Invariance principles for dependent variables

On complete convergence for Lp‐mixingales

“…Furthermore, McLeish [6,8] studied the invariance principles for mixingales. Yin [9] generalized McLeish's concept of mixingales to operator-valued mixingales, and proved the operator-valued mixingale convergence theorems.…”

“…is implied with summability conditions such as [6,7]). (1.6), and (1.9) are satisfied with the above λ and each…”

Fractional Integrals of Stationary Processes and the Central Limit Theorem

“…The standard central limit theorem is for (2) ,(,)-{i ", 0<f <1, otherwise, and is often proved under a number of possible strong mixing conditions (see [4] and [5]). Suppose that the process X(t) has an absolutely continuous spectral distribution function F with spectral density /.…”

“…Let X(t), -x <f <°°, be a strictly stationary real-valued separable random process with EX(t) = 0 and EX(t) 4 <*>. Our interest is in limit theorems for (…”

An Exponential Continuous-Time GARCH Process

“…The left-hand inequality of (3.9) is easy to see (cf. [12,Lemma B.2]) and the right-hand inequality follows from Lemma 3.5 of [17]. For a stochastic process Y , the correspondingα-mixing coefficient is defined as…”