Limit behaviour of convolution products of probability measures (Q5928183)

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.