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Hw was due yesterday solutions posted todayHonorow

self grade due ii 3

25 handed un HN all should be able Gunll

Plan todaythen tutti EZ more generally as martingale

Aaaaa Heffding as generalization of Halfdryfor sum of non uhdependent variables

Proof of McDiarmid via Asama Hoeffding

HoeffdayguCZ In 2 Zi EZ where 2 i were E abgaussian RV

independent

z n EEDi

galt O IF Di O

Sg II DiSgt 7 Sg s Dg gigadding

randomness to Sj

S E gutt 0 S i add D

Gutt Sn E Di adding rammodomness sequentiallyi 4

Here all D are wedependent

Sj E gue l Zi ZjElsie I Zi 2jJ ELECgnEEIIZi.ituJ1Zi ZjJ ElgnEH4Z

if El Sj Is in for all j Sj is aSJ

martingale sequenceWrt Ziti adapted to filtration

Dj Sj Sj is a martingale

generated by Ziti

difference sequence

guta.EE Ez EDi PIC

guyEigula the

just by G is so subgaussian candirectly bound

by Hoeffday's inequality P yDie

Nof assume get is an arbitrary function ofindependent 2 i

eamare war 6 h eEt EEEIbe lcaiifl

tUgz Sn So E Sj Sj uFnateF n je

2 Djje

Dj Elgula l Z 2JI E Ign 17 Zji

additional randomness by conditioning on 2J

If Eelguellen Sjfa is a doob

Sig El Cgu It zj3martingale sequence

Edgell za aj J Sj EIS ka

in order to show concentration on Ig i

need some conditions on Di

Ele Ifp j e ei

FIEFfor almost allof Cemil Zizi 4244 9 1 eZ Zin

Di l ti Z 1almost surely bounded

the event where Di IZ 44 E Cai bird

for some aEbi has probability d

Azunaltoeffdingareguality

If martingale difference sequence i Fi Din

satisfies Di IZ Zia E Cai bid for almost

all ti Zit then

CEN Dist ee i aifProof

T.ff.ar.ir is c Cai bi then it s big subganssian

For Ele iz z eeMb4I

at ri

e

a

Eaa big

i

Not If ga Z IR satisfiesh

bounded difference condition then

o.si 9giEssiEe

If

McDianidine quattyFor any ma

c Z defu Ehby27g zj if jelle and oink 2 u

If gu E712 satisfies for any zit

cZ

and all he 4 n we have

1gCal GE'T I E oucbdiffmd.de

u n condition

then Ip Capiz Earth He

Eep

e EEif Z Za are

independent RV

Proof by showingt Azuma Heeffetug

Define the Ezio 2 a

E Cguthlzi I It Rguczi IT because of independerof 7 Zn

i iz'T zi'll IE Lgu Iai Zi Elgin l z't IIM LEE gulti ZiZiel Ezingulai Zazie

mGSEP E g Ei Ziel inzf

Ezii.guizihitiimete.tk'sdoesn'tget f E IgaCt't izZiti gnat a'it pthe tight McDiarmidboundCapitol Earth ee EEFiiq.lguczi gu.tt l Oi

U aw E eczift

Define Gutt µf pRCH Ralf and l is uniformly

atbonded by E

262P Cga

CH Eg.CZt ee

C.FI ftRf2RRtt7eeE go.cz E 272 be

µ E aEZi eczi.fihoax2e 5h leche la f t feel