TY - JOUR
T1 - A note on the number of maxima in a discrete sample
AU - Qi, Yongcheng
PY - 1997/5/16
Y1 - 1997/5/16
N2 - Consider {Xj, j ≥ 1}, a sequence of i.i.d., positive, integer-valued random variables. Let Kn denote the number of the integer j ∈ {1,2,...,n} for which Xj = max1≤m≤n Xm. In this paper we prove that 1imn→∞ EKn = 1 if and only if Kn converges in probability to one, if and only if 1imn→∞ P(X1 = n)/P(X1 ≥ n)=0 and prove that Kn converges almost surely to one, if and only if ∑∞n=1(P(X1 = n)/P(X1 ≥ n))2 < ∞. Some of the results were shown by Baryshnikov et al. (1995) and Brands et al. (1994). AMS classification: Primary 60F15.
AB - Consider {Xj, j ≥ 1}, a sequence of i.i.d., positive, integer-valued random variables. Let Kn denote the number of the integer j ∈ {1,2,...,n} for which Xj = max1≤m≤n Xm. In this paper we prove that 1imn→∞ EKn = 1 if and only if Kn converges in probability to one, if and only if 1imn→∞ P(X1 = n)/P(X1 ≥ n)=0 and prove that Kn converges almost surely to one, if and only if ∑∞n=1(P(X1 = n)/P(X1 ≥ n))2 < ∞. Some of the results were shown by Baryshnikov et al. (1995) and Brands et al. (1994). AMS classification: Primary 60F15.
KW - Almost sure convergence
KW - Maxima
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U2 - 10.1016/s0167-7152(96)00150-2
DO - 10.1016/s0167-7152(96)00150-2
M3 - Article
AN - SCOPUS:0031574857
SN - 0167-7152
VL - 33
SP - 373
EP - 377
JO - Statistics and Probability Letters
JF - Statistics and Probability Letters
IS - 4
ER -