# Now assume that sup n n e x n then sup n n e x n

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Now assume that supn∈-NE[|Xn|]<; then supn∈-N|E[Xn]|<.Sincen7→E[Xn] is increasing,limn→-∞E[Xn] exists and is finite. Fixε >0; then there is anN0∈ -Nsuch thatE[Xn]>E[XN]-εifnN. For anyK >0, we calculate that for anynN,E|Xn|χ{|Xn|≥K}=EXnχ{XnK}-EXnχ{Xn≤-K}=EXn(χ{XnK}+χ{Xn>-K})-E[Xn]EXN(χ{XnK}+χ{Xn>-K})-E[XN] +ε/2 =E|XN|χ{|Xn|≥K}+ε/21Taken from Revuz and Yor30
In the last inequality, the first term comes from the submartingale inequality, and the last term comes fromthe choice ofN. By Markov’s inequality,supn∈-NP{|Xn| ≥K} ≤K-1supn∈-NE[|Xn|].SinceXNis integrable, we have thatlimK→∞supnNE|Xn|χ{|Xn|≥K}ε;since all of theXn’s are integrable, we also have thatlimK→∞supN<n0E|Xn|χ{|Xn|≥K}ε;putting these two together, we have that theXn’s are uniformly integrable. Hence, theP-a.s. convergenceofXntoX-∞also holds inL1. For anyAG-∞and anyn, we thus haveE[X-∞χA] =limm→-∞E[XmχA]E[XnχA].The last inequality follows from the submartingale inequality. Thus we get the final claim.We can use this result to easily prove the Strong Law of Large NumbersTheorem0.31 (Strong Law of Large Numbers).Let{ξ1, ξ2. . .}be a collection of independent andidentically distributed integrable random variables with common lawμ. Thenlimn→∞n-1nXk=1ξk=ZR(dx),this limit being both almost-sure and inL1.Proof.SetSndef=nXk=1ξknNFor eachnN, defineG-ndef=σ{Sk;kn}.SetXndef=E[ξ1|Gn].n∈ -NThenXis a martingale. Clearlysupn∈-NE[|Xn|]E[|ξ1|],soX-∞def= limn→-∞Xnexists both inL1andP-a.s. Note that for anyn1 and any 1kn,E[ξk|G-n] =E[ξ1|G-n];thusnXk=1X-n=nXk=1E[ξ1|G-n] =nXk=1E[ξk|G-n] =E[Sn|G-n] =Snso in factX-n=1nSnfor alln1. ThusSn=X-nfor allnN. HenceX-∞= limn→∞Snnthis limit being bothP-a.s. and inL1. We thus only need show thatX-∞=RR(dx). Note that for everyk,X-∞=limn→∞Snn=limn→∞nj=1ξk+jn.31
so by Kolmogorov’s zero-one law,X-∞is almost-surely constant. HenceX-∞=E[X-∞] = limn→∞n-1E[Sn] =ZR(dx)This completes the proof.Exercises(1) Show that for any stopping timeτ,Fτis indeed a sigma-algebra.(2) Show that for any stopping timeτ,τis itselfFτ-measurable.(3) Show that for any fixedtI, the mappingτ: Ω7→tis a stopping time. Show thatFτ=Ft.(4) Let{τ1, τ2. . .}be a countable collection of stopping times. Show that supn1τnis also a stopping time.Show that if{τ1, τ2. . . , τn}is a finite collection of stopping times, then min1knτkis also a stoppingtime. stopping times.

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Term
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Tags
Probability theory, lim, Xn, def