By A.M. Kagan
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Extra resources for Characterization Problems in Mathematical Statistics
Polya's in the year 1923, where it was established that only the normal law admits identically distributed linear statistics X1 and a 1 X1 + a2 X 2 from a random sample (X1 , X 2 ). The note  ofl. Marcinkiewicz's was published in 1938; it was established there that laws having moments of all orders and admitting the existence of a nontrivial pair of identically distributed linear statistics based on a random sample are normal. In 1953 appeared the work  of Yu. V. Linnik, in which a description was given of the class of symmetric laws admitting identically distributed linear statistics, the problem of characterizing the normal law through such statistics was studied in detail, and some applications were given to the theory of testing of hypotheses.
20) It is easy to see that rank (AO A*) = r, so that J.. 3 = 0, and so on. 5. (cf. ). 13) be satisfied for It; I < (), i = 1, ... , p, with rank A = r ~ p, and let every column of A have at least two nonzero elements. Then the degree of the polynomials t/1 1, ... , and e1, ... , eP is at most two. 3, considering q equations with p variables t' = (t1 , ... , tp). 21) j = 1, ... , q, where Itd < (), i = 1, ... , p; e 1 , ... , eP are the columns of the identity matrix. ;, i = 1, ... 6. (cf. ) Suppose that every column of the matrices A and B contains at least one nonzero element and that no column of A is proportional either to another column of A or to any column of IP.
The intersection theorem). Let Vi and V2 be complex algebraic varieties in the complex space en, having points in common. 5) For a proof, vide , pp. 4 INFORMATION FROM THE THEORY OF ENTIRE FUNCTIONS AND F~OM THE THEORY OF DIFFERENTIAL EQUATIONS The material given below will be used mainly in Chapters 2, 5, 6, and 12. 1. (the Paley-Wiener theorem). , for Jlf(x)l 2 dx < oo it is necessary and sufficient that it admits the representation f(z) = r eizt
Characterization Problems in Mathematical Statistics by A.M. Kagan