<正> §1.一个概率分布律叫作稳定的,如果它的特徵函数 f(t)适合函数方程式f(at)f(bt)=f(ct),(1)那就是说,对于任意给定的正数 a 同 b,常有正数 c 使(1)成立.稳定性分布律首先为保罗勒威(P.Lévy)所寻出.勒威解出方程式(1),其结果是

The stable law of distribution is one whose characteristic function isf(t)=e~(-(c_0-ic_1 sgn t)|t|~α) (0<α<2,c_0>0,|c_1|≤c_0|tan(?)|).(1)Without assuming that(1)represents a characteristic function,but preservingthe conditions α>0(excluding the easy case α=1)and c_0>0,we derive in thispaper the series expansions and asymptotic expansions of the following functions,corresponding to the“probability density function”and the“distribution function”:p(x)=(?)e(-itx)f(t)dt,F(x)=(?)p(y)dy.The summability of p(x),which no longer follows automatically as in theprobability-theoretic case,is proved in passing.Letc=(?),ω=tan(?) (|ω|<π/2)).The results about p(x)are(0 denoting any quantity with |0|<1):(i)0<α<1.p(x)=1/πsum from n=1 to ∞(?)sin n((?)+ωsgn x),(x≠0)(2)p(x)=(?)sum from n=0 to N(?)sin(n+1)(?)+ +(?)|x|~(N=1).(ii)α>1.p(x)=(?)sum from n=0 to ∞(?)sin(n+1)(?),p(x)=(?)sum from n=1 to N(?)sin n((?)+ωsgn x)++(?).(5)By an indirect method we obtain the formulaF(0)=(?).(6)Formulae (2),(3),(4),(5),(6)and F(∞)=1 are sufficient for the derivationof the series expansion and asymptotic expansion of F(x).The method used is elementary,including a contour integration and somelimit passages that can be readily dealt with.