hsu's work

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Hsu's Work in ProbabilityAuthor(s): K. L. ChungSource: The Annals of Statistics, Vol. 7, No. 3 (May, 1979), pp. 479-483Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2958733 .Accessed: 14/08/2011 19:24

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The Annals of Statistics1979, Vol. 7, No. 3, 479-483

HSU'S WORK IN PROBABILITY

BYK. L. CHUNGStanford University

Hsu retumed from England o Kunming n the middle of the Sino-Japanese war,apparently n 1940. He gave a long course beginning with the theory of measureand integration, hrough probability heory, and leading to mathematical tatistics.For the first part he based his lectures on Caratheodory's Vorlesungen ber ReelleFunktionen, or the second on Cramer's Random Variables nd Probability Distribu-tions. He was a polished and vivacious lecturer with complete notes written outbeforehand n a notebook he carried. He enjoyed making a fine point in class. Forinstance, when he was doing the inversion ormula or characteristic unctions, hetook delight in the fact that one could integrate over a single point by theLebesgue-Stieltjes ntegral. He had a tattered copy of Cramer's book with manymarginal notes and used to say that it was written in a more difficult way thannecessary but contained all that was essential for probability. He was a truevirtuoso in the method of characteristic unctions. His papers [17], [23], [25], [26],[31] and [35] all showed his fascination or, as well as mastery of, this precious ool.Mathematical iterature was hard to come by in Kunming during those years, and

some of the old volumes shipped from Peking (then called Peiping) were kept incaves to preserve them from air raids. (We did not actually live in caves, butfrequently had to run to them for hours at a stretch under raids or alerts.) Amongthe books so kept was, for example, Kolmogorov's Grundbegriffe er Wahrschein-lichkeitsrechnung. t my request Hsu had this book extracted rom the caves, but Iremember his saying of it: "This s another kind of mathematics". He was not, ofcourse, a probabilist per se at a time when such an appellation hardly existedanywhere in the world; by education and inclination he was more fond ofproblem-solving han generalities. However he never tried to dissuade others fromfollowing their own bent and getting interested in those other kinds of things.Indeed, if he could be drawn nto a new topic, he would work at it in earnest andquickly produce something of value. For instance, when Borel's work on what hecalled "denumerable probabilities" attracted my attention, Hsu took an interestalso, which led to the paper [21]. This was an early approach to what becameknown as "recurrent vents" under Feller's development. Papers 23] and [30] wereprobably written under similar circumstances.

But, of course, Hsu's major contribution to probability and its applicationsflowed from his consummate kill in manipulating haracteristic unctions. Let medescribe and comment on these papers n their chronological order.

Received June 1978.AMS 1970 subject lassifications. Primary OIA70; econdary 60E05.Key words nd phrases. Obituary, robability heory.

479


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480 K. L. CHUNG

Paper [17] is, without doubt, one of Hsu's most important works. He wrote itshortly after A. C. Berry obtained the correct order of magnitude of the errorestimate n the classical central imit theorem. Namely, if {(, n > 1} are indepen-dent and identically distributed random variables having distribution P(x) withmean 0 and variance 1, and finite third absolute moment /3, then for all x thedistribution

F(x) = Pr{ 2Er= r < x

differs from the standard normal distribution by less than Af33/n 2, where A is anabsolute constant. (This result was also proved by Esseen, but Esseen's paper waspublished ater

and not known to Hsu at the time.) Hsu extended Berry's methodto give a simpler proof of Cramer's heorem on the asymptotic expansion of F(x)when P(x) is nonsingular. But much deeper and farther reaching are the corre-sponding results when the sample mean

= zr= ltr

is replaced by the sample variance

In Hsu's own words: "so much for the known results for the approximatedistribution f {. By a purely formal operational method Cornish and Fisher obtainterms of successive approximation o the distribution of any random variable Xwith the help of semiinvariants. t is hardly necessary o emphasize he importanceof turning Cornish and Fisher's formal result (asymptotic expansion withoutappraisal of the remainder) nto a mathematical heorem of asymptotic expansionwhich gives the order of magnitude of the remainder. n this paper we achieve this

for the simplest function next to C,viz. q." The basic idea is as follows. LetX-1 En r2 _ I n=

n2 (a4- 1)2 n2

where a4 is the fourth moment of P(x). These are related to q by

( n 2 (11) = X- y24 - l ((a4 n)

and the crux of the method s to approximate he distribution f the random vector(X, Y) through its characteristic unction. Even a cursory perusal of the proofwould make it obvious that this is a task of a higher degree of complexity and notjust a routine analogue of the results of Berry and Cramer, since X and Y arehighly correlated. To achieve the goal Hsu literally added a new dimension to themethod of approximation used before him. He was still teaching the course


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HSU'S WORK IN PROBABILITY 481

mentioned above when he obtained the result, and was obviously elated. I recall hissaying that he made several previous attempts at the problem and the breakthroughcame when he realized that he must attack the bivariate approximation ather hanevading the difficulties by shortcuts. Hsu's method is applicable to many otherfunctions commonly used in statistics such as sample moments of higher order andStudent's tatistic. Some of these were carried out by his students. Although Hsu'spaper was cited by Bikjalis n his work on normal approximations, recent bookon the subject failed to mention it. In the course of writing this article I asked theauthors about this glaring omission and received the regretful eply that they werenot aware of Hsu's paper This is a serious omission and suggests hat the analyticpower generated by Hsu's method has yet to be fully explored.

In paper [23] the main result is that for a sequence of independent andidentically distributed random variables with zero mean and finite variance, wehave for every E > 0:

nP {I+XI + +Xnl > E}< X.

