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\noindent{\bf Jarl Waldemar LINDEBERG}\\
b. 4 August 1876 - d. 24 December 1932
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\noindent{\bf Summary}. Finnish mathematician and statistician, Lindeberg
is best known for his important proof of the central limit theorem.
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Jarl Waldemar Lindeberg was son of a teacher at the
Helsinki Polytechnical Institute; the family was well-to-do. He was 
aware early of his mathematical talent and interest. Having taken his
candidate's degree in 1897, he went for a year to study in Paris,
with the support of an aunt. Provided from there with a theme -
apparently essentially his own choice - for his doctoral
dissertation, on partial differential equations, he went for the
following winter with his family to Corfu, sharing his time between
scientific and social occupations. In December 1900 he was ready to
defend his thesis, with Ernst Linde1\"{o}f as the appreciative
opponent. In spring 1902 he was appointed docent and three years
later adjunct of Mathematics. In 1919 he was awarded the title of
professor. Being economically independent, and feeling the position
of adjunct best to his taste, he never cared for an ordinary
chair. Apart
from his teaching at the university he also taught at the Technical
University 1911-18, and in his younger days in a Finnish secondary
school. Among his pupils he found his future wife, Inez Beeker,
whom he married in 1905, barely a year after her graduation.

Lindeberg remained interested in the problems of school teaching,
and represented Mathematics in the Matriculation Examination Board
from 1902 to 1918. The problems that he gave were expressly aimed
at testing not only knowledge but also the ability to think;
afterwards, he often had to defend them against enraged school
teachers. What was wrong, according to him, was not the examination
in itself, but rather the widespread tendency to require this
examination for professions that had nothing to do with true
academic studies.

He was elected member of the Society of Sciences in 1909 and of the
Finnish Academy of Sciences in 1919. In the latter he acted for
many years as treasurer. 

Lindeberg's scientific activity comprises several different
periods. Having started out with partial differential equations, he
shifted to the calculus of variations, which occupied him from
about 1904 to 1915. Around 1918, there is a short interlude given
to the theory of functions. His later years, until his early death
in 1932, were devoted to the fields in which he was to become most
renowned, viz., probability and statistics. A first impulse in this
direction he probably received when he was appointed to the
administrative board of a short-lived life insurance company in
1912; his main achievment was actually its sound liquidation. He
was later an active member of the Finnish Actuarial Society. From
1916 he regularly lectured on probability, and was soon considered
a national expert in that field; in 1925, his adjunctship was
redefined so as to cover, in particular, the calculus of
probability and its applications. At about the same time
representatives of several scientific associations requested him to
write an elementary text-book on the subject, and in 1927 he
published his {\it Todennakoisyyslasku} (Calculus of Probability
with Applications to Statistics) which presupposes no calculus.
The ultimate objective is to provide research workers in various
fields with tools for critical judgement.

Lindeberg's scientific profile is that of an independent and
critical research worker, a man who is able to distinguish between
essentials and unessentials, and feels an urge to get to the bottom
of problems. These qualities were of particular importance in his
dealing with probability and statistics - fields on the border line
between pure and applied Mathematics, and fields in which the
fundamental notions were at that time just about to crystallize.
For instance, in choosing descriptive characteristics for
distributions, Lindeberg, with typical common sense, first of all
wanted them to give easily understandable answers to simple
questions.

In his thinking habits, he seems to have been a lone wolf; he
worked best when alone at his desk, trying to find out things for
himself rather than consulting predecessors. According to Lindel\"{o}f,
Lindeberg always found it laborious to study other people's
writings. He spoke with much modesty and joking irony of his own
relationship to science.
``You see", he once said, ``owning a country place with forest and
farming land, in Helsinki I can defend my laziness by saying that I
am really a farmer, and in the country by claiming really to be a
scientist". In his earlier years, he had serious doubts about his
ability.

As an academic teacher, Lindeberg was painstaking and helpful but
somewhat dry. In private life he was a friendly and sociable
person, with wide interests, and a large circle of relatives and
friends among whom he was much loved.

