On the strong law of large numbers and additive functions (Q653796)
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scientific article; zbMATH DE number 5990580
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the strong law of large numbers and additive functions |
scientific article; zbMATH DE number 5990580 |
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On the strong law of large numbers and additive functions (English)
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19 December 2011
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Let \(f\) be a strongly additive complex-valued arithmetic function satisfying some mild conditions. It is proved that if \(X,X_{1},X_{2},\dots \) is any sequence of integrable i.i.d.\ random variables, then \[ \lim_{N\rightarrow \infty } \frac{\sum_{n=1}^{N}f(n) X_{n}}{\sum_{n=1}^{N}f(n)}=\mathbb{E}X\quad \text{a.s.} \]
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strong law of large numbers
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weighted i.i.d.\ sums
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strongly additive functions
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