# Independent processes

In this subsection we discuss LLNs for independent processes.

Theorem 18.12 (Kolmogorov’s strong LLN for iid random variables) Let Zt be a sequence of identically and independently distributed (iid) random variables with E |Z1| < ro and EZ1 = p. Then Zn p (and hence Zn p) as n ^ ro.

We have the following trivial but useful corollary.

Corollary 6. Let Zt be a sequence of iid random variables, and let f be a Borel – measurable real function satisfying E |f(Z1)| < ro, then n_1 Xn=1 f (Zt) Ef (Z1) as

n ^ ro.

The corollary can, in particular, be used to establish convergence of sample moments of, say, order p to the corresponding population moment by choosing f(Zt) = Ztp.

We now derive the probability limit of the ordinary least squares estimator considered in Example 6 as an illustration.

Example 7. Assume the setup of Example 6. Assume furthermore that the processes xt and et are iid with Ex’2 = Qx, 0 < Qx < ro, E | et | < ro, and Eet = 0, and that the two processes are independent of each other. Then xtet is iid, has finite expectation and satisfies Extet = ExtEet = 0. Hence it follows from Theorem 18 that n1 X n=1 xtet 0. Corollary 6 implies n_1 X n=1 xt2 Qx. Applying Theorem 14

then yields Pn 0 + 0/Qx = 0.

The assumption in Theorem 18 that the random variables are identically distri­buted can be relaxed at the expense of maintaining additional assumptions on the second moments.

Theorem 19.13 (Kolmogorov’s strong LLN for ID random variables) Let Zt be a sequence of independently distributed (ID) random variables with EZt = pt and

var(Zt) = о;2 < ro. Suppose X T=1 c2/12 < ro. Then Zn – <n 0 as n ^ ro.

The condition X n=1 о;2/12 < ro puts a restriction on the permissible variation in the о;2. For example, it is satisfied if the sequence о2 is bounded.