On Unimodality and Rates of Convergence for Stable Laws

It is known that in the case of convergence to a normal law, the uniform rate of convergence is completely determined by the rate of convergence at points near the origin, up to terms of order n~; see [6, pp. 12, 46]. In this paper we show that results of this nature hold true for convergence to non-normal stable laws, provided the centering point is taken as the mode of the limit distribution. We concentrate on the case of a symmetric limit, which is the closest analogue of the case of normal convergence. We must stress at the outset that there are essential differences between convergence to normality and convergence to stable laws. It is shown in [7] that in certain pathological cases the rate of convergence on any compact set is faster than the uniform rate of convergence. This behaviour occurs when the tail sum function S behaves erratically towards infinity. However, it is to be hoped that if we impose regularity conditions to exclude such pathologies, then the rate of convergence on compact sets will convey considerable information about the uniform rate of convergence. We shall show in Section 3 that this is indeed the case. Very little is known about the mode of asymmetric stable laws. Indeed, it was comparatively recently [15, 20] that all stable laws were shown to be unimodal. In Section 2 we provide asymptotic formulae for the unique mode of a stable law when the law is close to being symmetric. We conjecture that not only are all stable laws unimodal, but they are also perfectly skew, in the sense that p(m — y) — p(m + y) is of the one sign for all y > 0, where p is an asymmetric stable density with mode m. Tables of asymmetric stable densities appear to support this proposition. If it is true, then the results in Section 3 can be partially extended to the asymmetric case. We close this section with notation for Section 3. Only the case of a symmetic limit is treated, although we permit the summand distribution to be asymmetric. Let Xi,X2,... be independent and identically distributed random variables in the domain of normal attraction of a stable law with exponent a, scaled and located so n that the normed sum, Sn = n~ X X}, has limit distribution function G with