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Limit behaviour of convolution products of probability measures. (English) Zbl 0987.60010
Denote by $${\mathbb R}^{d}$$ the $$d$$-dimensional Euclidean space with norm $$|\cdot|$$ and by $$B(a,r)$$ the open ball in $${\mathbb R}^{d}$$ with center $$a\in{\mathbb R}^{d}$$ and radius $$r>0$$. Given a probability measure $$\mu$$ on $${\mathbb R}^{d}$$ such that $$c(\mu)=\int_{{\mathbb R}^{d}}xd\mu(x)<\infty$$, put $$\|\mu\|_{2}=\left(\int_{{\mathbb R}^{d}}|x-c(\mu)|^{2} d\mu(x)\right)^{1/2}$$. For a probability measure $$\mu$$ on $${\mathbb R}^{d}$$ such that $$0<\|\mu\|_{2}<\infty$$ and for $$\alpha>0$$ define $V_\alpha(\mu)=\frac{1}{ \|\mu\|^{2}_{2}} \int_{B(c(\mu),\alpha\|\mu\|_{2})}|x-c(\mu)|^{2} d\mu(x).$ The authors show that the following dichotomy concerning limit behaviour of convolution products of measures occurs for a large class of probability measures on $${\mathbb R}^{d}$$.
Theorem 1 (The convergent case). Let $$(\mu_{n})$$ be a sequence of probability measures on $${\mathbb R}^{d}$$, $$\sum_{n=1}^{\infty} \|\mu_{n}\|_{2}^{2}<\infty$$, $$v_{n}=\sum_{j=1}^{n} c(\mu_{j})$$, $$v\in{\mathbb R}^{d}$$, and let $$(k_{n})$$ be a strictly increasing sequence of natural numbers. If $$\lim_{n\to\infty} |v_{k_{n}}|=\infty$$, then the sequence $$(\mu_{1}*\cdots*\mu_{k_{n}})$$ converges vaguely to zero. If $$\lim_{n\to\infty}v_{k_{n}}=v$$, then the sequence $$(\mu_{1}*\cdots*\mu_{k_{n}})$$ converges weakly to a probability measure $$\nu$$ on $${\mathbb R}^{d}$$ with $$c(\nu)=v$$.
Theorem 2 (The divergent case). Let $$(\mu_{n})$$ be a sequence of probability measures on $${\mathbb R}^{d}$$ such that $$0<\|\mu_{n}\|_{2}<\infty$$, $$\sum_{n=1}^{\infty} \|\mu_{n}\|_{2}^{2}=\infty$$ and $$\sup_{\alpha>0}\liminf_{n\to\infty}V_{\alpha}(\mu_{n})=1$$. Then $$(\mu_{n})$$ is dissipative in the sense that $$\lim_{n\to\infty}\|f*\mu_{1}*\cdots*\mu_{n}\|_{\infty}=0$$ whenever $$f$$ is a continuous real function on $${\mathbb R}^{d}$$ which vanishes at infinity. In particular, the sequence $$(\mu_{1}*\cdots*\mu_{n})$$ converges vaguely to zero.
MSC:
 60B10 Convergence of probability measures 28A33 Spaces of measures, convergence of measures
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