Research On Limiting Distribution Of Maximum(Minimum) Value And Related Topics

Limiting distribution of maximum?minimum?value can be regarded as a kind of extreme value problem.In recent years,this kind of problem has drawn attention of researchers from different fields,such as mathematics,climate prediction,geological disaster assessment,oceanic forecast,insurance actuarial and so on.Therefore,the study of the extreme value problem not only has great significance but also plays an important role in the realistic society.The limiting distributions of maximum?minimum?value of random variables and sums of random variables,probability measure are considered in this paper in the light of mathematical analysis,probability theory,maximum entropy theory and stochastic analysis.The main work accomplished includes the following:In the first chapter,the research background and significance as well as the current situation of studies both at home and abroad of the extreme value problem are introduced.Moreover,the main contents and invitation of this paper are briefly stated.In the second chapter,we consider the minimum-maximum models for the independent and identically distributed random sequence and stationary sequence,respectively.With Taylor's expansions of the distribution functions and some important probability formulas,the limiting distributions for these two kinds of sequences are obtained and convergence analysis is carried out for the limiting distributions.Numerical experiments are conducted to confirm our theoretical analysis.The third chapter dedicates itself to considering two special kinds of path in transport problems and proving the limiting distributions of maximum sums of these two kinds of path on condition that every path is a discrete time,homogeneous and irreducible ergodic Markov chain with a finite number of states.Combining total probability formula and central limit theorem with the decomposition of Markov chain,we not only derive the extreme value distribution for the first kind but also establish the equivalence between limit distributions of a sum and of a maximum sum for the second kind.Numerical experiments are conducted to confirm our results.In the forth chapter,we show that the control chart with the charting statistics of the sum of log likelihood ratios with a special dynamic control limits for detecting the change in distribution of the finite numbers of dependent observations is optimal,in the sense that the optimality of this kind of control chart shows it has the smallest out-of-control average run length?ARL?among all control charts with a given probability of a false alarm no greater than a preset level,or,among those with a given false alarm rate no less than a given value.Moreover,we prove that the ARL₁ is approximately equal to 1 for any ARL₀ with a negative control limit when the number of observations tends to be infinity.By the equivalence of limit distribution between a longest path sum and a maximum sum of Markov chain,we give the estimation of the ARL₁ with a large enough positive control limit.The numerical simulations illustrate the detection performance of the optimal control chart.The last chapter is devoted to converting the evaluation of nonlinear expectation into seeking some probability density functions satisfied the maximum and minimum moment conditions.Combining the moment method and the maximum entropy method to present a necessary and sufficient condition for the existence of probability density functions.Moreover,we carry out a kind of maximum entropy solutions and analyze the convergence for theses solutions.Numerical experiments are presented to compute the two-dimensional maximum entropy density functions.