On asymptotic behaviour of probabilities of large and moderate deviations for some iterated stochastic processes

We derive logarithmic asymptotics for probabilities of large deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables. When these conditions do not hold, the asymptotics of large deviations for the iterated processes are quite different. When the iterated process is a homogeneous process with independent increments, in which time is replaced by a random one, the behaviour of large and moderate deviations are investigated in the case of finite variance. For this case, the following one-sided moment restriction are considered: the Cramèr condition, the Linnik condition, the existence of moment of order $p>2$ for a positive part. Bibl. – 6 titles.