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June 2, 1980 Dear Professor Kallenberge Thank you for your letter. I assume that you have discussed your comra;ents with Lindvall. I certainly did not know that the con ditions were equivalent. In one dimension I believe the equivalence of (a) and (b) is trivial and given in some textbook. Is it trivial also in two dimensions? The main reason I suggested the paper was the mistake in the paper by Ornstein and rme reproduced in my Course. For 10 years nobody told me the mistake, perhaps nobody tried it (Esseen said he used the book.) When the mistake was found I corrected it in a direct way and Lindvall extended the result. Actually a student re cently showed me a relatively simple Porrection of the problem which will be given in the next printing. Somewhere in the proof all use the uniform convergence. Now the condition you indicated with the limsup, and "for a fixed x>O", is more interesting to me. Do you riean for one single value of x? I must say I am so remote from such results now that I must ask you to send me the proof. The syrrinetric case may still be difficult but intuitively it makes recurrence more likely. On the other hand, you probably know that so far no proof has been given o the transience result in dimension >2 without use of Fourier transform. i aybe you have a combinatorail proof in the same spirit of the method in d=l and 2. I think it is interest ting to have a condition weaker than the CLT in the case d=2, and it seems that your limsup result is of that kind. Sincerely, CHALMERS TEKNISKA HOGSKOLA OCH G'TEBORGS UNIVERSITET Matematiska institutionen June 10, 1980 Dear Professor Chung, Many thanks for your letter of June 2, which just arrived. Yes, I made these comments to iindvall also (but he refused to see my point, "since everything in the paper is correct"). I shall try to be more explicit about the equivalence of the following conditions: (a) EX.xO and E (X. 2 (b) n1/2 S is asymptotically normal, (c') limsup P{ Sn ( x n1/2 >0 for some x>O np (for one single value of x, yes). Since (a)=> (b)=# (c') is obvious, it remains to prove that (o')> (a). Assume that E Xi i2.oo. Then there exists a unit vector e such that E(e,X.)2=Co (where (',') denotes the inner product). By Theorem 4.1 of Esseen in ZfW 2 (1968), 290308, the concentration function of (e,S ) must then be of the order o(n 1/2), which implies that lim P S n x n 1/2 < lim Pl(e,S ) ( x n1/2 0 n+oo nPo 2 for all x>0. This contradiction to (c') shows that (c') implies EI Xi n1/2S n1/2EX. n I is asymptotically normal, and then the probability in (c') must tend to zero unless EX =O. This completes the proof of the implication (c')==(a). The equivalence of (a) and (b) can certainly be proved directly in many ways (though I am not aware of any trivial proof). The explicit statement in one dimension may be found, e.g., in the English version of Gnedenko Kolmogorov (Theorem 4, p.181). The multidimensional result is an immediate corollary. One may look at the transience result for d>3 as a consequence of the BorelCantelli lemma and the fact that the concentration function of S is of the order n2 (provided that X. is truly ddimensional). The latter estimate follows for d=1 from the KolmogorovRogozin inequality (see Esseen's paper quoted above, formula (A)), which was indeed first proved by a combinatorial argument (before Esseen gave a new proof based on Fourier transforms). I haven't seen the combinatorial proof, so I don't Postadress Gatuadress Telefon t o Fack Sven Hultins gata 6 031 81 01 00 402 20 GOTEBORG 81 0200 Olav Kallenberg, PTRF Div. of Mathematics, 120 Math. Annex, Auburn Univ., Auburn, AL 368495307, USA Professor K.L. Chung Department of Mathematics Stanford University Stanford, CA 94305 March 10, 1991 Dear Professor Chung, You probably received some time ago a letter from Hermann Rost dated January 26, 1991, where he announces his decision to retire as the main editor of Probability Theory and Related Fields and informs about Springer's choice of me as his successor. I am certainly extremely flattered by their choice, and at the same time I feel a great responsibility to maintain the high standards of the journal. Ever since I was asked to take over, thus for several months by now, I have been thinking about what changes might be necessary or desirable with respect to the composition of the editorial board, the editorial policy, routines for handling submitted manuscripts, etc. I have also had extensive discussions with Hermann Rost and with representatives for SpringerVerlag, and I have received advice from a number of people with editorial experience that I have consulted with. After all this thinking and consulting, I have come to the conclusion that some rather drastic changes ought to be made in a number of different ways. Some of those are discussed below. Let me begin with the composition of the Editorial Board. Presently, we have a rather strange mixture of people who are listed as editors for a variety of different reasons: Schmetterer, the founder of the journal back in 1962, is just a honorary member of the board and is hardly active any more. You and Frank Spitzer, great senior people, whose reputation is such that it is an honor for the journal to have you as members of the board, whether you are actually taking part in the daily editorial work or not. About 15 probabilists who are doing a great job for the journal and are eminent enough to