7. Probability and Statistics Soviet Essays - Sheynin, Oscar

will be sufficient, although not necessary, for normal stability. This condition was the basisof Chebyshev’s and Markov’s classical theorems.A remark made in 1918 by Bernstein [2] may be considered as a point of departure for thework of Soviet mathematicians. He noted that the equality2( ζn− Cn)limE21+( ζ − C )nn= 0 as n (1.1.4)can serve as a necessary and sufficient condition of stability for given constants C n . In 1925,Slutsky [5] provided similar but more developed ideas.If the sequence of n is stable, their medians m n can always be chosen as the constants C n .Therefore, owing to Bernstein’s finding, the necessary and sufficient condition for stability ofthe sequence of n can be written down as2( ζn− mζn)limE21+( ζ − mζ)nn= 0 as n (1.1.5).It is easy to apply the classical condition (1.1.3) to sums of independent terms n = 1 (n) + 2 (n) + … + n (n) (1.1.6)having finite variances var k (n) = b k (n) ; indeed, the variances of the sums n are the sums of thevariances of the appropriate terms:B n = b 1 (n) + b 2 (n) + … + b n (n) .Unlike the variances B n , the expectations included in conditions (1.1.4) and (1.1.5) cannotbe expressed in any easy manner through magnitudes describing the separate terms k (n) . In1928 Kolmogorov [2; 5] discovered a necessary and sufficient condition of stability for sumsof independent terms which was easily expressed through the properties of the separate terms.It can be written down as (Gnedenko[24])nlimk = 1( n)( n)2( ξk− mξk)E( n)( n)21+( ξk− mξk)= 0 as n . (1.1.7)A necessary and sufficient condition of normal stability for the sums of independent termsis somewhat more complicated (Gnedenko [24]), but, once (1.1.7) is established, itsderivation does not present any great difficulties. Of special interest is the case k (n) = k /n, n = ( 1 + 2 + … + n )/n,where 1 , 2 , …, n … is a sequence of identically distributed independent variables. Here,the necessary and sufficient condition of stability becomes extremely simple:lim nP(|| > n) = 0 as n . (1.1.8)This condition {the authors have not specified } is satisfied if such variables k have finiteexpectations, and the stability is here certainly normal. This fact constitutes the essence of