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# Olav Kallenberg, 1980-1996

## Material Information

Title:
Olav Kallenberg, 1980-1996
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Unknown
Language:
English
Creator:
Kallenberg, Olav
Physical Location:
 Box: 1 Folder: Olav Kallenberg, 1980-1996

## Subjects

Subjects / Keywords:
Mathematics -- History -- 20th century

## Record Information

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University of Florida
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All applicable rights reserved by the source institution and holding location.
System ID:
AA00007243:00001

Full Text

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
n-p
(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), 290-308, 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 But under this condition, the CLT implies that
n-1/2S 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 multi-dimensional result is an
immediate corollary.

One may look at the transience result for d>3 as a consequence of the
Borel-Cantelli lemma and the fact that the concentration function of S
is of the order n-2 (provided that X. is truly d-dimensional). The latter
estimate follows for d=1 from the Kolmogorov-Rogozin 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
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 36849-5307, 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 Springer-Verlag, 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
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 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
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 e-mail, whatever is most
convenient for you. My e-mail address is

OLAVK@AUDUCVAX.BITNET.

Yours sincerely

Olav Kallenberg

Olav Kallenberg, Department of Mathematics, Auburn University,
Auburn, AL 36849-5310 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
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
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 WOULD-ipT 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 proof-readers, but you and/
I know how bad the lattefc6uld be., My co-author 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.
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 ej-t 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 Perron-Frobenius. 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 wi-i o-e 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 integrable-R f be of compact support. Otherwise I am.not sure
the Cauchy-Darboux 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 construct-a --integr-
able f-to violate the conclusion, what would be much more interesting is
a positive result. Nothing really to do with probability. Best regards*