contribute to our prestige. Some of those people are in fact outstanding mathematicians, such as Jacod and Watanabe. Two (!) first class (?) statisticians, Beran and Donoho, who seem to be doing a great job for the journal, even though they are both are very busy with other duties. The rest no individual comments necessary. Some of those people were appointed long ago, sometimes for geographical reasons to represent their country, in other cases to represent some specific field that was fashionable then but perhaps not anymore. In the old days, the associate editors were appointed for an unlimited time, and nobody dared to ask them to leave. With Schmetterer I am planning, with Springer's consent, to have him listed together with the other former editors on the back cover of the journal. As for you and Spitzer, I have been thinking of creating a special 'Advisory Board', comprising about five of the world's most distinguished probabilists. Those people would not be expected to take part in the daily editorial work, but hopefully they should be available for advice concerning appointments of new associate editors, regarding editorial policy in general, and on some other matters of a more principal nature. Needless to say, it would also continue to give an enormous prestige to the journal, if names like yours would remain on the cover. So my primary reason for writing is to ask whether you would be interested in such an arrangement. (The matter is contingent upon the agreement of some other people I am planning to ask, which might include Meyer, Liggett, and Dynkin. They have not yet been asked.) Regarding the other fifteen probabilists who are presently doing an excellent job on the board, things are very straightforward. During the last few days, I have been writing long personal letters to each of them, to ask whether they are willing to continue as editors. The first replies could be expected within a couple of weeks. Hopefully there will not be too many resignations. A more delicate task is to get rid of the 'deadwood'. Nobody certainly likes to make enemies ... Next comes the extremely difficult and important issue of the role of Statistics within PTRF. The problem has in fact been discussed for ages. Some thirty years ago, when the 'Zeitschrift' was founded, it was certainly natural and desirable to keep Probability and Statistics together, and this was also a major concern during Klaus Krickeberg's term of editorship. Perhaps it is less natural today, because the two subjects have grown so immensely and have developed in rather different directions. With the increasing emphasis on computers in statistics, the gap seems to be widening more than ever. The problem is then, if we want to continue with statistics, to keep a sufficient number of statisticians on the board, which will necessarily be at the expense of our coverage of 'core' probability. Conceivably we are running the risk that, by trying to cover too broad an area, we may lose in depth and strength all over. (My remarks on statistics are intended to be neutral, and I am not trying to convince in either direction.) The matter of statistics is part of the general question of how to interpret the phrase 'Related Fields' (formerly 'verwandte Gebiete') in the name of the journal. I was surprised to discover that no Editorial Policy exists or has ever existed for the 'ZW'. Our only guidelines are in some historical documents that Springer have kindly dug up for me. The issue is largely unresolved, and it seems to be very much up to us (Editors, Advisors, Springer,...) to decide upon a policy. Let me now turn to some changes in the procedures that I am planning to make, and which will partly change the character of the editorial work. I feel that something rather radical has to be done to shorten our comparatively very long processing times for submitted manuscripts. My intention is then to send practically all papers out to referees myself, to obtain a first technical expert opinion. According to my experience, it is usually rather easy to find a suitable referee, so the work is mainly secretarial (mailing, filing, keeping records, taking copies, sending reminders, etc.) and can probably be done quite efficiently, once the routines have been carefully worked out. If the referee recommends rejection and I agree, then I shall take care of the paper myself and write to the author, so no associate editor will be involved at all. It is only when a paper receives a favorable review that I shall have to mail it out, together with the report, for a second opinion from some associate editor. (Springer requires that no paper should be accepted without at least two favorable opinions, which I think is a good policy.) Assuming that the paper has already been carefully refereed, there will normally be no need for any further detailed reading or additional refereeing. All that is needed will be the editor's general judgement and recommendations. It should hardly be necessary, perhaps not even desirable, for the paper to be in his field of expertise, since the editor's job will be mainly to comment on quality and publishability in general, rather than to make any specific technical remarks. By this arrangement, the associate editor will never see those really boring poor papers, and for the others only the most pleasant part of the work will remain. This seems to me the most rational way of using the Editorial Board, and at the same time it should