# Asymptotic Behaviour of Linearly Transformed Sums of Random by Valery Buldygin, Serguei Solntsev (auth.)

By Valery Buldygin, Serguei Solntsev (auth.)

Limit theorems for random sequences may well conventionally be divided into huge components, one in every of them facing convergence of distributions (weak restrict theorems) and the opposite, with nearly certain convergence, that's to assert, with asymptotic prop erties of just about all pattern paths of the sequences concerned (strong restrict theorems). even if both of those instructions is heavily with regards to one other one, each one of them has its personal variety of particular difficulties, in addition to the personal method for fixing the underlying difficulties. This e-book is dedicated to the second one of the above pointed out strains, which means we research asymptotic behaviour of virtually all pattern paths of linearly remodeled sums of self sufficient random variables, vectors, and components taking values in topological vector areas. within the classical works of P.Levy, A.Ya.Khintchine, A.N.Kolmogorov, P.Hartman, A.Wintner, W.Feller, Yu.V.Prokhorov, and M.Loeve, the speculation of virtually convinced asymptotic behaviour of accelerating scalar-normed sums of autonomous random vari ables was once developed. This concept not just presents stipulations of the virtually yes convergence of sequence of self sustaining random variables, but additionally reports diverse ver sions of the powerful legislation of huge numbers and the legislations of the iterated logarithm. One should still indicate that, even during this conventional framework, there are nonetheless difficulties which stay open, whereas many definitive effects were acquired particularly recently.

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3) obey the Lebesgue theorem on passage to the limit: given that Yn n_oo ~ Y and E sUPn~ 1 IIYnll < 00, the mean EY exists and limn-oo EYn = EY, that is to say, lim IIEYn - EYII = O. 2 Convergence in probability The sequence (Yn, n ~ 1) is said to be convergent in probability to an X-mndom element Y if for any e > 0 lim P {llYn - YII > e} = 0, n--+oo which is written as Y = P- n-+oo lim Yn , Yn n-+oo ~ Y. We shall say that the sequence (Yn , n ~ 1) con verges in probability if there exists an X-valued random element Y such that Y" ~ Y.

T g_(t) - 1, t get) - 1, dt g+(t) ~ t g+(t) - 1. 38 CHAPTER O. RANDOM ELEMENTS AND THEIR CONVERGENCE... This is why for all t > 0 d dt (g(t) - g_(t» < t (g(t) - g_(t», d dt (g+(t) - g(t» < t (g+(t) - g(t». Since lim Ig(t) - g_(t)1 = tlim Ig+(t) - g(t)1 = 0 ....... oo t-+oo then, by virtue of the Lagrange theorem on the increments of differentiable functions, for all t > 0 0:5 get) - g_(t), One should only observe that for t 0:5 g+(t) - get). 0 ~ P{bl > t} = 1 . rn=exp(t 2 j2)g(t). 1) immediately imply the conditions of the almost sure boundedness and almost sure convergence to zero of a sequence of Gaussian random variables.

In particular, given that II . II is the intrinsic quasi norm on X, there exists some a > such that Eexp(allYID < 00. PROOF. With minor modifications, the proof follows the lines of that of the Fcrnique theorem. Suppose that (YI , Y2 ) is a pair of independent copies of the random element Y. 6. GAUSSIAN RANDOM ELEMENTS 45 forms the pair of independent Gaussian random elements similar to the pair (Y1 , Y2 ). This is why one has for any s, t E (0, (0) P{q(Y) ~ s} P{q(Y) Y2) ~ s, q (Yi v'2 + Y2) > t }.