Almost sure max-limits for nonstationary Gaussian sequence

Almost sure convergence for the maxima of strongly dependent stationary Gaussian vector sequences

“…The last inequality comes from Lemma 3.1 and the arguments in [1]. For P (2) i, j , noting that the {X s , s 1} and {Y (n) m } are independent, for i < j n we can get…”

“…For the weakly dependent stationary Gaussian sequence {X n , n 1} with E X 1 = 0, Var X 1 = 1, Csáki and Gonchigdanzan [5] obtained the ASLT for the maxima if their correlation r n = E X 1 X n+1 satisfies r n log n(log log n) 1+ε = O (1) for some ε > 0. For some extensions see [2] and [9]. Chen et al [3] studied the ASLT of extremes for weakly dependent stationary Gaussian vector sequences.…”

Almost sure central limit theorem for the maxima and sums of stationary Gaussian sequences

“…Csáki and Gonchigdanzan (2002) and Lin (2009) have extended (2) for weakly dependent and strongly dependent stationary Gaussian sequences, respectively. For the non-stationary case, we refer to Chen and Lin (2006) and Tan and Peng (2009). For the stationary Gaussian random fields, we refer to Choi (2010).…”

Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima

“…The topics pertaining to the ASCLTs have attracted an immense attention since the publication of the two above mentioned papers and a great deal of works devoted to the proofs of (1) for various classes of functions f n and random sequences .X i / have appeared throughout the last two decades or so. We cite in this context the articles by: Berkes and Csáki [3], Chen and Lin [4], Cheng et al [5], Csáki and Gonchigdanzan [6], Dudziński [7], Dudziński and Górka [8], Gonchigdanzan and Rempała [9], Ho and Hsing [10], Lacey and Philipp [11], Matuła [12], Mielniczuk [13], Peligrad and Shao [14], Stadtmüller [15], and Zhao et al [16], among others. Functions f n included different kinds of functions of r.v.…”