reduce the average handling time considerably. The latter is extremely important to us in the competition with other journals, since the market is much more sensitive to the efficiency of practical routines than to the quality of published papers. (Why is it so? Well, let us face it: Nobody reads papers anyway, unless they are in the person's 'field'.) This brings up the matter of my special aims and ambitions regarding the journal. Here are some of my main objectives:  To maintain our high standards of quality  To shorten the average handling time of manuscripts  To increase the submission rate of high quality papers  To encourage submissions from outstanding people  To strengthen our coverage of 'core' probability  To encourage interaction between subfields  To improve the readability of published papers  To be open for suggestions and encourage discussion  To please authors, readers, editors, referees, & Springer  To make/maintain PTRF as the #1 Probability journal Whether you are interested in the proposed arrangement with an Advisory Board or not, I would certainly appreciate all kind of advice from you, both regarding matters that I have discussed here and perhaps other things of importance. Your comments will be extremely valuable to me and will carry a high weight, because of the great respect and admiration that I have always felt for you and your work, ever since I was a student. In any case, I hope you are willing to give your continued support to the journal, by allowing your name to appear on the cover, and by offering your advice in one form or another. Finally it might be appropriate with a brief presentation of myself. Although my Bachelor degree was actually in Physics and my Masters and Doctoral degrees in Statistics, my heart has always been with Pure Mathematics, especially theoretical Probability. (This doesn't mean that I intend to neglect applications for the PTRF.) Ever since my Ph.D. from 1972, I have been teaching, almost exclusively, a wide variety of graduate courses in Analysis and Probability, first in Gothenburg, and later (since 1986) at Auburn. I never liked to specialize (beyond choosing Probability), so my work has been scattered over the entire field of Probability Theory. I guess my taste is more for general questions of form and structure than for specific technical problems. Notions of symmetry and invariance have been recurrent themes. Now, why Auburn? (A lot of people wonder.) Well, I was appointed to Esseen's chair at Uppsala, hence automatically dismissed from my Gothenburg position, and then I couldn't afford to move. So we came to the US and Auburn as a kind of emergency. After some time we realized that the South is in fact one of the best parts of America, with a minimum of pollution, beautiful nature, friendly people, and an almost perfect climate (except for the summers which are awful; but then I am usually not here anyway). My main interests, apart from family and mathematics, include classical music, cultural history, and downhill skiing. Well, I think I had better quit here. I hope to get your reply soon, by ordinary mail or email, whatever is most convenient for you. My email address is OLAVK@AUDUCVAX.BITNET. Yours sincerely Olav Kallenberg Olav Kallenberg, Department of Mathematics, Auburn University, Auburn, AL 368495310 olavk@mail.auburn.edu. February 24, 1996 Professor Kai Lai Chung 903 Lathrop Drive Stanford, California 94305 Dear Professor Chung, Many thanks for your letter of February 13, 1996. I shall try to answer your questions: No, Kolmogorov didn't use the coupling method of proof. The main results for a finite state space were obtained already by Markov. Kolmogorov announced his results in the countable case in two short notes published in 1936, and the full proofs were later published in 1937 in Russian. I believe that the account in Feller, Vol. 1, follows rather closely Kolmogorov's original approach. The coupling method was first used by Doeblin in 1938, but he seems only to treat the case of a finite state space. With Doeblin's death in 1940 the method fell into oblivion, but it was revived by Harris 1955, and was later used in elementary textbooks by Breiman 1969 and Hoel, Port, and Stone 1972. Since then, the method has been explored by many authors, including Pitman 1974, Griffeath 1975, and Lindvall 1977. I am enclosing the relevant part of Dynkin's memorial article, and also an extract from Kolmogorov's list of publications. My statement of the basic ergodic theorem is equivalent to that of Kolmogorov, but the formulation is mine, as with most results in my book. About the use of fail in a mathematical context, I changed the formula tion as you suggested. Thanks for pointing this out to me. Sincere regards Olav Kallenberg May 31. 1996 Dear Kallenberg: Thank you and Jinsu very much for your kindnesses. The let1t4z>alibi [I think you wrote Alibi?] took 2 weeks ,to arrive* ergo. Anyway, please tell Jinsu that I got the list back, so all is well. I believe I owe you an exp nation about the general renewal theorem in Feller Vol. 2. I gavq a full course on the topic in U. of Illinois in 1970, and kept notes on it.