Some Problems About Record Numbers Of Random Walk

In this paper,we study record numbers of the left continuous integer-valued random walks.A record at some moment means that the value of the random walk at this moment is bigger than those before,and record numbers is to count the numbers of record by some moment.Evidently,record number is a random variable about renewal process,also we can regard it as occupying time in markov chain.The first part of the thesis is to find the large deviation and the moderate deviation of the record number.First,we transform the problem into the Markov chain occupancy problem,and then use a very clever method to obtain the characteristic function of the interval time,and then use the Cramér theorem in large deviation theorem to derives the large deviation of the record number;for the meoderate deviation problem,we need to use Tauberian theorem,and then use the Markov chain Occupation time theory;The second part of the thesis gives a theorem on the calculation of the Laplace transform of the number of records.We decompose event{A_n=m}as a series of events by the last record point,and then transform it into the relationship of the generating function,and at last find it is all about the generating function of the initial positive value,which is related to the famous Sparre Andersen theorem;Finally,we also give an accurate expression of the number of records under a symmetrical distribution formula.