Sample Properties Of The Lévy Sheet

Lévy sheet is a kind of important two-parameter stochastic process with zero initial value,steady independent increment and stochastic continuity.Poisson sheet,Brown sheet and so on are all special Lévy sheet.It is very important in the fields of economy,finance,communication,insurance and so on.Therefore,the study of Lévy sheet has important academic value and application value.For the sample distribution,uniform stochastic continuity and so on of Lévy sheet,in this paper,the following results are obtained:(1)Sample distribution.The related conclusions of homogeneous distribution,independent and same distribution of Lévy sheet are given in this paper.(2)Uniform stochastic continuity.It is proved that Lévy sheet is uniformly random continuous on a bounded closed interval.And the rectangular increment of Lévy sheet on arbitrary small left side half open rectangular square converges to 0 according to probability 1.At the same time,the relationship between these two properties is established.(3)Random bounded.Lévy processes obtains the maximum value with probability 1 on the bounded closed interval.Lévy sheet is randomly bounded on a bounded closed interval.(4)Local property.The asymptotic behavior of Lévy sheet at the original point and the local growth at a point away from the axes are obtained.By means of the small ball probability of the two-parameter Ornstein-Uhlenbeck process derived from Brown sheet,the properties of the Brown sheet are studied.