59.20. GENERALIZED MULTIVARIATE NORMAL 1933

So what if det(Σ) = 0? Is there a probability measure having characteristic equation

eit·me−12 t∗Σt?

Let Σn→ Σ in the Frobenius norm, det(Σn)> 0. That is the i jth components converge. LetXn be the random variable which is associated with m and Σn. Thus for φ ∈C0 (Rp) ,

|λ Xn (φ)| ≡

∣∣∣∣∣∫Rp

φ (x)1

(2π)p/2 det(Σn)1/2 e

−12 (x−m)∗Σ−1

n (x−m)dx

∣∣∣∣∣≤ ∥φ∥C0(Rp)

Thus these λ Xn are bounded in the weak ∗ topology of C0 (Rp)′ which is the space of signedmeasures. By the separability of C0 (Rp) and the Banach Alaoglu theorem and the Rieszrepresentation theorem for C0 (Rp)′, there is a subsequence still denoted as λ Xn whichconverges weak ∗ to a finite measure µ . Is µ a probability measure? Is the characteristicfunction of this measure eit·me−

12 t∗Σt?

Note that E(eit·Xn

)= eit·me−

12 t∗Σnt→ eit·me−

12 t∗Σt and this last function of t is contin-

uous at 0. Therefore, by Lemma 59.18.2, these measures λ Xn are also tight. Let ε > 0 begiven. Then there is a compact set Kε such that λ Xn (x /∈ Kε)< ε. Now let φ = 1 on Kε andφ ∈Cc (Rp), φ ≥ 0,φ (x) ∈ [0,1]. Then

(1− ε)≤∫Rp

φdλ Xn →∫Rp

φdµ ≤ µ (Rp)

and so, since ε is arbitrary, this shows that µ (Rp)≥ 1. However, µ (Rp)≤ 1 because

µ (Rn)≤∫Rp

ψdµ + ε ≤∫Rp

ψdλ Xn +2ε ≤ 1+2ε

for suitable ψ ∈ Cc (Rp) having values in [0,1] and n. Thus µ is indeed a probabilitymeasure.

Now what of its characteristic function?

eit·me−12 t∗Σt = lim

n→∞eit·me−

12 t∗Σnt = lim

n→∞

∫Rp

eit·xdλ Xn (x) (59.20.49)

Is this equal to ∫Rp

eit·xdµ (x)?

Using tightness again,∣∣∣∣∫Rpeit·xdµ (x)−

∫Rp

eit·xdλ Xn (x)∣∣∣∣≤ ∣∣∣∣∫Rp

eit·xdµ (x)−∫Rp

ψeit·xdµ (x)∣∣∣∣

+

∣∣∣∣∫Rpψeit·xdµ (x)−

∫Rp

ψeit·xdλ Xn (x)∣∣∣∣+ ∣∣∣∣∫Rp

ψeit·xdλ Xn (x)−∫Rp

eit·xdλ Xn (x)∣∣∣∣

≤ ε +

∣∣∣∣∫Rpψeit·xdµ (x)−

∫Rp

ψeit·xdλ Xn (x)∣∣∣∣+ ε