We prove several results about the asymptotics of the distributions of nonnormalized CRPs Z(t) and Y(t). These results, known as integro-local theorems, are sharper than the central limit theorem and are concerned with the probabilities of Z(t) and Y(t) hitting intervals of small length in the normal deviation zone.

We continue the study of integro-local probabilities that was initiated in Chapter 2 in the normal deviation zone. Now, assuming that the vector (?, ?) satisfies the Cramér moment condition, we study the integro-local probability in a wider zone, which in analogy with random walks can be called the Cramér deviation zone. This zone includes the zones of normal, moderately large, and "usual" large deviations.

For the case where the jump distributions vary regularly at infinity (slow decay), for the sake of completeness we present without proof a number of results from A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks. Vol. I: Slowly Decaying Jump Distributions (in Russian; Moscow: Fizmatlit, 2008).

We obtain limit theorems describing sharp asymptotics in boundary crossing problems for CRPs.

We introduce the objects of study and outline the contents of the book.

We study the nature of the distributions of CRPs in the large deviation zone and establish the corresponding LDPs. We clarify the relation of the distribution of a CRP with the renewal measure for the sequence {Tn, Zn}. We investigate some properties of the deviation functions of the renewal measures that appear in this problem. We prove the LDPs for the CRPs Z(t). The definition of the fundamental function is given, and we also study its properties and relations to the deviation functions. We present several results on LDPs for the process Y(t) and for Markov additive processes.

We continue the study of large deviation principles for compound renewal processes. We are mostly concerned with probabilities of large deviations of the trajectories of CRPs. We establish "partial" LDPs (local LDPs applicable only in the space of absolutely continuous functions). "Complete" LDPs are obtained under rather restrictive conditions. It proves possible to obtain LDPs for boundary crossing problems under broader conditions and with explicit deviation functional. A moderately large deviation principle for CRPs is established.

We present the basic limit theorems for CRPs in the domain of normal deviations (with the functional limit theorems), including the case of infinite variance of the jumps of the process. We also present the law of the iterated logarithm and its analogs.

We extend the invariance principle for CRPs to the domain of moderately large and small deviations. The results in this chapter turn out to be new for random walks as well.