7. Probability and Statistics Soviet Essays - Sheynin, Oscar

nlimk = 1P(| k (n) – m k (n) | > H n ) = 0 as n (1.2.3)for any > 0 is necessary and sufficient forF(t) = (1/t2 π ) −∞exp[– (t – a) 2 /2 2 ] dt.Feller was the first to publish the necessary and sufficient conditions themselves in anexplicit form for attraction to the Gauss law under restriction (1.2.2). They can be expressedin the following way: For the existence of constants C n and H n > 0 such that conditions(1.2.1) and (1.2.2) are satisfied, it is necessary and sufficient that such constants K n exist forwhichlim inf P(| n – m n | > K n ) > 0 as n , (1.2.4a)nlimk = 1P(| k (n) – m k (n) | > K n ) = 0 as n . (1.2.4b)The former relation concerns properties of the sums rather than that of separate terms and itcan be written down in an equivalent formnlim inf K n2k = 1K( n)( ξk−2 ( n)n( ξkmξ− m( n)2k)( n)2ξk)> 0 as n , (1.2.4a)free, as (1.2.4b) also is, from the indicated shortcomings but somewhat less obvious.Bernstein [13; 40] originated a similar investigation of the conditions for the attraction ofsums of independent vectors to n-dimensional Gauss laws. In the Soviet Union, Khinchin,Romanovsky and Gnedenko followed up this direction; the last-mentioned author offered themost polished formulation [25].1.3. Specifying the Classical Limit Theorem. Even now, normal attractionlim F n (t) = (t) as n ,F n (t) = P[( n – A n )/Bn< t], (t) = (1/t2 π ) −∞exp[– (t 2 /2) dt,remains the most important case of attraction to the Gauss law. The desire to estimate asprecisely as possible the difference F n (t) – (t) is natural, both theoretically and practically.If this difference is not sufficiently small, it is also natural to add to (t) some correctionterms expressed simply enough through the distributions of the terms k (n) so that F n (t) willthen be estimated sufficiently precisely.In 1911, Bernstein discovered the most effective method of precisely estimating F n (t) forthe particular instance of the Laplace limit theorem; later he [39] somewhat strengthened hisfindings. The foundation of most of the subsequent investigations of the general case isChebyshev’s method. It approximately represents F n (t) as a series of the type of(t) + C 3 (n) (3) (t) + … + C s (n) (s) (t) + …