This is an interesting trengthening f the classical strong law of large numbers nthe direction of the Borel-Cantelli emma. The idea of such a result s probably dueto Robbins, but the method of proof is vintage Hsu. The problem s reduced, after

a convoluted Fourier inversion, to the estimation of several pieces of an integralinvolving the characteristic unction-a remarkable our de force. Erdos (1949)later sharpened the result and showed that the conditions on the moments arenecessary as well as sufficient. (According to Robbins, their original manuscriptalso contained a proof of the necessity of the conditions which was not publishedas being too complicated.) Erdos used a combinatorial ype of argument. Baumand Katz extended the result to moments of fractional and higher order.

Paper [35] forms Appendix III in the translation of Gnedenko-Kolmogorov(1968) and its story may be worth telling here. Hsu had always been impressed withthe necessary and sufficient conditions for the general form of the central limittheorem due to Feller, if only for the adroit use of characteristic unctions there. Ina letter dated May 12, 1947, from Chapel Hill, he told me that he had just solvedthe same problem for all symmetric stable laws. He wrote: "I believe that in adomain of the weak laws this is the best theorem yet. The method in this work alsopoints to solution of the most general problem of weak law, namely [when the limitis] an infinitely divisible aw. But further difficulty s great.... The problem n myhand is so natural that I am in constant worry of clashing with people. So,

whenever you communicate with probability people, please spread the news." Alatter dated May 26, 1947, announced the final result to me. It gives the necessaryand sufficient conditions under which the row sums of a triangular array ofinfinitesimal random variables, ndependent n each row, converge n distributionto a given infinitely divisible distribution. His conditions differ somewhat fromthose in Gnedenko (1944) in that the "two tails" are combined. The proof is direct


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482 K. L. CHUNG

and makes use of the kernel (sin t/t)3. The complete manuscript containing thisresult and its self-contained proof is paper [35]. It was given to me shortly beforeHsu's departure or China in the summer of 1947. (I have erroneously placed theyear to be 1946 in the preface to the translation cited above.) Despite hispremonition, Hsu did not learn of Gnedenko's 1944 paper until later. In the onlyletter he wrote me after his return to China, dated March 19, 1950, he acknowl-edged Gnedenko's priority and urged me in most emphatic erms to keep the onlycopy of his paper for him. He had clearly staked his own strength on solving thischallenging problem, he culmination of a series of papers by such names as Levy,Khintchine, Kolmogorov, Marcinkiewicz whom Hsu met), Feller and Gnedenko.He toiled hard and triumphed quickly. Does it matter that another striver has

climbed the same peak before? Hsu's method is direct and "starts rom nothing".The last three words give a free rendering of a more elusive Chinese saying whichHsu used on another occasion, meaning minimal reliance on precedents. He hadalways considered this nonreliance as a virtue of mathematical work, and it isapparent n all his papers where use of previous results for intermediate teps areshunned so far as possible. The new edition of the translation of Gnedenko-Kolmogorov 1968) appeared n the middle of the cultural revolution n China; Ido not know if Hsu ever saw this paper in print.

Papers [25] and [26] were reviewed in some detail by me in the MathematicalReviews, Vol. 17 (1956), page 274. Since this is easily available, I need not repeatthe contents here. Suffice t to say that they are again products of his expertise withcharacteristic unctions. Paper [31] falls in the same category but apparently wasnever reviewed. n it it is proved that every distribution n the class L is absolutelycontinuous. So far as I know this was a new result. It was rediscovered y Fisz andVaradarajan in 1963 (Z. Wahrscheinlichkeitstheorie nd Verw. Gebiete, Vol. 1).

Paper [30] stands out in Hsu's work in probability as it has nothing to do with acharacteristic unction. Like [31], it is published in Chinese with an English

abstract, appeared in the same issue of the journal, and apparently was neverreviewed. It deals with the differentiability roperties of the transition probabilityfunction of a homogeneous Markov process whose state space is Euclidean andwhose sample paths are purely jumping. Such processes were studied by D. G.Kendall in 1955, who analysed the differentiation of the transition probabilityfunction P(t, x, E) at t = 0. In the case of Markov chains, where the state space isdiscrete, he term-by-term ifferentiability f the Chapman-Kolmogorov quations:

P'(s + t) = P'(s)P(t), s > 0,

P'(s + t) = P(s)P'(t), t > 0,was proved by Austin analytically, and by myself probabilistically for referencesto D. G. Kendall and Austin see Chung (1967).) Hsu extended these equations tothe Eucidean case, "using more elementary methods and obtaining more preciseresults". For the first equation above he actually derived an integral representationof P(t, x, E) in terms of the post-exit process, thus generalizing my approach but


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HSU'S WORK IN PROBABILITY 483

by purely analytic means. The intimate relation between the analytic and stochasticstructures s implicit n his work, and can easily be made explicit.

I did not learn of Hsu's death until my arrival n Peking n June 1972. Hsiao-fuTuan told me that Hsu spent his last years re-doing a lot of classical combinatorialanalysis in his own way and left copious notes. So far the contents of these notesremain unknown.

REFERENCES

CHUNG, K. L. (1967). Markov Chains With Stationary Transition Probabilities, 2nd ed. Springer-Verlag.ERDos, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Statist. 20 286-291.GNEDENKO, . V. (1944). Limit theorems or sums of independent andom variables. UspehiMat. Nauk.

10 115-165. English ranslation: Translation No. 45, Amer. Math. Soc., New York, 1951.GNEDENKO, . V. and KOLMOGOROV, . N. (1968). Limit Distribution of Sums of Independent Random

Variables, ev. ed. (translated by K. L. Chung, with Appendices by J. L. Doob and P. L.Hsu). Addison-Wesley.

DEPARTMENT OF MATHEMATICSSTANFORD UNIVERSrrYSTANFORD, ALIFORNiA4305

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