Phi-mixing is a strong asymptotic independence condition for a strictly stationary sequence 
of real-valued random
variables. If 
and 
are the sigma-algebras
generated by the past 
and the future 
, respectively, for a positive
integer 
, define
 |
The sequence is phi-mixing when 
. This controls conditional probabilities uniformly over
past events of arbitrarily small positive probability
(Bradley 2005). A sequence of independent
random variables has 
for every 
.
The Ibragimov-Iosifescu conjecture asks whether this mixing condition, finite variance,
and divergence of the variance of partial
sums suffice for a central limit theorem
without an additional rate assumption on 
.
