Weak Convergence of a Two-Sample Empirical Process and a Chernoff-Savage Theorem for $|phi$-Mixing Sequences

On the chernoff-savage theorem for dependent sequences

“…This result is a generalization of Fears and Mehra [4] in that the stationarity is not required and that the condition imposed on the mixing numbers is substantially relaxed. A similar result is shown to hold for strong mixing sequences under slightly stronger conditions on the mixing numbers.…”

“…Assume that (i) {X~} and {Y~} have absolutely continuous finite dimensional distributions with respect to Lebesgue measure, and that either (ii) both sequences are uniformly mixing, or (ii)' both sequences are strongly mixing. For ease of comparison between our result for the case of uniform mixing sequences and that of Fears and Mehra [4] we shall try to be consistent with their presentation and closely follow their notations. Note that the result of Fears and Mehra is obtained assuming strict stationarity for both sequences and ~n2[r while we waive n=l the stationarity assumption and only assume r Let {X~} and {Y~} be two independent sequences of random variables satisfying Conditions (i) and (ii) above.…”

“…It should be noted here that the arguments of Fears and Mehra [4] can be adopted to show the weak convergence result for a strictly stationary strong mixing sequence with mixing number a(m) satisfying ~]m2[a(m)]l/~-'<co for some re (0, 1/2). In this section we will again waive the stationarity assumption and assume a(m)=O(m -Sn-~) for some 3>0.…”

“…Their proof shows that the theorem holds even for a larger class of score functions than initially established. Motivated by interest in the robustness of the two-sample linear rank statistics, Fears and Mehra [4], using the approach of Pyke and Shorack, establish the asymptotic normality of linear rank statistics for strictly stationary C-mixing sequences of random variables satisfying certain regularity conditions.…”

“…For C-mixing sequences we waive the stationarity assumption and considerably weaken the condition imposed by Fears and Mehra [4] on the mixing numbers. Secondly, we establish, under slightly stronger assumptions on the mixing numbers, analogous results for strong mixing sequences of random variables.…”