DEMONSTRATIO MATHEMATICA Vol. XLI No 3 2008 Marcin Dudzinski, Konrad Furmanczyk ON THE ALMOST SURE CENTRAL LIMIT THEOREMS IN THE JOINT VERSION FOR THE MAXIMA AND MINIMA OF SOME RANDOM VARIABLES Abstract. Let: {X} be a sequence of r.v.'s, and: M n := max (Xi, ... , X n ), m n : = min (Xi,... ,X n )- Our goal is to prove the almost sure central limit theorem for the properly normalized vector {M n ,m n }, provided: 1) {X,} is an i.i.d. sequence, 2) {Xi} is a certain standardized stationary Gaussian sequence. 1. Introduction The almost sure central limit theorem (ASCLT) has become an inten- sively studied subject in recent years. The simplest versions of the ASCLT have been proved in the papers of Brosamler [3], Schatte [19], Fisher [12], Lacey and Philipp [13] and Berkes and Dehling [1], They relate to the AS- CLT for the sums of independent r.v.'s. The ASCLT for sums has been later generalized by Peligrad and Shao [18], Matula [15], [16], Mielniczuk [17] and Dudzinski [8], for the sums of some weakly dependent r.v.'s. Starting from Cheng et al. [4] and Fahrner and Stadtmueller [11], the ASCLT for maxima of r.v.'s has become another popular direction of the research concerning the topic. The ASCLT for the maxima of independent r.v.'s has been proved in the mentioned papers. These results have been later extended by Csaki and Gonchigdanzan [6] and Dudzinski [7] to the cases of maxima of certain stationary Gaussian sequences. Some other valuable results concerning the issue of the ASCLT's are due to Berkes and Csaki [2], Stadtmueller [20] and Dudzinski [9], [10]. In Berkes and Csaki [2], the ASCLT's for the variety of functions of independent r.v.'s, such as maxima of partial sums, the Kol- mogorov statistics, U-statistics etc. have been proved. In [20], the proof of the ASCLT for certain order statistics of i.i.d. r.v.'s has been given. In turn, Key words and phrases: Almost sure central limit theorem; Vectors of maxima and minima of some random variables; Normal Comparison Lemma. 2000 Mathematics Subject Classification: Primary 60F15; Secondary 60F05.