ON THE DENSITY OF THE DISTRIBUTION OF p-STABLE

Let p. be a symmetric p-stable measure, 0 < p < 1, on a locally convex separable linear metric space E and let q be a lower semicontinuous seminorm on E. It is known that F(t) = u{x : q(x) < t) is absolutely continuous with respect to the Lebesgue measure. We prove an explicit formula for the density F'(t) and give an asymptotic estimate of it at infinity. 1. Let X be a symmetric Gaussian random element with values in a locally convex metric linear space E with a measurable seminorm q. Then the distribution function F(t) — P{q(X) < t} is absolutely continuous apart from one possible jump [3, 4]. In these papers some properties of the density F'(t) were also investigated. Recently it has been shown [2, 6] that if X is a symmetric p-stable random element with values in a linear metric separable space E and q is a lower semicontinuous seminorm on E such that P{q(X) < oo} = 1, then the distribution function F(t) = P{q(X) < t} is absolutely continuous (apart from one possible jump for 1 0 and is continuous. The asymptotic behavior of F'(t) is also given. 2. In this paper E will stand for a locally convex separable linear metric space. By q we denote a measurable seminorm on E, i.e. a measurable function q: E —> [0, oo] such that q(x + y) < q(x) + q(y) and q(ax) = \a\q(x) for all x, y G E and a G R. We will always assume in the sequel that q is lower semicontinuous. A probability measure p on E is called strictly p-stable if for every two independent random vectors X and Y with distributions p and for every a,ß > 0, ap + ßp = 1, the distribution aX + ßY is equal to p. We will assume in addition that p is symmetric. If p is a symmetric p-stable measure on E, then there exists a symmetric <7-finite measure v on E, v(Uc) < oo for every open neighborhood U of the orgin, and such that p — lim exp(i/|<7£) for Un \ {0}. The measure v is called the Levy measure of p and has the following property: v(sA) = s~vu(A) for every Borel set A. If q < oo p-a.s., then there exists a finite symmetric measure o on Si = {x: q(x) = 1} such that if r(x) = q(x) and s(x) = x/q(x) then (1) C C°° fir ^\{q>e}{x:q(x + y)GA} = J I lA(q(rs + y))^j^a (ds) Received by the editors May 20, 1985 and, in revised form, April 2, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 60B05; Secondary 60E07.