Extremes Of Mixed Distributions And Almost Sure Convergence Theorem Of Maxima Of Multivariate Gaussian Sequence

There are two main parts in this thesis. The first part is for the limiting dis-tributions of finite mixed distributions, and the almost sure convergence of maxima of a class of multivariate Gaussian sequence for the second part.The finite mixed distribution has been widely applied to fit the complex dis-tribution for its flexibility. The finite mixed distribution is defined by F(x)= (?)piFi(x), where pi> 0,i= 1,…,r,(?)pi= 1,and Fi,i= 1,…,r are different distribution functions. In the first part of this thesis, we consider limiting distri-butions of extremes with parent following finite mixed distributions.There are two case:One is the components Fi follow some special distributions. The other is the case of tail equivalent classes.In the second part, we consider the almost sure limit theorem of maxima of a class of multivariate Gaussian sequence. The limit distributions of Gaussian se-quences always depend heavily on the convergence rates of their correlations. Under Berman's condition and the conditions restricted on the multivariate levels, we get the weak convergence of the maxima of multivariate non-stationary Gaussian vector sequence and its almost sure convergence theorem.