butfor five semesters. With Ronald (and two more Argentinean mathematicians, like myself visiting the institute)we coauthored an article and most importantly, not from probability theory but from analysis. My lecturesfor the foreign students were related to the restoration of the independence. First course in English was atGothenburg University in the Spring semester, 1993, for PhD students in probability. The same occurredduring the Spring semester of 1994 at Cornell University, where I gave two courses. Also, I spent 1996–1998as a Visiting Professor at the University of California, Santa Barbara. During those two years, I read morecourses, and most importantly, there I started reading two new (for me) courses, actuarial mathematics andfinancial calculations. It was important to me, because in Vilnius, we had already introduced a new studyprogram of Financial and Actuarial Mathematics. So, after returning to Vilnius for some time, I read those twocourses. Life in Santa Barbara, which Americans often call paradise on Earth, was memorable. I got to knowCalifornia and the surrounding states very well during those two years: from San Francisco to San Diego, fromLos Angeles to Las Vegas, and the Grand Canyon. With my wife we even visited Maui Island in Hawaii fora week.

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434 M. Bloznelis and A. Rackauskas

You have been chairing several Vilnius conferences. Could you share your personal impressions about theseconferences.

It can be said without exaggeration that the Vilnius school of probability partly received its world recognitiondue to the Vilnius International Conferences on Probability and Mathematical Statistics. On the other hand,theory of probability under the leadership of Jonas Kubilius, Vytautas Statulevicius, and Bronius Grigelionisdeveloped very rapidly in Lithuania in the 1960s and 1970s (Statulevicius alone supervised 17 candidate dis-sertations in a decade since 1962), and this was a prerequisite for the emergence of the Vilnius Conferences.Such a legend is flying about the emergence of Vilnius Conferences (it was repeatedly told by my scientificsupervisor V. Statulevicius). From 1945 to 1966 in USA, Berkeley Symposiums on Mathematical Statisticsand Probability were organized at Berkeley University, with Jerzy Neyman as the main organizer. The pro-ceedings of these conferences were impressive, and even now I remember how I looked with respect at thelarge-format volumes of the proceedings of the Berkeley Conferences on the shelf in the library of the Instituteof Mathematics. There were 5 volumes of the last, fifth, conference; additionally, the second volume, dedi-cated to the Theory of Probability, has two parts. The number of pages of these volumes is really impressive:666, 447, 483, 324, 934, and 451! However, these grand conferences had one drawback—there were very fewrepresentatives of the Soviet Union, whereas the school of probability in this country was one of the strongest(probably the strongest) in the world; it suffices to mention the corifiers of this science A.N. Kolmogorov,Yu.V. Prohorov, A.V. Skorokhod, E.B. Dynkin, and many other famous names. In those post-war decades, therelationship between the two world powers was very tense (enough to remember the Cuban rocket crisis, spacerace), the travel to USA was very limited; moreover, among scientists in USSR, there were such as A.V. Sko-rokhod, who were not allowed to leave the country at all for political reasons. Therefore, and perhaps becauseit became too difficult for him to organize such large conferences, Jerzy Neyman (72 years old in 1966) sug-gested A.N. Kolmogorov and Yu.V. Prohorov, the leaders of probability school in USSR, to organize the nextconferences in the Soviet Union. In Moscow, it was decided to organize conferences in Vilnius, because inVilnius there was a large and strong group of probabilists. In 1973, the first Vilnius International Conferenceon Probability Theory and Statistics was organized. Although the first conference was not very large (about250 participants), it became clear that the conference prospects were good, as the conference became a uniquemeeting place for East and West researchers. Indeed, for two decades, every four years, Vilnius Conferenceswere held; the number of participants had steadily increased. In the last conference, organized in Soviet Unionin 1989, there were about 800 participants. The world-renowned scientists, such as the Russians A.N. Kol-mogorov (who is recognized as one of the most famous mathematicians of the 20th century), Yu.V. Prokhorov,the Ukrainian A.V. Skorokhod, the American R.M. Dudley, Englishman D. Kendall, French X. Fernique,P. Meyer, Indian R.C. Rao, Australian P. Hall, and others, participated in Vilnius Conferences. Since Lithuaniaregained its independence in 1990, seven conferences have already been held in the capital of the country,and although the main idea of the conference as a meeting place between Eastern and Western scholars haveceased to exist, the conference has already acquired a solid reputation. Moreover, two Vilnius Conferenceswere organized together with other important Probability and Statistics conferences: in 1998 together with theEuropean Meeting of Statisticians, and in 2018, with the Annual Conference of the Institute of MathematicalStatistics. However, the sixth (and the first in independent Lithuania) Vilnius Conference in 1993 might nothave happened if not for the stubbornness of my teacher Professor Vytautas Statulevicius. The first years ofindependence in Lithuania were difficult; there was an economic blockade by USSR, the Lithuanian economywas in a deep pit. Even some of organizers suggested to write letters to foreign colleagues apologizing thatthe conference would not take place. The governments changed every few months. Statulevicius agreed onsupport with one Prime Minister, and a few months later he had to talk to another Prime Minister. And theconference happened! V. Statulevicius deservedly is considered as spiritus movens of the first eight confer-ences. I am proud to say that not only have I attended all Vilnius Conferences, but that I have participated inthe organization of all twelve conferences. I started career in this job in 1973 from a fairly “low” position—I was the manager of the team responsible for welcoming conference participants at the train station and airportand transporting them to hotels. Now it is hard to believe that we were meeting all participants of the confer-ence (I no longer remember when we stopped doing that; maybe only when we regained independence), but in1973 this was probably a necessary decision. For a foreigner who did not speak Lithuanian or Russian, it was

