The Convolutions Of Non-identical Distributions And The Tail Probability For The Supremum Of A Random Walk

It is well known that the distribution theory is one foundations of probability,and it have very important applications in random walk,and so in risk theory,queueing theory,branching process theory and so on.So they have been paid wide attentions. While one key problem of the distribution theory is so called the closure and asymptotic of convolution,including the convolution roots.In the first part of the second chapter,we obtain sufficient conditions and necessary conditions of local closure and local asymptotics for the convolutions(including convolution roots)of non-identical distributions on[0,∞),which reveal the relations between convolutions and pairwise convolutions of non-identical distributions.The sufficient part of the result extends the corresponding,non-local result of Geluk and De Vries(2006),and the methods we use are different from theirs;whatever,the necessary part of the result is not included in Geluk and De Vries(2006).In the last part of the second chapter,we discuss the local closure and local asymptotics for the convolutions of non-identical distributions on (-∞,∞).In the third chapter,we obtain an upper bound inequality of tail probability for the supremum of a random walk that is formed by a class of negative associated increments.The result extends the corresponding result of Leipus and(?)iaulys(2007) which is formed by a class of independent increments and the method we Use is different from their.The inequality prepaires for discssing the asymptotic of rum probability for some negative associated claims and waited time.Finally,we point out that the result of(?)iaulys(2007),in fact,is included by Theorem2(A)of Veraverbeke(1977)...