| If 4 is a non-negative integrable function on R , then the behavior of the n-fold convolutions, 4n , as n → infinity is described by the Central Limit Theorem. However, the problem of describing the behavior of 4n rises when 4 is not a positive function.; Recently B. Baishanski has argued that in case ReA = 0, the "natural" scaling for 4n in complex-valued case should be the same as the essentially unique scaling of | 4n |/‖ 4n ‖ L1 . He has considered two examples and shown that under the natural scaling one obtains analogues of the Central Limit Theorem of a new kind when ReA = 0.; We have obtained more general results of the same nature. Our main theorem is:; Theorem. Suppose that 4 is a complex-valued function such that 4n∈ L1R ∩ LsR for some s>1 ; | 4&d14;t | < | 4&d14;0 | = 1 for every t ≠ 0; n24n ∈L1R ; ln 4&d14;t =ir=pqart r-btq+Gt , t∈U -- a neighborhood of zero, where p, q∈N , 2≤p<q , ap,ap+1, ..., aq,b are real, ap>0,b>0, G∈C2U , G″(t) = Otq-1 , G(t) = Otq+1 , t → 0. Let the scaled n-fold convolutions be given by lyl c , where ylc =sn4n snlc ,sn=n 1q,l =n1-pq . Then lyl will exhibit one of two types of very regular divergence: (1) If p is even, then l4l c=Fcu lc+rl c , where F(c) is an integrable function given by Fc=B |c|--gamma exp {lcub}-- E|c|rho{rcub}, (1); ulc =1 and | rlc | ≤Cle ∣c∣ for |c| <ln2l, l>L (2); (2) If p is odd, then l | ylc | =Fctl c+rl c , where Fc=&cubl0; 0,ifc<0 2F c,if c>0 , F(c) being defined by (1), 0≤tlc ≤1 and rlc satisfies (2). (Abstract shortened by UMI.)... |
