Kenneth Kuttler

1910 CHAPTER 59. BASIC PROBABILITY

Then in fact,Sk (ω)→ S (ω) a.e.ω (59.15.30)

and off a set of measure zero, the convergence of Sk to S is uniform.

Proof: Let nk ≤ l ≤ m. Then by Lemma 59.15.5

([sup

nk<l≤m

∣∣∣∣Sl−Snk

∣∣∣∣> 2−k

])≤ 2P

([∣∣∣∣Sm−Snk

∣∣∣∣> 2−k])

In using this lemma, you could renumber the ζ i so that the sum

∑j=nk+1

ζ j

corresponds tol−nk

∑j=1

ξ j

where ξ j = ζ j+nk.

Then using 59.15.29,

([sup

nk<l≤m

∣∣∣∣Sl−Snk

∣∣∣∣> 2−k

])≤ 2P

([∣∣∣∣Sm−Snk

∣∣∣∣> 2−k])

< 2−(k−1)

If Sl (ω) fails to converge then ω must be in infinitely many of the sets,[supnk<l

∣∣∣∣Sl−Snk

∣∣∣∣> 2−k

]

each of which has measure no more than 2−(k−1). Thus ω must be in a set of measure zero.This proves the lemma.

59.16 The Multivariate Normal DistributionDefinition 59.16.1 A random vector, X, with values in Rp has a multivariate normal dis-tribution written as X∼Np (m,Σ) if for all Borel E ⊆ Rp,

λ X (E) =∫Rp

XE (x)1

(2π)p/2 det(Σ)1/2 e−12 (x−m)∗Σ−1(x−m)dx

for µ a given vector and Σ a given positive definite symmetric matrix.

Theorem 59.16.2 For X∼ Np (m,Σ) ,m = E (X) and

Σ = E((X−m)(X−m)∗

1910 CHAPTER 59. BASIC PROBABILITYThen in fact,Sx(@) > S(@) ae@ (59.15.30)and off a set of measure zero, the convergence of S; to S is uniform.Proof: Let n, <1 <m. Then by Lemma 59.15.5o( ) <2”(([se—sull>2-)In using this lemma, you could renumber the ¢; so that the sumlLS;j=n+lsup ||Si —Sn,|| >2kng<l<mcorresponds tol—ngXs;J=1where § ; = jiny:Then using 59.15.29,|If S;(@) fails to converge then @ must be in infinitely many of the sets,sup ||S;—Sn,|| > >) < 2P (|| |Sn— Sn > 2*)) < 27)ng<l<m| | [Si — Sr, || > 2ny<l(k-1)each of which has measure no more than 2~ . Thus @ must be in a set of measure zero.This proves the lemma.59.16 The Multivariate Normal DistributionDefinition 59.16.1 A random vector, X, with values in R? has a multivariate normal dis-tribution written as X ~N, (m,~) if for all Borel E CR’,1 =1 aAy(E)= XG “Fe (Km )*E (x—m) 7x(E) RP #(x) (2m)? det (=) *for Lt a given vector and X a given positive definite symmetric matrix.Theorem 59.16.2 For X ~ N, (m,x),m = E (X) andX= E ((X—m)(X—m)").