In this section is a further explanation of generalized multivariable normal random
variables. Recall that these have characteristic function equal to e^{it⋅m}e^{−}
1
2
t^{∗}Σt
where
Σ ≥ 0,Σ = Σ^{∗}. The new detail is the case that det
(Σ)
= 0.
Definition 58.20.1A random vector,X,with values in ℝ^{p}has a multivariate normaldistributionwritten as
X ∼N (m, Σ)
p
if for all Borel E ⊆ ℝ^{p}, the distribution measure is given by
∫
λ (E ) = X (x)-------1-------e−12 (x−m)∗Σ−1(x−m )dx
X ℝp E (2π)p∕2det(Σ )1∕2
for m a given vector and Σ a given positive definite symmetric matrix. Recall also thatthe characteristic function of this random variable is
( )
E eit⋅X = eit⋅me− 12t∗Σt (58.20.48)
(58.20.48)
So what if det
(Σ )
= 0? Is there a probability measure having characteristic
equation
eit⋅me− 12t∗Σt?
Let Σ_{n}→ Σ in the Frobenius norm, det
(Σn)
> 0. That is the ij^{th} components converge.
Let X_{n} be the random variable which is associated with m and Σ_{n}. Thus for
ϕ ∈ C_{0}
Thus these λ_{Xn} are bounded in the weak ∗ topology of C_{0}
p
(ℝ )
^{′} which is the space of
signed measures. By the separability of C_{0}
p
(ℝ )
and the Banach Alaoglu theorem and the
Riesz representation theorem for C_{0}
p
(ℝ )
^{′}, there is a subsequence still denoted as λ_{Xn}
which converges weak ∗ to a finite measure μ. Is μ a probability measure? Is the
characteristic function of this measure e^{it⋅m}e^{−1
2
t∗Σt
}?
Note that E
(eit⋅Xn)
= e^{it⋅m}e^{−12
t∗Σ
nt}→ e^{it⋅m}e^{−12
t∗Σt
} and this last function of t is
continuous at 0. Therefore, by Lemma 58.18.2, these measures λ_{Xn} are also tight. Let
ε > 0 be given. Then there is a compact set K_{ε} such that λ_{Xn}
. The middle term is less than ε
if n large enough thanks to the weak ∗ convergence of λ_{Xn} to μ. Hence the
last limit in 58.20.49 equals ∫_{ℝp}e^{it⋅x}dμ
(x)
as hoped. Letting X be a random
variable having μ as its distribution measure, (You could take Ω = ℝ^{p} and
the measurable sets the Borel sets.) what about E
((X − m )(X − m )∗)
? Is it
equal to Σ? What about the question whether X ∈ L^{q}
(Ω;ℝp )
for all q > 1?
This is clearly true for the case where Σ^{−1} exists, but what of the case where
det
If m≠0, the same kind of argument holds with a little more details. This proves the
following theorem.
Theorem 58.20.2Let Σ be nonnegative and self adjoint p × p matrix. Then thereexists a random variable X whose distribution measure λ_{X}has characteristicfunction
∗
eit⋅me− 12t Σt
Also
E((X − m )(X − m)∗) = Σ
that is
( )
E (X − m )i(X − m )j = Σij
This is generalized normally distributed random variable.
There is an interesting corollary to this theorem.
Corollary 58.20.3Let H be a real Hilbert space. Then there exist random variablesW
(h)
for h ∈ H such that each is normally distributed with mean 0 and for everyh,g,
(W (h),W (g))
is normally distributed and
E (W (h)W (g)) = (h,g)H
Furthermore, if
{e}
i
is an orthogonal set of vectors of H, then
{W (e )}
i
are independentrandom variables. Also for any finite set
{f ,f ,⋅⋅⋅,f }
1 2 n
,
(W (f1),W (f2),⋅⋅⋅,W (fn))
is normally distributed.
Proof: Let μ_{h1}
⋅⋅⋅
h_{m} be a multivariate normal distribution with covariance
Σ_{ij} =
(h,h )
i j
and mean 0. Thus the characteristic function of this measure
is
− 1t∗Σt
e 2
Now suppose μ_{k1}
⋅⋅⋅
k_{n} is another such measure where for simplicity,
{h1⋅⋅⋅hm,km+1 ⋅⋅⋅kn} = {k1⋅⋅⋅kn}
Let ν be a measure on ℬ
(ℝm )
which is given by
( )
ν (E ) ≡ μk1⋅⋅⋅kn E × ℝn− m
Then does it follow that ν = μ_{h1}
⋅⋅⋅
h_{m}? If so, then the Kolmogorov consistency
condition will hold for these measures μ_{h1⋅⋅⋅
hm}. To determine whether this is so,
take the characteristic function of ν. Let Σ_{1} be the n × n matrix which comes
from the
which is the characteristic function for μ_{h1⋅⋅⋅
hm}. Therefore, these two measures are the
same and the Kolmogorov consistency condition holds. It follows that there exists a
measure μ defined on the Borel sets of ∏_{h∈H}ℝ which extends all of these measures. This
argument also shows that if a random vector X has characteristic function e^{−1
2
t∗Σt
}, then
if X_{k} is one of its components, then the characteristic function of X_{k} is e^{−1
2
}