On a Uniqueness Property of Second Convolutions

converges uniformly on compact subsets of the upper half-plane C+ := {z ∈ C : Im z > 0} to a function analytic in C+. Let l(μ) := inf suppμ denote the left boundary of the support of μ, and μn∗ the nth convolution power of μ. The following uniqueness property of nth convolutions of measures from M∞ was discovered in connection with some probabilistic results (see for example [1], [7], [8], [9], [10] and the literature therein): Let n ≥ 3 be an integer, and let μ ∈ M∞ be such that l(μ) = −∞. Then every half-line (−∞, a), a ∈ R, is a uniqueness set for the nth convolution μn∗, in the sense that the implication holds: Suppose ν ∈ M∞ and (2) there exists a ∈ R such that μn∗|(−∞,a) = νn∗|(−∞,a). Then μn∗ = νn∗. It is also known that property (2) does not hold for n = 2. An easy way to check this is to take two measures ξ1, ξ2 ∈ M∞ such that l(ξ1 + ξ2) = −∞ and ξ1 ∗ ξ2 = 0 on some half-line (−∞, a). Then the measures μ = ξ1 + ξ2 and ν = ξ1 − ξ2 belong to M∞, l(μ) = −∞ and we have (μ2∗ − ν2∗)|(−∞,a) = 4ξ1 ∗ ξ2|(−∞,a) = 0.