Lindeberg's first and most famous contribution to probability and 
statistics was his new proof of the {\it central
limit theorem}. That is, roughly speaking, the sum of a large number
of independent random variables is approximately normally
distributed. A proposition to this effect was first formulated by
Laplace in 1812. The first rigorous proof of it was given by
Ljapunov in 1901. The conditions imposed in his proof are fairly
wide, but it suffers from the drawback that it requires the
existence of the third absolute moments of the component
distributions. When Lindeberg first attacked the problem he was not
aware of Ljapunov's proof, only of a much weaker result by von
Mises (q.v.). His first paper on the matter, {\it \"{U}ber das
Exponentialgesetz in der Wahrscheinlichkeitsrechnung}
(Ann. Acad. Sci. Fenn., 1920) essentially leads up to Ljapunov's
results, but uses entirely different methods. Whereas Liapunov had
used the characteristic function, Lindeberg's proof, although
rather intricate, is in principle elementary: using the ordinary
convolution formula, he step by step compares the distributions of
the cumulative sums of (a) the arbitrary random variables under
consideration, to those of (b) normal variables with the same
variances as in (a). An ingenious device is to start off both
series with an auxiliary normal term. Two years later, having
learnt about Ljapunov's proof, Lindeberg modified and improved his
own, in {\it Eine neue Herleitung des Exponentialgesetzes in der
Wahrscheinlichkeitsrechnung} (Math. Zeitschr., 1922). His ultimate
result, which goes beyond Liapunov's, can in a slightly modified
version be formulated as follows: Let $X_1,X_2,...$ be independent random
variables with means 0, with variances ${\sigma_i}^2$, and with cumulative
distribution functions ${F_i}(x), i= 1,2,..$.  
Let ${s_n}^2 = {\sigma_1}^2 + ... + {\sigma_n}^2$ be the variance
of the sum $S_n = X_1+X_2+... +X_n$. Then a {\it sufficient} condition for
the c.d.f. of $S_n/s_n$ approaching uniformly the normed c.d.f. with
mean 0 and variance 1, is that the "normed tail sum" of the
variances tends to zero, i.e. that the following condition (L) holds:

$$ {s_n}^{-2} \sum_{k=1}^{n} \int_{|x| > t s_n} x^2 dF_i (x)
\rightarrow 0$$
as $n \rightarrow \infty$
for each $t > 0$. In 1935, Feller proved that the criterion is, in a
sense, the sharpest possible under the plausible restriction to
situations in which $s_n \rightarrow \infty$,
${\sigma_n}/s_n \rightarrow 0$ the {\it Lindeberg condition}
(L) is actually {\it necessary}.

Once Lindeberg had become known as an expert on probability, he was
approached as consultant in various fields of application: biology,
medicine, linguistics, and above all, forestry. He was thus led to
an active interest in mathematical statistics, a branch of science
which at that time went through a most dynamic evolution, launched
by the British statisticians K. Pearson (q.v.) and R. A. Fisher (q.v.). In
far-away Finland, the impact of this development was hardly yet to
be noted. While one of the main objectives of the British school
was the derivation of exact distributions (like the $\chi^2$, $t$, and
$F$ laws) for sample statistics connected with normal
distributions, Lindeberg essentially stuck to the classical ``large
sample" technique, i.e., the approximate calculation of mean errors
on which statistical inference could be based if there was
enough material to warrant normality. With his characteristic urge for
common sense in the applications, Lindeberg was particularly
interested in the choice of sensible statistics to describe
relevant properties of the sampled population; that is, statistics
that gave simple answers to simple questions and were intuitively
explainable to laymen. 

In a paper read at the Scandinavian Congress in Copenhagen, {\it
Uber die Korrelation} (1925), Lindeberg 
objects to Pearson's correlation coefficient which ``does not answer
any natural question", instead, he supports an association
statistic called ``percentage of correlation", proposed by Esscher
and equivalent to what has later been called {\it Kendall's tau}; its
mean error is announced, and in a later publication derived.
Several of his papers are actually early contributions,
probably not much noted, to the theory of {\it non-parametric
methods} in statistics.

Lindeberg, also became interested in linear multivariate analysis.
His paper {\it Zur Korrelationstheorie} (1929) is concerned with
partial regression and correlation coefficients, and notably with
the mean errors of the corresponding sample statistics; in a
subsequent paper there is a more streamlined treatment of the same
subject. 

The only consultation work of Lindeberg that led
to publications was his dealing with forestry but here his
contributions were of lasting importance. They are published in two
small papers, {\it \"{U}ber die Berechnung des Mittelfehlers des
Resultates einer Linientaxierting} (1924) and Z{\it ur Theoria der
Linientaxierung} (1926), both in {\it Acta Forestalia Fennica}.
They are concerned with line transect
sampling procedures for estimating such quantities as (say) the
lumber volume $p$ per hectare, taken in the mean over a large area.
In the second of the 
papers, it is interesting to note an attempt at small
sample theory: Lindeberg actually derives Student's distribution,
presumably wholly unaware of the achievements of the contemporary
British school.
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\noindent{\bf Bibliography}

\noindent Elfving, G. (1981), {\it The History of Mathematics in Finland
1828-1918} Societas Scientarium Fennica, Helsinki. [The above article
is an edited abstract of the article on Lindeberg in this book.]

\noindent Schweder, T. (1980), Scandinavian statistics, some early
lines of development. {\it Scandinavian Journal of
Statistics}, {\bf 7}, 113-129.
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\hfill{G. Elfving}


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