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almost impossible to communicate on the street or to use the public transport. And my team members, dividedin pairs, for about two days were staying at the train station and at the airport with the conference banners.When a sufficient number of incoming participants was formed, they called (from street phone; at that timemobile phones were in fiction novels only) to the conference organizers’ headquarters to get a vehicle (big orsmall bus). Then a team for accommodation (there was such a team!) took care of the conference participants.There were teams that were responsible for the catering of the conference participants and for the conferenceaudiences. I cannot tell you much about their work because I had not worked in these teams. After defendingmy second dissertation and becoming a professor my duties also increased, in 2002 at the eighth conferencein Vilnius, I was Vice-Chair of the Program Committee. The Chair was my friend, Peter Jagers from the Uni-versity of Gothenburg. I have been acquainted with him since 1976, when I spent a year at the University ofGothenburg. In general, I have sentiments for Sweden, the first capitalist country I visited, and for Gothen-burg, where I gave my first course in English for students abroad. P. Jagers chaired the Program Committeeat three conferences (2002, 2006, 2010). After Statulevicius’ death in 2003, I had to take over the positionof the Chairman of the organizing committee. So I chaired the organizing committee in 2006 and 2010, andchaired the program committee in 2014 and 2018. As I mentioned, the 2018 Vilnius Conference was orga-nized in conjunction with the IMS Annual Conference, therefore the Program Committee and the OrganizingCommittee were chaired by two co-chairs, one from Vilnius and the other from IMS. I chaired the ProgramCommittee with Peter Bülmann from Zürich, and the organizing committee was chaired by Remigijus Leipusand Erwin Bolthausen (also from Zürich). I was familiar with Bülmann; we had met at previous meetings atvarious conferences, so working together was easy and enjoyable. Vilnius Conferences were important, notonly as each served as a meeting place for Eastern and Western scholars, but also very important for Lithua-nian probabilists. As a result of these conferences, new connections were made, and our work became moreknown to the broad probability community. I want to mention that the Presidents of the Republic of LithuaniaA. Brazauskas and V. Adamkus personally attended the opening ceremonies of the Vilnius Conferences.

As our conversation is coming to a close, are there questions that you would like to address?

OK, you could have asked about what seems important to me in my professional life. It goes without sayingthat I like some of my mathematical results (but only time will tell how much they are worth). Also impor-tant is the collaboration with outstanding mathematicians while I was working at good foreign universitiesfor about eight years. However, I think that my greatest scientific value is my students. Even though only 16dissertations (not a large number) have been prepared and defended under my supervision, five of the studentsare now habilitated doctors and professors, three of them, Mindaugas Bloznelis, Rimas Norvaiša, and AlfredasRackauskas are working in our faculty in Vilnius; Gyula Pap is working in his native Hungary, and RicardasZitikis has been working in Canada for many years. Gintaras Bakštys holds an important position in an inter-national insurance company, and at the same time he teaches master courses in actuarial mathematics in ourdepartment. And most importantly, the students already have their students. The American Mathematical So-ciety created the so-called Mathematical Genealogy Tree (https://genealogy.math.ndsu.nodak.edu/id.php?id=88154); according to it, I have 27 mathematical grandchildren (by comparison, I haveonly four true grandchildren). Alfredas and Ricardas each gave me eight mathematical grandchildren, andGyula added seven! Now all I have to do is to wait for great-grandchildren, both real and mathematical.

References

1. V. Bentkus and V. Paulauskas, Optimal error estimates in operator-norm approximations of semigroups, Lett. Math.Phys., 68(3):131–138, 2004.

2. M. Bloznelis and V. Paulauskas, A note on the central limit theorem for stochastically continuous processes, StochasticProcesses Appl., 53:351–361, 1994.

3. X. Fernique, Les fonctions aléatoires càdlàg, la compacité de leurs lois, Liet. Mat. Rink., 34(3):288–306, 1994. Englishtransl.: Compactness of distributions of cadlag random functions, Lith. Math. J., 34(3):231–243, 1994.

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436 M. Bloznelis and A. Rackauskas

4. V. Paulauskas, On the reinforcement of the Lyapunov theorem, Litov. Mat. Sb., 9(2):323–328, 1969 (in Russian).

5. V. Paulauskas, Some remarks on multivariate stable distributions, J. Multivariate Anal., 6(3):356–368, 1976.

6. V. Paulauskas, On operator-norm approximation of some semigroups by quasi-sectorial operators, J. Funct. Anal.,207:58–67, 2004.

7. V. Paulauskas and A. Rackauskas, Accuracy of the Approximation in Central Limit Theorem in Banach Spaces,Mokslas, Vilnius, 1987 (in Russian). English transl.: Approximation Theory in the Central Limit Theorem. ExactResults in Banach Spaces, Kluwer, Dordrecht, Boston, London, 1989.