^ The edition Rnot the last one) I used then in class wa.o "full of (a%74 errors and mispri nts" (as you said). The former were mostlya'orrectable (qorrectible?) but Irecall there was one or two far from tiast, Unfortunat ely I do not wish to go back to look at my notes and since Orey is dead nobody might answer the questions, perhaps they had been already corrected in a later edition. Hence, CONCLUSION: any serious mathn' who wants to cite or use that theorem .owes it to the math. public to go thru it thoroughly and CORRECT ALL THE ERRORS before he can agree with the result. I assume you have done so* bu/t noi'ddays who will? In an earlier letter I recall you said something to the ,effet that altho Feller's book (which vol ? without my help in 1948/9 Wryj would have been many more in the lst ed. but Feller acknowledged it I thAnk) is "full of ...", TET h you) "loved it". I must warn you since you have/ai excellent reputation of being puncjiliouse a statement of this kind from/you might well have ill effects on the less itt) scrupulous readers! I WOULDipT SAY sugh a thing. A book published should \j have PEW ERRORS. and those ich cpnnot'be helped should be EASILY CORREHT, able. That is the DUTY oft/he author (and the proofreaders, but you and/ I know how bad the lattefc6uld be., My coauthor Zhao in our book 'eft about 260 errors in the galleys/befoe I coPreoted most (only?) of them. Orey's early pder on Markov chaiis wpre ep p11 of errors that Jain (my PF4. D. student who wrote his the is /on rented stuff) had to rewrite it (ask him if you eper see him). /In my book: n M. 0. there AW many misTaints and at least one big ERROR ;woh w's caught by, and only by, Paul Levy. oob was the editor and McKeanralso read thenms bu, id not see it. Your book is "4 oundaions'ie and m comments Wee mostly iastgrical and anecdotal. I ve forfottex the dis sion but recall tf'at one of my comments was that OST AMAf jPRQBABILI did not know that from discre. te time and disor te space th rd is a great 'difficulty going to continuous time owing to the tremendous dif ic6lty of f oving tFFERENTIABILITY. Kol. had his equation but he vdd any diWerent ilitys nor i the ar rogant and able '6blin ever did X under if 1 tried?). For this reasons authors like Fel er proceed quick to generl\fspace in order to gloss over this unsolved problem. Oce ne in say, R oe might as well atsume some smoothness as one must. Your boedoea not tre t the continuous times dis crete space case and may bypass se problems to obtain one of those dif fusion equations. That iq the a oach of nearly& all author o# large books incl. the Russians, and Doob (19 ,From an an ecotal viewpoint isn't it a wonderful exercise in advanced calculus to sho that one can dieferentiate term by term the equation 1 = I 1(t)  now dio it! None of the above mentioned authors did it until 1b/4 by D. G. Austin who was ,ere a month ago. Of course no book can e everything. What may be to th knowing is: differential equations ,4iifferentiability which cannot be proved without either hard work or (as ller and everybody does) assuming i  fine for physicists. Best reirh0. (f te L..ScAdvhL L Schwaria August 5,, 1996 Dear Kallenberg: U Ik rfl'e ejt IyI .. I I think Kolmogorov's 19cmuch cited long paper was the first one con itaining a partial differential equation for the "continuous" space2 ntt '."Atime case which everybody including Peller "copied"in later texts, even Etob)Doob4 rben, I first read it as a student I was greatly disappoaCted in its banality in comparahvon with the other Kol GREAT papers on random series and large numbers. Because there was hardly a trick in it and all depends on strong analytic hypothesA which made the whole thing an exercise in advanced calculus. The Markovian character is almost concealed there. That is one reason why the "discrete" case attracted so much attention at one time: (no Kol this time until later) Doob, Levy, .... The theorem that in a standard Markov chain piJ is either identically zero or never 0, proved first by Db Gi Austin, is a marvellous probabilistic result totally beyond the reach of matrix experts such as PerronFrobenius. Alas it is so little known beca it is regarded as "special"! Similarly EpOa(t) =0 as I wrote last time. It is this kind of #speciality" $that elevates probabi lity above mundane analysls Perhaps you wii oe mSerested in another ancient results which might possibly have something to do with point processes [after randomizing]. Let f be a density function. Let Pn be a sequence of equal partitions of 1n R into alternate red and black intervals and denote by A the union of all redd/ ones. Under what most general (neo. and suff. if one can) will / f(x)dx converge to 1/2? I hope you recognize this as Poincare's famous An functionn arbitraire"(Oaloul des probabilites., 1912* p. 148) by means of which he solved the roulette problem. See Hostinsky and Pre'het's books Xt lI of 'QRt&pontain those two words above in quotes. Poincare proved the result when f has a bounded derivative, ngt apparent knowing or oaring for his precedessor Cauchyls ........ ....... emrltt??j. Hostinsky proudly ex e4  nuuMP4aOWWW to any integrable function. The latter of course (193 book) never thought of Lebesgue integrability and apparently forgot to assume that his integrableR f be of compact support. Otherwise I am.not sure the CauchyDarboux criterion is valid, and since you are such a fan of Pel ler* you may check to see if one really needs what Peller calls directt #R)integrable" which he needs badly for his renewal extension. 11 this from memory, not verified]. I "'i airly suretone can constructa integr able fto violate the conclusion, what would be much more interesting is a positive result. Nothing really to do with probability. Best regards* 