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We propose a discrete-time discrete-space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the proposed approximation with the Brownian bridge correction is $O(n^{-2})$ in the case of $C^2$ boundaries and a uniform time grid with n steps.
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 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.
Compound renewal processes (CRPs) are among the most ubiquitous models arising in applications of probability. At the same time, they are a natural generalization of random walks, the most well-studied classical objects in probability theory. This monograph, written for researchers and graduate students, presents the general asymptotic theory and generalizes many well-known results concerning random walks. The book contains the key limit theorems for CRPs, functional limit theorems, integro-local limit theorems, large and moderately large deviation principles for CRPs in the state space and in the space of trajectories, including large deviation principles in boundary crossing problems for CRPs, with an explicit form of the rate functionals, and an extension of the invariance principle for CRPs to the domain of moderately large and small deviations. Applications establish the key limit laws for Markov additive processes, including limit theorems in the domains of normal and large deviations.
Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.