Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 1 , pp. 44--53

ON BAHADUR REPRESENTATION OF SAMPLE QUANTILES FOR m_{n}- DECOMPOSABLE PROCESSES

By

KAMAL C. CHANDA and FRITS H. RUYMGAART, *Texas Tech University*

SUMMARY. Let X_{1}, … , X_{n} be random variables with a common distribution function F. Assume that these random variables form a m_{n}-decomposable set in the sense that we can write $X_t=X_{t, m_n} +X^{*}_{t, m_n}(1 \leq t \leq n)$ for some (< n) such that (i) $X_{t, m_n}(1 \leq t \leq n)$ are identically distributed and (ii) { $X_{t, m_n}$; 1£ t £ n} is m_{n}-dependent and for every $t(1 \leq t \leq n) : X^{*}_{t, m_n}\rightarrow 0$ in probability as n ® ¥ . Let V_{n} be the k_{n}-th order statistic for X_{1}, …, X_{n}, with k_{n}/n ® p as n ® ¥ , where p is a given number e (0, 1). Under certain regularity conditions on F and m_{n} it can then be proved that $n^{1/2}(V_n-\zeta)=-\sum_{t=1}^{n}(I(X \leq \zeta)-p)/n^{1/2}+ O(n^{1/2}l_{n}^{3/4}log l_{n})$ *a.s.* as n ® ¥ , where F(z ) =p and we assume that ln=[n/mn] ® ¥ and $n^{-1/2}n^{1/2}l_{n}^{-3/4}\log{l_{n}} \rightarrow 0$ as n ® ¥ . Further, it is shown that ${\mathcal{L}} (n^{1/2}(V_n-\zeta)) \rightarrow {\mathcal{N}}(0, \sigma^{2})$ where 0 < s ^{2} < ¥ .

*AMS (1980) subject classification.* 60F15, 62G30.

*Key words and phrases.* Order statistic, Bahadur representation, $m_n$ - decomposable processes, asymptotic normality.