• Journal of the Korean Mathematical Society
• /
• v.51 no.3
• /
• pp.609-633
• /
• 2014

From the ordinary notion of uniformly strong mixing for a sequence of random variables, a new concept called conditionally uniformly strong mixing is proposed and the relation between uniformly strong mixing and conditionally uniformly strong mixing is answered by examples, that is, uniformly strong mixing neither implies nor is implied by conditionally uniformly strong mixing. A couple of equivalent definitions and some of basic properties of conditionally uniformly strong mixing random variables are derived, and several conditional covariance inequalities are obtained. By means of these properties and conditional covariance inequalities, a conditional central limit theorem stated in terms of conditional characteristic functions is established, which is a conditional version of the earlier result under the non-conditional case.

• Journal of the Korean Mathematical Society
• /
• v.61 no.4
• /
• pp.713-742
• /
• 2024

From the ordinary notion of upper-tail quantitle function, a new concept called conditionally upper-tail quantitle function given a σ-algebra is proposed. Some basic properties of this terminology and further properties of conditionally strictly stationary sequences are derived. By means of these properties, several conditional central limit theorems for a sequence of conditionally strong mixing and conditionally strictly stationary random variables are established, some of which are the conditional versions corresponding to earlier results under non-conditional case.