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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
##### Keywords:
local limit theorem
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