3 Martingales Some very useful limit theorems pertain to martingale sequence Let iXt, t T be a stochastic process defined on(S, F, P() and let Ft be a sequence of a-fields Ft C F for all t(i.e. Ft is an increasing sequence of g-fields)satisfying the following conditions (i). X, is a random variable relatives to Ft for all tE T (ii). E(LXtD)<oo for all tE T (iii. E(XLIFt-1)=Xt-1, for alltET Then (Xt, tET is said to be a martingale with respect to Ft, tET) Example(of increasing sequence of a-fields Define the function X-the number of heads", then X(HH=2, X(THD) 1, X(HTD)=l, and X(TT))=0. Further we see that X-(2)=I(HH) X-(1)=I(TH), (HT)) and X-(0)=i(Tr). In fact, it can be shown that the o-field related to the random variables, X, so defined is F={S,,{(HH)},{(①T)},{(TH),(HT)},{(HH),(TT)}, {(HT),(H),(HH)},{(HT),(TH),(TT)} We further define the function X1-at least one head", then XI(HHI XI(THD=XIHT))=l, and X1(TT=0. Further we see that XI (1) [(HH), (TH), (HT))E F and X-(0)=I(TT)E F. In fact, it can be shown that the a-field related to the random variables, X1, so defined is 万1={S,0,{(HH),(TH),(HT)},{(TT)} Finally we define the function X2-"two heads", then X2(HH=1, X2(TH)) 2(HTD= X2(TT)=0. Further we see that X2(1)=I(HHIE F, X-(0)=I(TH), (HT), (TT)) E F. In fact, it can be shown that the a-field related to the random variables, X2, so defined is 2={S,0,{(HH)},{(HT),(TH),(TT)}3 Martingales Some very useful limit theorems pertain to martingale sequence. Definition: Let {Xt ,t ∈ T } be a stochastic process defined on (S, F, P(·)) and let {Ft} be a sequence of σ − fields Ft ⊂ F for all t (i.e.{Ft} is an increasing sequence of σ − fields) satisfying the following conditions: (i). Xt is a random variable relatives to {Ft} for all t ∈ T . (ii). E(|Xt |) < ∞ for all t ∈ T . (iii). E(Xt |Ft−1) = Xt−1, for all t ∈ T . Then {Xt ,t ∈ T } is said to be a martingale with respect to {Ft ,t ∈ T }. Example (of increasing sequence of σ − fields): Define the function X—”the number of heads”, then X({HH}) = 2, X({T H}) = 1, X({HT}) = 1, and X({TT}) = 0. Further we see that X −1 (2) = {(HH)}, X−1 (1) = {(T H),(HT)} and X−1 (0) = {(TT)}. In fact, it can be shown that the σ − field related to the random variables, X, so defined is F = {S, ∅, {(HH)}, {(TT)}, {(T H),(HT)}, {(HH),(TT)}, {(HT),(T H),(HH)}, {(HT),(T H),(TT)}}. We further define the function X1—”at least one head”, then X1({HH}) = X1({T H}) = X1({HT}) = 1, and X1({TT}) = 0. Further we see that X −1 1 (1) = {(HH),(T H),(HT)} ∈ F and X−1 (0) = {(TT)} ∈ F. In fact, it can be shown that the σ − field related to the random variables, X1, so defined is F1 = {S, ∅, {(HH),(T H),(HT)}, {(TT)}}. Finally we define the function X2—”two heads”, then X2({HH}) = 1, X2({T H}) = X2({HT}) = X2({TT}) = 0. Further we see that X −1 2 (1) = {(HH)} ∈ F, X−1 (0) = {(T H),(HT),(TT)} ∈ F. In fact, it can be shown that the σ − field related to the random variables, X2, so defined is F2 = {S, ∅, {(HH)}, {(HT),(T H),(TT)}}. 10