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A Functional Equation from Probability Theory

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A Functional Equation from Probability Theory proceedings of the american mathematical society Volume 121, Number 3, July 1994 A FUNCTIONAL EQUATION FROM PROBABILITYTHEORY JOHN A. BAKER (Communicated by J. Marshall Ash) Abstract. The functional equation IV (1) f(x) = T\[f(ßix)yi 7=1 has been used by Laha and Lukacs (Aequationes Math. 16 (1977), 259-274) to characterize normal distributions. The aim of the present paper is to study (1) under somewhat different assumptions than those assumed by Laha and Lukacs by using techniques which, in the author's opinion, are simpler than those employed by the afore-mentioned authors. We will prove, for example, that if 0 < ßj < 1 and y¡ > 0 for 1 < j < N, £* , ßfyj = 1 , where it is a natural number, /: R -» [0, +co), (1) holds for x € R and /^'(O) exists then either / = 0 or there exists a real constant c such that f(x) = e\p(cxk) for all x e R. Introduction In [3, p. 237] we find the following theorem attributed to Vincze [6]. Theorem A. Suppose a, a, b>0, a2 + b2 = 1, çjgR-+C, (2) cp(x) = o\/2ñcp(ax)ip(bx) forallx£R, cp"(0) exists, and ¡&<p(x)dx = 1. Then (3) cp(x) = (l/crv/27t)exp(-x2/fT2) forallx£R. Notice that if we let f(x) = oV2ñ<p(x) for x G R then (2) is equivalent to (1) with N = 2, ßx = a, ß2 = b, yx = y2 = 1 and, in this instance, ß2x7i + ß\y2 = 1 - Suppose f:R -» R, f is nondecreasing and continuous on the right, limr__00 F(f) - 0, and lim,_+00 F(t) = 1. If /oo eixtdF(t) forxG -oo then / is called the characteristic function of the distribution function F (see [5]). Thus a characteristic function / is, by definition, the Fourier-Stieltjes Received by the editors June 23, 1992 and, in revised form, October 9, 1992. 1991 Mathematics Subject Classification.Primary 39B12, 60E10; Secondary 39B22. Key words and phrases. Functional equation, probability. ©1994 American Mathematical Society 0002-9939/94 $1.00+ $.25 per page 767 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 768 J. A. BAKER transform of a distribution function and therefore, as is well known, / is con- tinuous, /(0) = 1 and f is positive definite in the sense of Bochner. In [5], Laha and Lukacs proved Theorem B. If 0 < ßj < I and y} > 0 for I <j <N, £*=1 ßfy = I, f is a characteristic function, and (1) holds for all x £ R then f is the characteristic function of a normal distribution with zero mean. The conclusion is equivalent to asserting that f = cp where <p is defined by (3) for some a > 0. Theorem A is proved in [3] by using general results concerning certain classes of functional equations. Theorem B is deduced in [5] from a part of the theory of characteristic functions involving the Fourier transform. Our results will be based on the comparatively simple propositions of the next section. Throughout this paper N is a given natural number and ßx, ... , ß^, yi,..., yN are given real numbers such that 0 < ß} < 1 and y¡ > 0 for 1 < j < N. We also let (*) Pi = -Inßj > 0 for 1 < j < N. If we let p(s) = 53 _j ßpj for s £ R then p is continuous, strictly decreas- ing, limJ_+00 p(s) = 0, and limJ_(_0o p(s) — +oo . Hence there is a unique real number k such that (#) E^ =i- 7=1 Let Pi = ßfyj > 0 for 1 < j < N so that £?=i p}■=1. Notice that k = 2 in both of the above theorems. We will be mainly con- cerned with cases in which k is a natural number and / satisfies some regularity condition at (or near) zero. We denote the natural numbers by N, the integers by Z, the real numbers by R, and the complex numbers by C. Although we will eventually consider cases of ( 1) for functions / from R to R (or C or even a Banach algebra), we will begin by concentrating on functions / from (0, +oo) to (0, +00) ; the main ideas of our proofs are more easily understood in this setting. TWO BASIC PROPOSITIONS Proposition 1. Suppose / : (0, +00) -+ (0, +00). Then (1) holds for all x > 0 if and only if there exists a function cp: R —»R such that f(x) = exp[xV(lnx)] far all x > 0, i.e., <p(t)= e~kllnf(e') for allt£R, and N (5) cp(t)= YJPi<P(t-pj) forallt£R. 7=1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A FUNCTIONALEQUATION FROM PROBABILITYTHEORY 769 Proof. Suppose (1) holds for all x > 0 and let y/(t) = lnf(e') for t £ R. Then,by (1), y/(t)= ^{Y^Mißj^W) = E?