Limit theorem conditioning on information set

[sihag]

Would anyone be able to throw some light on the convergence of
var[x_t | F_t] , where x_t is a stochastic time-series process, and F_t is the information (it's past history) uptil time t.

What I mean by the question is that
var[x_t | F_t] can be regarded as a function of F_t
say, g(F_t)

Now, does
1/T*g(F_t) ----> converge in probability (a sort of Weak Law of Large Numbers) to its expectation (over the distribution of F_t)?
(possibly under some assumptions on the dependence of F_t through time ?)

Thanks for any help

 

[Valkarie]

. The answer to your question is yes, under some assumptions on the dependence of F_t through time. Specifically, if F_t is a Markov process, then it has been shown that 1/T*g(F_t) converges in probability to its expectation (over the distribution of F_t). This result is known as the law of large numbers for Markov processes.