Abstract
In this paper, we study the large deviation behavior of sums \(S_{{Z}_{n}}\) of i.i.d. random variables X i , where Z n is the nth generation of a supercritical Galton–Watson process. We assume the finiteness of the moments \(EX_{1}^{2}\) and EZ 1 logZ 1 . The underlying interplay of large deviation probabilities of partial sums of the X i and of lower deviation probabilities of Z is clarified. Here, we heavily use lower deviation probability results on Z we recently published in [7].
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Supported by the German Science Foundation.
This paper has been written during the time the second author was a staff member of the WIAS Berlin.
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Fleischmann, K., Wachtel, V. Large deviations for sums indexed by the generations of a Galton–Watson process. Probab. Theory Relat. Fields 141, 445–470 (2008). https://doi.org/10.1007/s00440-007-0090-1
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DOI: https://doi.org/10.1007/s00440-007-0090-1
Keywords
- Large deviation probabilities
- Supercritical Galton–Watson processes
- Lower deviation probabilities
- Schröder case
- Böttcher case
- Lotka–Nagaev estimator