# Contiguity of probability measures by George G. Roussas

By George G. Roussas

This Tract offers an elaboration of the concept of 'contiguity', that's an idea of 'nearness' of sequences of chance measures. It presents a strong mathematical instrument for developing yes theoretical effects with functions in statistics, relatively in huge pattern idea difficulties, the place it simplifies derivations and issues easy methods to vital effects. the opportunity of this idea has thus far merely been touched upon within the present literature, and this booklet offers the 1st systematic dialogue of it. replacement characterizations of contiguity are first defined and on the topic of extra usual mathematical rules of an analogous nature. a couple of normal theorems are formulated and proved. those effects, which offer the technique of acquiring asymptotic expansions and distributions of chance capabilities, are necessary to the functions which keep on with.

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Contiguity of probability measures

This Tract offers an elaboration of the concept of 'contiguity', that's an idea of 'nearness' of sequences of likelihood measures. It presents a robust mathematical software for developing sure theoretical effects with functions in records, fairly in huge pattern concept difficulties, the place it simplifies derivations and issues how one can vital effects.

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The first three sections deal with a topic suggested to NIELSEN by LANDSBERG involving differential geometry. The last section deals with the problem of determining the minimal number of fixed points of a one-to-one topological self-mapping of a torus. It arose from a question raised by M. DEHN who became NIELSEN'S thesis advisor after the death of LANDSBERG. The homogeneous modular group GL(2, Z) plays a role in this paper. However, its main importance is derived from the fact that NIELSEN pursued the fixedpoint pFoblem for topological self-mappings of two-dimensional manifolds in many later publications, some of which also contained important grouptheoretical contributions to the theory of mapping class groups.

The survey by WEHRFRITZ [1973] shows how much work has been done here since the beginning of the Twentieth Century. These groups have some surprisingly simple properties, particularly if the entries of the generating matrices belong to a field of characteristic zero. In tills case (but not only in this case), MAL'CEV [1940] showed that the groups are residually finite. Later, SELBERG [1960] proved that the groups contain a torsion-free normal subgroup of finite index. And TITS [1972] proved that these groups either contain a free subgroup of rank 2 (and, therefore, of infinite rank) or they are finite extensions of solvable groups.

So far, it is not possible to prove this theorem without using the theory of algebraic groups. We have quoted this result as an illustration of the coherence of mathematics. We cannot try to give a full account of the development which led to it or do justice to the many mathematicians who contributed to it. We merely quote the original papers by PICARD [1896] and VESSIOT [1892], an early and very lucid systematic foundation of the theory by LOEWY [1908], an important survey by RIrr [1950], and the lecture notes of HUMPHREYS [1975] which contain references to later papers.