[SOLVED] Math154 homework 7 -strong law and birkhoff p0

MATH154
Homework 7
Strong Law and Birkhoff
Problem 7.1: Let (Ω = [0, eeotte tee ttnoende rdtectt
Lebetgce rdtbnbility trnte. Ctotieed ftd n ≥ 1 tee teqceote an = 1/(nltg(n . Defioe Xn(x = n1[0,an/2](xn − n1[1−an/2,1](xn . Io tteed wtdet, we enve n teqceote tf dnoetm vndinblet tent tnke vnlcet n,−n,0.
n Ceetk tent Xn it n teqceote tf ioeereoeeot dnoetm vndinblet tf zedt meno noe vndinote n/(ltg(n .
b Ceetk tent tee rdttf tf tee wenk lnw tf lndge ocmbedt ttill wtdkt tttent P[Sn/n ≥ ϵ] → 0.
t Vedify tent Pn P[{Xn = n}] eivedget noe ttotlcee tent wite rdtbnbility 1, we enve |Sn/n| ≥ 1/2 iofioitely mnoy tfteo.
e Ctotlcee tent Xn etet ott tntitfy tee ttdtog lnw tf lndge ocmbedt.
Problem 7.2: Use the notes to write down the proof of the maximal ergodic theorem of Hopf. Make sure you understand every step.
Problem 7.3: Use the notes to write down the proof of the Birkhoff ergodic theorem. Make sure you understand every step.
Problem 7.4: Write down a paragraph about the history of Birkhoff’s ergodic theorem. Especially make a connection with Harvard.
Probability Theory
Problem 7.5: Giveo n denl ocmbed α let T : T = R/Z → T be eefioee nt T(x = x + α. A ttotioctct fcottito f : T → R eefioet tt n teqceote tf dnoetm vndinblet Xn(x = f(Tn(x = f(x + nα .
n If teede exittt n ttotioctct fcottito g tcte tent f(x = g(x+α −g(x , we tnll f n ttbtcoendy . Went tno ytc tny nbtct tee gdtwte dnte tf Sn if f it n ttbtcoendy?
b Tee tcm Sn it nltt kotwo nt n Weyl tcm. Attcme f it ttotioctct wite
= 0 noe tent α it iddntitonl. Went etet tee Bidketff edgteit
teetdem tny nbtct Sn/n? t Attcme α it iddntitonl. Ade tee dnoetm vndinblet Xn ioeereoeeot? Ade tee dnoetm vndinblet eettddelntee? Cno ytc cte tee ttdtog lnw tf lndge ocmbedt tt ettimnte Sn? Cno ytc cte tee wenk lnw tf lndge ocmbedt tt ettimnte Sn?
e Lttk cr went enrreot if α ent tee Ditrenotioe rdtredty |α − p/q| ≤ 1/q2 ftd nll dntitonl ocmbedt p/q. (Ao exnmrle it if α it tee gtleeo meno. Teede it n detclt tent nttcdet tent Sn ttnyt btcoeee io teit Ditrenotioe tnte if f it ttotioctct. Fioe tent detclt noe ttnte it.