A note on the number of maxima in a discrete sample

Yongcheng Qi

doi:10.1016/s0167-7152(96)00150-2

A note on the number of maxima in a discrete sample

Yongcheng Qi

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

Consider {X_j, j ≥ 1}, a sequence of i.i.d., positive, integer-valued random variables. Let K_n denote the number of the integer j ∈ {1,2,...,n} for which X_j = max_1≤m≤n X_m. In this paper we prove that 1im_n→∞ EK_n = 1 if and only if K_n converges in probability to one, if and only if 1im_n→∞ P(X₁ = n)/P(X₁ ≥ n)=0 and prove that K_n 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.

Original language	English (US)
Pages (from-to)	373-377
Number of pages	5
Journal	Statistics and Probability Letters
Volume	33
Issue number	4
DOIs	https://doi.org/10.1016/s0167-7152(96)00150-2
State	Published - May 16 1997

Keywords

Almost sure convergence
Maxima

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10.1016/s0167-7152(96)00150-2

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title = "A note on the number of maxima in a discrete sample",

abstract = "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.",

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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

UR - https://www.scopus.com/pages/publications/0031574857

UR - https://www.scopus.com/pages/publications/0031574857#tab=citedBy

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 -

A note on the number of maxima in a discrete sample

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