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 ¡&

/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.

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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 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.,

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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 -oo. Given e > 0, choose a £ R such that \cp(t)-l\ 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 \ 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)]*

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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. Then either f = 0 or there exists eel such that f(x) = exp[cxk] for all x £ R. Proof. Since f'(0) exists, / is continuous at 0. By Proposition 3 (applied to the restriction of / to (0, +00)), either f(x) = 0 for all x > 0 or /(0) = 1 and f(x) > 0 for all x > 0. Assume the latter. Let F(x) = lnf(x) for x G R. Choose cp: R -» R such that (4) and (5) hold. Then f(x) = xk 0, F(0) = 0, and F'(0) - /'(0)//(0) = /'(0). Suppose k = 1. Let c = F'(0) = lim F(x) ~ f(0) = lim 0+ X x-»0+ i-»-oo By Proposition 2, cp(t) —c for all t eR. Hence F(x) - ex for all x > 0 and thus f(x) = exp[cx] for all x > 0. The continuity of / at 0 thus implies that

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f(x) = exp[cx] for all x > 0. Let g(x) = f(-x) for x G R. Then N N g(x)= fi-x) = Hm(-x))7i = llg(ßjxy> 7=1 7=1 for all x G R and g'(0) = -f'(0) = -c. Hence g(x) = e~cx for all x > 0. That is f(x) —ecx for all x < 0. Thus our assertion is true in the case k = 1. Now suppose that k > 2. By assumption, there exists A > 0 such that fik~x\x) is defined for all x G (-A, A). Hence ç>(fc_1)(i)is defined for all t £ (-oo, In A) and hence, by (5), for all (el. It then follows from (4) that flk-V(x) and F<*-0(x) are defined for all x > 0. Now for all x > 0 Fix) - xktpilnx) and F'(x) = kxk~l 0. Similarly, if k > 3 we have, for all x > 0, 7"'(x) = xk~2i[D + k- l][D + k]

F^k-X\x) = x([D - 2] • • • [D + k]0. Now F(x) = lnf(x) = £jl, yjlnf(ßjx) = ¿$mí yjF(ßjX) for all x G R so that N F'(x) = Y, ßjYjF'(ßjx) for all x G R. 7=1

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 772 J. A. BAKER Hence F'(0) = {£*, ßj7j}F'(fl). But £?=1ßy; > T$.iß}n = 1 ñu* k > 2 so F'(0) = 0. If k > 3 then N F"(x) = J] ß2yjF"(ßjX) for all x G R 7=1 so that F"(0) = 0 since £? i ßJYj > EjLi $7./ = ! • Continuing this argu- ment we find that 0 = F^(0) for m = 1, 2, ... , k - I. But F^(0) exists since f{k)(0) exists and we know that F^k-X\0) = 0. Hence, by (8), F^(0) = lim ^"'H^-T-^-'HO) = lim v{lnx) x-»0+ X x->0+ By (7) and Proposition 2, ^ is constant, say if/(t) = X £ R for all í G R. That is, for all t G R, f ífc-!)(í) + ak_2cp«-V(t) + ... + axcpx(t) + (k\)cp(t) = X. Hence there exist real constants c2, ... , ck such that cp(t) = c2e'2t + ■■■+ cke~ki + (X/k\) for all t e R. Thus Fix) = xV(lnx) = xk -4c2 + --- + -TCk + X1 XK a: X xk + ck -I-h c2xfc~2 for all x > 0.

Since 0 - 7"(m)(0) for m = 1, 2, ... , k - 1, we have c2 = ■■■= ck -0. Thus, if c = X/k\, then F(x) = cxk for all x > 0 and hence, for all x > 0, (9) /(x) = exp[cx*]. Arguing as we did in the case k = 1 we find that (9) holds for all real x . D

Generalizations If S is a (multiplicative) semigroup and yx, ... , yjv G N then one can make sense of (1) for functions / from R (or (0, +oo)) into S. Suppose that A is a (real or complex) commutative Banach algebra with identity 1. For a e A define CO exp(a) = £(l/W.

As is well known, exp is a C°° map of A into A , exp(0) = 1, exp'(0) is the identity map of A onto itself, and exp(a + b) = (expa)(expb) whenever a, b e A. By the Inverse Function Theorem (see, e.g., [7, p. 172]) there exists a neighbourhood U of 0 in A and a neighbourhood V of 1 in A such that the restriction of exp to f7 is a bijection of U onto V whose inverse, call it L, is C°°. Suppose yj G N for 1 < j < N and let y = £>=i y¡ . Choose a neighbour- hood W of 1 in A such that wxw2 ---Wy e V whenever wx, w2, ... , wy e W. It follows that L(tüi • • • w7) = Y7j=xL(wj) for all wx, ... ,wyeW. With this machinery we can generalize some of the above results as follows.

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Theorem 2. With the above assumptions, suppose f:R->A satisfies (I) for all x e R. If k £ N, /(/c) = 0 exists, and f(0) is invertible then there exist a, c £ A such that a = ay and f(x) - a expfx^c] for all x £ R. Sketch of proof. First note that if a = /(0) then, by (1), a = a1. If we let fx(x) = a~lf(x) for x G R, then /,(0) = 1, fax) = a~xnJLil/GW = \~["=x[a-xf(ßjX)]y>, and /W(0) exists. Thus we may assume that f(0) = 1. The existence of fxk\0) implies the continuity of / at 0. This, in turn, in light of (1), implies that f(x) is invertible for all x > 0. Moreover there exists A > 0 such that f(x) £ W whenever -A < x < A. Let cp(t) = e~ktL[f(e')] for -oo < t < In A. Then cp(t) = Y$=i ßkYj 0 for 1 < j < N, and Y,f=l yj = y < M for some M £ N. Let W = {w £ V : pL(w) £ U for all p £ (0, M)} and define wß = exp[pL(w)] for w £ W and p £ (0, M). Suppose A > 0, /: (-A, A) -» W , (1) holds for all x G (-A, A), and /(0) = 1. Then by slightly modifying the above ideas one can prove that if k £ N and /(fc)(0) exists then there exists c £ A such that f(x) = exp[xkc] for all xg(-A,A). 2. In Theorem 2 it is not necessary to assume that A is commutative. Instead one could assume that f(x)f(y) - f(y)f(x) for all x, y £ R and replace A by the closed subalgebra of A generated by {/(x) : x G R} U {1} . For if this subalgebra is denoted by B then / maps R into B and B is commutative. 3. Our results were obtained by reducing (1) to (6). Equation (6) has also been studied under a variety of conditions different from those we have assumed (see, for example, [1,2, 4]).

References

1. John A. Baker, Functional equations, tempered distributions and Fourier transforms, Trans. Amer. Math. Soc. 315 (1989), 57-68. 2. W. Jarczyk, A recurrent method of solving iterative functional equations, Uniwersytet Slaski, Katowice, 1991. 3. M. Kuczma, B. Choczewski, and R. Ger, Iterative functional equations, Cambridge Univ. Press, London and New York, 1990. 4. M. Laczkovich, Non-negative measurable solutions of a difference equation, J. London Math. Soc. (2) 34 (1986), 139-147. 5. R. G. Laha and E. Lukacs, On a functional equation which occurs in a characterization problem, Aequationes Math. 16 (1977), 259-274. 6. E. Vincze, Bemerkung zur Charakterisierung der Gauss'sehen Fehlergesetzes, Magyar Tud. Acad. Mat. Kutató Int. Kösl. 7 (1962), 357-61. 7. E. Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, 1986.

Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

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