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],