6-ConceptsProbability Page 191 Wednesday, February 4, 2004 3:00 PM
Concepts of Probability 191d
Xi →Xiflim FX x () ∈ , C
i
()= FXx x
i → ∞where C is the set of points where all the functions FXi and FX are con-
tinuous.
It can be demonstrated that if a sequence converges almost surely
(and thus converges in probability) it also converges in distribution
while the converse is not true in general.INDEPENDENT AND IDENTICALLY DISTRIBUTED SEQUENCES
Consider a probability space (Ω,ℑ,P). A sequence of random variables Xi
on (Ω,ℑ,P) is called a sequence of independent and identically distributed
(IID) sequence if the variables Xi have all the same distribution and are
all mutually independent. An IID sequence is the strongest form of white
noise, that is, of a completely random sequence of variables. Note that in
many applications white noise is defined as a sequence of uncorrelated
variables. This is a weaker definition as an uncorrelated sequence might
be forecastable.
An IID sequence is completely unforecastable in the sense that the
past does not influence the present or the future in any possible sense. In
an IID sequence all conditional distributions are identical to uncondi-
tional distributions. Note, however, that an IID sequence presents a sim-
ple form of reversion to the mean. In fact, suppose that a sequence Xi
assumes at a given time t a value larger than the common mean of all
variables: Xt > E[X]. By definition of mean it is more likely that Xt be
followed by a smaller value: P(Xt+1 < Xt) > P(Xt+1 > Xt).
Note that this type of mean reversion does not imply forecastability
as the probability distribution of asset returns at time t + 1 is indepen-
dent from the distribution at time t.SUM OF VARIABLES
Given two random variables X(ω), Y(ω) on the same probability space
(Ω,ℑ,P), the sum of variables Z(ω) = X(ω) + Y(ω) is another random
variable. The sum associates to each state ωa value Z(ω) equal to the