=, y,-In/(*"»*'), i.e., N (6) yr(t) = Y, 7j¥(t -Pj) for all f e R. 7=1 For s £ R let y/s(t) = e~sty/(t) for t £ R. By (6) and (*), N N Kit) = e~stY YjV(t- Pj) = £ rje-We-l'-M wit - pj) 7=1 7=1 N = Y ßj?j Vs(t - Pj) for 5, t £ R. 7=1 If we let cp = y/k then we have (5) and i//(t) = ekty/k(t) = (e')ktp(t) for all ÍGR. Thus f(x) = exp[(/(lnx)] = exp[xktp(lnx)] for all x > 0, i.e., (4) holds. The converse involves only a simple computation. The next result is a generalization of the simple fact that if cp: R -+ R, cp is periodic, and lim,-,.^ cp(t) exists, then cp is constant. Proposition 2. Suppose cp: R —►R (or any normal linear space), (5) holds, and lim,-.-,*, cp(t) exists. Then cp is constant. Proof. Suppose <p(t) —»/ G R as f -> -oo. Given e > 0, choose a £ R such that \cp(t)-l\<e whenever t < a. Let p = min{p!, ... , pN} > 0. If t < a+p then t - Pj < a for 1 < j < N so that \cp(t- pj) - l\ < e . Hence if t < a+ p then, by (#) and the definition of pj, N N \<P(t)~l\ = YM<p(t-Pj)-i} 7 = 1 7 = 1 By induction, \<p(t)- l\ < s provided t < a + np for some « G N. It follows that \cpit)-l\<e for all? G R. Since this is so for every e > 0 we must conclude that cpit) = I for all t G R. D Proposition 3. Suppose /: (0, +oo) —>[0, +00), (1) holds for all x > 0, ^2f=x y¡ > 1 (equivalently, k > 0), and limx_o+ fix) exists. Then either f = 0 or fix) > 0 for all x > 0. Moreover assuming / ==0, // fe = 0 i«tf« / is constant and if k > 0 then limx_>0+fix) = 1. Proof. Let c = limJC_0+fix). By (1), c = c» where y = £?, y¡ > 1. Suppose first that c = 0. Let 0 < e < 1 and choose S > 0 such that 0 < fix) < e whenever 0 < x < S. Let ß = max{/3i, ß2,... , ßN} so that 0 < /? < 1. If 0 < x < S/ß then 0 < ßjx <ßx<6 for 1 < j < N and hence N 0 < /(x) = HtfißjX)]* <ey<e 7=1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 770 J. A. BAKER since 0 < e < 1 and y = Y,f=i 7j > ! • That is 0 < f(x) < e whenever 0 < x < o/ß . By induction, for any natural number « , 0 < f(x) < e whenever 0 < x < S/ß" from which it follows that 0 < f(x) < e for all x > 0. Since this is so for any e G (0, 1) we conclude that / = 0 if c = 0. Suppose next that c > 0 and y = ¿~^J=ly¡ = 1 (i.e., k = 0). Since limJC_o+/(•*) > 0, it follows from (1) that f(x) > 0 for all x > 0. Let y/(t) = lnf(e') for í G R so that, by (6), N y/(t) = 'V\yjV/(t~Pj) for all í G R and lim y/(t) = Inc =: d. 7=1 By Proposition 2, ^ is constant and hence / is constant. Suppose finally that c > 0 and 3D/=i7j > I >i-e-> fc > 0. To complete the proof it suffices to show that f(x) > 0 for all x > 0 and c = 1. But c = c5", c> 0, and y = £/Li y, > 1, so c = 1. Hence /is positive in (0, ô) for some r5 > 0 and it follows from (1) that f must be positive on (0, +00). D Corollary. Suppose that f: [0, +00) -» [0, +00) (or f: R -» [0, +00)), (1) holds for all x > 0, (res1/?,all x eR), k = 0, and f is continuous at 0. 7«e« / is constant. Proof. Proposition 3 implies that / is constant on (0, +00) and hence / is constant on [0, +00) so the proof is complete if the domain of / is [0, +00). Suppose /: R —»[0, +00). Let g(x) = f(-x) for all x > 0. Then N N g(x)= f(-x) = Y[[f(ßj(-xW= ll[g(ßjX)Y> 7=1 7=1 for all x > 0 and limx_0+ ¿?(-*)= limx_0 f(x) ■ Hence by Proposition 3, g is constant on (0, +00). Thus / is constant on (-00, 0). Using the continuity of / at 0 once again we conclude the / is constant on R. D The main RESULTS Theorem 1. Suppose f: R -» [0, +00), (1) holds for all x £ R, k £ N, and flk\0) exists.
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