On the asymptotic behaviour of the number of maximum points of a simple random walk

For a sequence (Xi)i≥1 of independent and identically distributed random variables, taking the values -1, 0 and 1, we define S0 = 0 and Sk = Pk i=1 Xi, for k ≥ 1. We study the asymptotic behaviour of the sequence of random variables (Qn)n≥1, where Qn indicates the number of absolute maximum points of the simple random walk S0, S1, · · · , Sn. The paper extends some results of Dwass [2], Rev´ esz [11], Katzenbeisser and Panny [7], [8].