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Strong laws for the largest ratio of adjacent order statistics

Vol. 12 No. 4 (2017) P.315~P.323
 DOI: https://doi.org/10.21915/BIMAS.2017402 10.21915/BIMAS.2017402

ABSTRACT

Consider independent and identically distributed random variables
$\{X_{n,k}$, $1\le k\le m_n, n \ge 1\}$. We order this data set, $X_{n(1)}< X_{n(2)}< X_{n(3)}< \cdots < X_{n(m_n-1)}< X_{n(m_n)}$. Then we find the ratio of these adjacent order statistics. Our random variable of interest is the largest of these adjacent ratios, $\max_{2\le k \le m_n} X_{n(k)}/X_{n(k-1)}$. We obtain various limit theorems for this random variable.

KEYWORDS
Almost sure convergence, strong law of large numbers, order statistics, exact laws.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60F15.

MILESTONES