Introduction to the Mathematical and Statistical Foundations of Econometrics
Mixing Conditions
Inspection of the proof of Theorem 7.5 reveals that the independence assumption can be relaxed. We only need independence of an arbitrary set A in F—T and an arbitrary set C in Ft—k = a (Xt, Xt—1, Xt—2,Xt—k) for k > 1. A sufficient condition for this is that the process Xt is a-mixing or y-mixing:
Definition 7.5: Let F— T = a (Xt, Xt-1, Xt -2, ■■■), = a (Xt, X+1,
Xt +2,...) and
a(m) = sup sup |P(A n B) — P(A) ■ P(B)|,
1 AeF”, Be^—T
y(m) = sup sup | P (A| B) — P (A)|.
1 ^ , bc F—m
If limm^Ta(m) = 0, then the time series process Xt involved is said to be а-mixing; iflimm^Ty(m) = 0, Xt is said to be y-mixing.
Note in the a-mixing case that
sup |P(A n B) — P(A) ■ P(B)|
Ae. F‘t—k, BeF— t
< limsupsup sup |P(A n B) — P(A) ■ P(B)|
m^T t, cz” D cz-t —k—m
Ae^ t —k, Be^ — T
= limsup a(m) = 0;
hence, the sets A e Ft—k, B e F— T are independent. Moreover, note that a(m) < y(m), and thus y-mixing implies а-mixing. Consequently, the latter is the weaker condition, which is sufficient for a zero-one law:
Theorem 7.6: Theorem 7.5 carries over for а-mixing processes.
Therefore, the following theorem is another version of the weak law of large numbers:
Theorem 7.7: IfXt is a strictly stationary time series process with an a-mixing base and E[|X1|] < to, then plimn^TO(1 / n)^f"= 1 Xt = E[X1],
