Almost Sure Central Limit Theorem For The Maximum And The Limiting Distribution Of Extremes With Random Index

In this thesis, we considered almost surely central limit theorem for the maximum of weakly dependent stationary sequence and a kind of quasi-stationary sequence, and also considered the limiting distribution of the maximum of independent identically distributed sequence with random index which is regular variation. The limiting distribution of the maximum of weakly dependent Gaussian sequence with random index also discussed. The main results are:Theorem 1. Let be stationary random variable sequence with Mn = be real sequences with{u_n}. Under conditions D'(u_n)and D₂ ({u_k, u_n}), if there exists non-degenerated distribution function G (x),such that as n→∞And for someTheorem 2. Let {Ψi} be a quasi-stationary sequence satisfying conditions (a) - (e), and let and be bounded. Supposehold such thatThen for some , as n→∞, we haveTheorem 3. Let {Xi} be independent identical distributed random sequence with be a positive integer valued random variables such that . If there exists non-degenerated distribution function G such thatas n→∞. ThenTheorem 4. Let {ξi} be a standard stationary Gaussian sequence with co-variance . Let and {N(n)} bea positive integer valued random variables satisfying Thenprovided...