STABILITY ESTIMATES IN THE THEOREM OF J. MARCINKIEWICZ L. B. Golinskii 1. Introduction and statement of the main results A weU-known theorem of J. Marcinkiewicz asserts that a function ~0(t) = exp{Q(t)}, where Q(t) = n E aktk is a polynomial, is the characteristic function (c.f.) of a probability distribution F only when F is a k=l (possib!y degenerate) normal distribution, in other words, when al = -~, a2 = a2 ~ 0, a3 ..... an = 0. This theorem has found application in a wide variety of characterization problems of mathematical statistics [9]. In studying the stability of such problems the question of the stability in Marcinkiewicz' theorem itself arises: what can be said about the coefficients of the polynomial Q(t) if the function ~0(t) is in some sense close to a c.f.? This problem was posed by N. A. Sapogov [1, 2] as follows: suppose ~(t, F) is the c.f. of some distribution F and l exp{Q(t)} - ~(t,F)l < c, It[ < T. (1.1) Regarding c and T -1 as sufficiently small, estimate the distance from the polynomial to the set of quadratic polynomials of the form iat - 7t:; a = a, 7 k 0. Sapogov [2] obtained an estimate for the leading coefficient an for polynomials Q(t) of a special form and described a procedure that makes it possible to estimate the subsequent coefficients inductively. Zinger and Yanushkyavichyus [3, 4] have studied the stability of a characterization theorem of Yu. V. Linnik that leads to the stability problem (1.1) in the theorem of Marcinkiewicz with T = In ~- . Using the procedure of [2], they obtained in [3] an estimate for the coeffieients ak: max lakl < 3<k<n - - n where the constants C and q are positive and depend only on n. In the present article we obtain an estimate that is in a sense nearly optimal for the coefficients ak of a polynomial Q(t) of general form in the ease T = In , fl > 0 being a fixed number. Since the inequality (1.1) does not imply that the quantities vl = Imal and U 2 = Rea 2 are small, we shall assume that the following normalizations hold: 1 731 : Imal : 0, u2 = Rea2 : -5" (1.2) The basic results of this article are the following two theorems. n Theorem 1. Let Q(t) = Eaktk be a complex polynomial normalized by the conditions (1.2), ~(t,F) k----1 the c.s os some distribution F, and ( [exp{Q(t)}-~(t,F)[~r Itl__T= ln~ , (1.3) Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 8-24, 1988. 0090-4104/91/4001-3193512.50 Q1991 Plenum Publishing Corporation 3193