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Limit theorems for arrays of ratios of order statistics

Vol. 33 No. 4 (2005) P.327~P.344

ABSTRACT

Let $\{X_{nk}, 1 \le k \le m_n, n \ge 1 \}$ be independent random variables from the Pareto distribution. Let $X_{n(k)}$ be the $k^{th}$ largest order statistic from the $n^{th}$ row of our array. Then set $R_{nij} = X_{n(j)} / X_{n(i)}$ where $j < i$. This paper establishes limit theorems involving weighted sums from the sequence $\{R_{nij}, n \ge 1 \}$.

KEYWORDS
Almost sure convergence, strong law of large numbers, weak law of large numbers, generalized law of the iterated logarithm

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60F05, 60F15

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