Geodesic
Limit theorems for small deviation probabilities of some iterated stochastic processes
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 218-232

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We derive logarithmic asymptotics for probabilities of small deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those of the processes which generate the iterated processes. When these conditions do not hold, the asymptotics of small deviations for iterated processes are quite different. We apply our results to the iterated processes generated by the compound Cox processes and the compound renewal processes. 
@article{ZNSL_2011_396_a15,
     author = {A. N. Frolov},
     title = {Limit theorems for small deviation probabilities of some iterated stochastic processes},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {218--232},
     publisher = {mathdoc},
     volume = {396},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a15/}
}
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A. N. Frolov. Limit theorems for small deviation probabilities of some iterated stochastic processes. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 218-232. http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a15/