Kenneth Kuttler

59.15. THE CONVERGENCE OF SUMS 1909

Now ∩ j−1i=1 [||Si|| ≤ r]∩

[∣∣∣∣S j∣∣∣∣> r

]is of the form[(

ξ 1, · · · ,ξ j

)∈ A]

for some Borel set, A. Then letting Y = ∑Ni= j+1 ξ i in Lemma 59.15.4 and Xi = ξ i, 59.15.28

equals

(∩ j−1

i=1 [||Si|| ≤ r]∩[∣∣∣∣S j

∣∣∣∣> r]∩

[∣∣∣∣∣∣∣∣∣∣S j−

∑i= j+1

ξ i

∣∣∣∣∣∣∣∣∣∣≤ r

])= P

(∩ j−1

i=1 [||Si|| ≤ r]∩[∣∣∣∣S j

∣∣∣∣> r]∩[∣∣∣∣S j− (SN−S j)

∣∣∣∣≤ r])

= P(∩ j−1

i=1 [||Si|| ≤ r]∩[∣∣∣∣S j

∣∣∣∣> r]∩[∣∣∣∣2S j−SN

∣∣∣∣≤ r])

Now since∣∣∣∣S j (ω)

∣∣∣∣> r,[∣∣∣∣2S j−SN∣∣∣∣≤ r

]⊆

[2∣∣∣∣S j∣∣∣∣−||SN || ≤ r

]⊆ [2r−||SN ||< r]

= [||SN ||> r]

and so, referring to 59.15.27, this has shown

P(A j ∩ [||SN || ≤ r])

= P(∩ j−1

i=1 [||Si|| ≤ r]∩[∣∣∣∣S j

∣∣∣∣> r]∩[∣∣∣∣2S j−SN

∣∣∣∣≤ r])

≤ P(∩ j−1

i=1 [||Si|| ≤ r]∩[∣∣∣∣S j

∣∣∣∣> r]∩ [||SN ||> r]

)= P(A j ∩ [||SN ||> r]) .

It follows that

([sup

k≤N−1||Sk||> r and ||SN || ≤ r

])=

N−1

∑i=1

P(A j ∩ [||SN || ≤ r])

≤N−1

∑i=1

P(A j ∩ [||SN ||> r])≤ P([||SN ||> r])

and using 59.15.26, this proves the lemma.This interesting lemma will now be used to prove the following which concludes a

sequence of partial sums converges given a subsequence of the sequence of partial sumsconverges.

Lemma 59.15.6 Let {ζ k} be a sequence of independent random variables having values ina separable real Banach space, E whose distributions are symmetric. Letting Sk ≡∑

ki=1 ζ i,

suppose{

Snk

}converges a.e. Also suppose that for every m > nk,

P([∣∣∣∣Sm−Snk

∣∣∣∣E > 2−k

])< 2−k. (59.15.29)

59.15. THE CONVERGENCE OF SUMS 1909Now nz [Sil] <r]. [|]S;|| > 7] is of the form(En )) €4)for some Borel set, A. Then letting Y = it €, in Lemma 59.15.4 and X; = €;, 59.15.28equalsP(A} Llisill $71 [l|S)]] > 7] 9 [||8)— (Sw —S) || <7)Sj- y gii=j+1(oft usit<aintisil> do]= P(r Uisill <rin[IIS)l| > r]9 [I]25;—Svl] <1)Now since ||S;(@)]|| >r(Il28;-Sw|] <r] [2|ISi|| =liswil <7][2r — ||Snv|| <7](Swv|| > 7]INIAand so, referring to 59.15.27, this has shownP(AjO[||Swl] S7})P(r llsill <1 [||S7|] > +] [|]25)—Sw|] <])P (OT MlSill <1 [|]$al] > *] Olliswll > x4)= P(A;N[||Swil > 71).It follows that(|and using 59.15.26, this proves the lemma.This interesting lemma will now be used to prove the following which concludes asequence of partial sums converges given a subsequence of the sequence of partial sumsconverges.N-1sup ||Sx|| > rand ||Sy]| </|) =) Paz ntllSll <7)<N-1 i=l< Y Pastis > r|) <P (| Sw] > 7)Lemma 59.15.6 Let {¢,} be a sequence of independent random variables having values ina separable real Banach space, E whose distributions are symmetric. Letting S; = ar Ci,suppose {Sn } converges a.e. Also suppose that for every m > nx,P({|ISn—Snelle>2*]) <2. (59.15.29)