Limit behaviors for ratios of order statistics from exponential distributions. Issue 24 (11th November 2021) - Xu, Shoufang; Mei, Changlin
Abstract: Let { X n i, 1 ≤ i ≤ m n, n ≥ 1 } be an array of independent exponentially distributed random variables with mean λn, and { X n ( 1 ) ≤ X n ( 2 ) ≤ ⋯ ≤ X n ( m n ) } be the order statistics of the random variables { X n i, 1 ≤ i ≤ m n } . Limit properties of the ratios { R nij = X n ( j ) / X n ( i ), 1 ≤ i < j ≤ m n } are investigated for fixed sample size mn  =  m . Specifically, the weak law of large numbers is derived for R n 1 j under some more general conditions, followed by the establishment of its complete convergence. Furthermore, the large deviation principles for both R n 1 j and R n 2 j are formulated.