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A local limit theorem on the semi-direct product of \(\mathbb{R}^{*+}\) and \(\mathbb{R}^ d\). (English) Zbl 0881.60018
Fix a norm \(|\cdot |\) on \(\mathbb{R}^d\) \((d\geq 1)\) and consider the connected group \(G\) of transformations \(g:\mathbb{R}^d\to\mathbb{R}^d\), \(x\mapsto g.x=a x+b\), where \((a,b) \in \mathbb{R}^{*+} \times \mathbb{R}^d\). Let \(a\) (resp. \(b)\) be the projection from \(G\) on \(\mathbb{R}^{*+}\) (resp. on \(\mathbb{R}^d)\). Let \(\mu\) be a probability measure on \(G\), \(\mu^{*n}\) its \(n\)-th power of convolution, \(\widetilde \mu\) the image of \(\mu\) by the map \(g= (a,b) \mapsto (1/a,1/b)\) and \(\overline \mu\) the image of \(\mu\) by the map \(g\mapsto g^{-1}\). The paper proves under suitable hypotheses that \(\mu\) satisfies a local limit theorem: there exists a sequence \((\alpha_n)_{n \geq 0}\) of positive real numbers, depending only on the group when \(\mu\) is centered, such that the sequence \((\alpha_n \mu^{*n})_{n\geq 0}\) converges weakly to a non-degenerate measure.

MSC:
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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