Limit distribution functions for sums of the reciprocals of a power of tangent of random variables. (English) Zbl 1463.60022
Summary: Let \(r\in[\frac12,\infty)\) and \((X_n)\) be a sequence of independent continuous random variables such that \(\operatorname{Im}(X_n)\subseteq\mathbb{R}-\{\frac{j\pi}{2}|j\in\mathbb{Z}\}\). This paper provides the sufficient conditions guaranteeing the existence of real constants \((A_n)\), \((A_n(r))\) and \((B_n(r))\) such that the sequences of the distribution functions of \(\frac1n\sum\limits^n_{k=1}\frac1{\tan X_k}-A_n\) and \(\frac1{B_n(r)}\sum\limits^n_{k=1}\frac{1}{|\tan X_k|^r}-A_n(r)\) converge.
MSC:
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60F05 | Central limit and other weak theorems |
| 60G50 | Sums of independent random variables; random walks |
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