Limit theorems for multidimensional renewal sets
| Autor: | Andrii Ilienko, Ilya Molchanov |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
General Mathematics Probability (math.PR) 010102 general mathematics Excursion Law of the iterated logarithm Positive expectation 16. Peace & justice 01 natural sciences 60F15 60K05 Set (abstract data type) 010104 statistics & probability 510 Mathematics Law of large numbers FOS: Mathematics Limit (mathematics) 0101 mathematics Random variable Mathematics - Probability Integer grid Mathematics |
| Zdroj: | Ilienko, Andrii; Molchanov, Ilya (2018). Limit theorems for multidimensional renewal sets. Acta mathematica hungarica, 156(1), pp. 56-81. Springer 10.1007/s10474-018-0806-y |
| Popis: | Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d.\ random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional limit theorem for random sets ${\mathcal M}_t$ that appear as inversion of the multiple sums, that is, as the set of all arguments $x\in{\mathbb R}_+^d$ such that the interpolated multiple sum $S_x$ exceeds $t$. The moment conditions are identical to those imposed in the almost sure limit theorems for multiple sums. The results are expressed in terms of set inclusions and using distances between sets. 24 pages. The results are extended to the lower limit in the law of the iterated logarithm |
| Databáze: | OpenAIRE |
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