an:04026447
Zbl 0631.60031
Haiman, George
??tude des extr??mes d'une suite stationnaire m-d??pendante avec une application relative aux accroissements du processus de Wiener. (Study of the extremes of a stationary m-dependent sequence with an application to the increments of Wiener processes)
FR
Ann. Inst. Henri Poincar??, Probab. Stat. 23, 425-457 (1987).
00152603
1987
j
60F10 62G30 60J65
extreme value theory; increments of the Wiener process; sequence of record times; record values
The author investigates a stationary m-dependent sequence \(\{X_ i\}\) of random variables. In the first part, a uniform bound for
\[
\mu^{- n}(u)P\{\max (X_ 1,...,X_ n)\leq n\}-1
\]
is given for a certain \(\mu\) (u) and for large x with \(x<\sup \{y;P(X<y)<1\}\). This completes a previous result obtained by the author [Ann. Inst. Henri Poincar??, Nouv. S??r., Sect. B 17, 309-330 (1981; Zbl 0479.60044)]. The result is applied to improve an evaluation of the probability that the increments of the Wiener process remain under a given threshold.
In the second part the records for \(\{X_ n\}\) are considered. Let \(\{(T_ n,O_ n)\}\) be the sequence of record times and record values associated with \(\{X_ n\}\), and let \(\{(S_ n,R_ n)\}\) be the corresponding sequence associated with the sequence \(\{X_ n'\}\) of independent random variables having the same distribution as \(X_ 1\). It is shown that \(\{X_ n'\}\) can be defined so that \(S_ n=T_{n-q}\) and \(R_ n=O_{n-q}\) for an integer q and for all large n.
T.Mori
Zbl 0479.60044