Kenneth Kuttler

1908 CHAPTER 59. BASIC PROBABILITY

Lemma 59.15.5 Let E be a real separable Banach space. Assume ξ 1, · · · ,ξ N are indepen-dent random variables having values in E, a separable Banach space which have symmet-ric distributions. Also let Sk = ∑

ki=1 ξ i. Then for any r > 0,

([supk≤N||Sk||> r

])≤ 2P([||SN ||> r]) .

Proof: First of all,

([supk≤N||Sk||> r

])

= P

([supk≤N||Sk||> r and ||SN ||> r

])

([sup

k≤N−1||Sk||> r and ||SN || ≤ r

])

≤ P([||SN ||> r])+P

([sup

k≤N−1||Sk||> r and ||SN || ≤ r

]). (59.15.26)

I need to estimate the second of these terms. Let

A1 ≡ [||S1||> r] , · · · ,Ak ≡[||Sk||> r,

∣∣∣∣S j∣∣∣∣≤ r for j < k

Thus Ak consists of those ω where ||Sk (ω)||> r for the first time at k. Thus[sup

k≤N−1||Sk||> r and ||SN || ≤ r

]= ∪N−1

j=1 A j ∩ [||SN || ≤ r]

and the sets in the above union are disjoint. Consider A j ∩ [||SN || ≤ r] . For ω in this set,∣∣∣∣S j (ω)∣∣∣∣> r, ||Si (ω)|| ≤ r if i < j.

Since ||SN (ω)|| ≤ r in this set, it follows

||SN (ω)||=

∣∣∣∣∣∣∣∣∣∣S j (ω)+

∑i= j+1

ξ i (ω)

∣∣∣∣∣∣∣∣∣∣≤ r

ThusP(A j ∩ [||SN || ≤ r]) (59.15.27)

= P

(∩ j−1

i=1 [||Si|| ≤ r]∩[∣∣∣∣S j

∣∣∣∣> r]∩

[∣∣∣∣∣∣∣∣∣∣S j +

∑i= j+1

ξ i

∣∣∣∣∣∣∣∣∣∣≤ r

])(59.15.28)

1908 CHAPTER 59. BASIC PROBABILITYLemma 59.15.5 Let E be a real separable Banach space. Assume & ,,--- ,& y are indepen-dent random variables having values in E, a separable Banach space which have symmet-ric distributions. Also let Sx = a &€,. Then for any r > 0,P (os ise > 4) < 2P((||Sw|| > 7).k<No( sup ||Sx|| > 4)k<N= P( [apis >rand ||Sy|| > 4)k<N+Pk< P(({|Swl] > aeProof: First of all,sup ||S,|| > rand ||Sv|| <r<N-1sup ||S_|| > rand ||Sy|| <r] }. (69.15.26)k<N-1I need to estimate the second of these terms. LetAy = [|[Sil] > r],-++ An = [l]Sell >S)|| <r for j<k].Thus A, consists of those @ where ||5; (@)|| > r for the first time at k. Thussup ||S«|| > rand ||Sv|| < 1 = UPA; l|Svll <7]k<N-1and the sets in the above union are disjoint. Consider A; [||Sw|| <7]. For @ in this set,||S;(@)|| > 4, ||Si(@)|| <rifi< j.Since ||Sy (@)|| <r in this set, it followsNI|Sv(@)|] = ]/Sj)(@)+ YY §;(@)|| <ri=j+l1ThusP(A; [[ISy|| <7) (59.15.27). N=»(nflsil<dnflisil =o Si+ YE; </|) (59.15.28)i=j+l1