Renewal Theory And Tail Asymptotic Theory Of Random Walks

It is well known that random walk theory, including the renewaltheory, tail asymptotic theory of random walks and related objects, hasalways been the important part of probability theory. It has not only im-portant theoretical value, but also important applications in many fieldssuch as queuing system, risk theory, branching process, etc. For exam-ple, in risk theory, we may describe the ruin probability of an insurancecompany through the probability of the supremum of a random walkexceeds the initial capital. So, random walk theory also has practicalsignificance in the finance work.This paper will investigate random walk theory from the followingthree aspects.The elementary renewal theorems of renewal counting processes,more generally, of counting processes generated by random walks, areone of the important parts of renewal theory. For the case where theincrements are independent and identically distributed, the classical re-newal theory has been mature. However, in practice, related randomvariables are usually not independent to each other. So, for the countingprocesses generated by random walks with some dependent increments,is it possible to establish corresponding results? For a long time, therehas not been any substantial progress in this field. In Chapter2, wewill discuss these problems systematically. We will deliver elementaryrenewal theorems of counting processes generated by random walks withwidely dependent increments. In proving them, we will first establisha weak law of large numbers for random walks with widely dependentincrements. Under some further conditions, a strong law of large num-bers for the same object will also be established. Besides, the obtainedresults will be applied to presenting an asymptotic estimate for precise large deviation of a random sum, where the random index is a nonstan-dard renewal counting process with widely dependent increments. Atthe end of this chapter, we will discuss a strong law of large numbersfor the first passage time process of random walks with a tend. Theobtained results will extend and improve the existing results to a greatextent.On the other hand, tail asymptotic theory of random walks hasclose relation with distribution theory. An important fact of distribu-tion theory, in particular, heavy-tailed distribution theory is that, whenthe increments of a random walk are independent and have commonsubexponential distribution, these increments are max-sum equivalent,which means that the tail probability of the sum of some random vari-ables is equivalent to that of the maximum of these random variables. Inrisk theory, this property may be interpreted as the ruin of an insurancecompany is usually caused by a big claim. Hence this property attractsmuch attention. In the study, the distributions of the increments areusually assumed to be in some specific distribution class. However, thiscondition seems too strict. Li and Tang (2010)[1]noticed this problemand they imposed the corresponding condition on the distribution of themaximum of the increments, thus expanding the scope of application.However, as usual, they assumed that the distribution of the maximumof the increments were subexponential. Since there exist many distribu-tions are long-tailed and O-subexponential, but not subexponential, sowe will in Chapter3extend the above-mentioned investigation to the in-tersection of the long-tailed distribution class and the O-subexponentialdistribution class. It is worth mentioning that when the distributionof the maximum of the increments is subexponential, the oscillationcaused by the sum of the increments is the smallest. In other words,when the distribution of the maximum of the increments is long-tailedand O-subexponential, but not subexponential, there may appear bigoscillation or risk. Thus, it is necessary to investigate related theory of random walks with the distribution of the maximum of the incrementsis long-tailed O-subexponential.At last, in Chapter4, we will apply the random walk theory to aclass of non-standard discrete-time risk models, which have both financeand insurance risks. We will present asymptotic estimates for finite-timeruin probabilities in this kind of risk models, where the distributions ofthe insurance risk or the product of the two risks may not belong tothe convolution equivalence distribution class. In doing so, we will firststudy the properties of a larger distribution class than the convolutionequivalence distribution class, namely the exponential distribution class.We will derive some equivalent conditions and some sufcient conditionseasy to check for closure property under convolution and tail equivalenceof distributions from the exponential distribution class. Since the pastwork presents only sufcient results, so our work generalizes some exist-ing sufcient conditions. Besides, in the study of tail equivalence, it iscommon to assume that the related distributions belong to the convo-lution equivalence distribution class. Our results show that the abovecondition is not necessary, thus expanding its application and enrich-ing the corresponding distribution theory and tail asymptotic theory ofrandom walks substantially.