extbf {Last Time} extbf {Branching Processes} extbf {Interpretation} extbf {Martingale Properties} extbf {Probability of Extinction} extbf {Probability of Extinction: Sub-critical Case} extbf {Probability of Extinction: Critical Case}Last Time•Doob decomposition (discrete time)•Doob-Meyer decomposition (continuous time)•Quadratic variation of stochastic process•Quadratic variation of Brownian motion•Martingale characterization of Brownian motionToday’s lecture: Section 4.6MATH136/STAT219 Lecture 22, November 14, 2008 – p. 1/7Branching Processes•Let N and N(n)jbe i.i.d. non-negative integer valued RV’swith finite mean m = IE(N) < ∞.•A Branching Process is a discrete time SP {Zn} takingnonnegative integer values such that Z0= 1 and for anyn = 1, 2, . . .Zn=Zn−1Xj=1N(n)j,•We take Zn+1= 0 if Zn= 0MATH136/STAT219 Lecture 22, November 14, 2008 – p. 2/7Interpretation•Znrepresents the size of the n-th generation of somepopulation•N(n)jrepresents the number of offspring of the j-thindividual of the (n − 1)-st generation•m = IE(N) is mean number of offspring for each individualMATH136/STAT219 Lecture 22, November 14, 2008 – p. 3/7Martingale Properties•Let Fn= σ(N(k)j, k ≤ n, j = 0, 1, 2, . . .)•Then IE(Zn+1|Fn) = mZn. In particular,◦IE(Zn) = mn◦If m = 1, {(Zn, Fn)} is a martingale•{(m−nZn, Fn)} is a martingaleMATH136/STAT219 Lecture 22, November 14, 2008 – p. 4/7Probability of Extinction•Let pexdenote the probability of extinction:pex= IP (Zn= 0 for some n ≥ 0)•Sub-critical process dies off:if m < 1 then pex= 1•Critical process dies off:if m = 1 and IP (N = 1) < 1 then pex= 1•Super-critical process can survive forever:if m > 1 then pex< 1MATH136/STAT219 Lecture 22, November 14, 2008 – p. 5/7Probability of Extinction: Sub-critical Case•Suppose m < 1 and let Xn= m−nZn•Since {Xn} is a nonnegative martingale•By the martingale convergence theorem, there exists arandom variable X∞such thatXn→ X∞a.s. as n → ∞•Since X∞< ∞ a.s. and m < 1, Zn→ 0 a.s.•Since Zntakes integer values, must have Zn= 0 for somen, so pex= 1MATH136/STAT219 Lecture 22, November 14, 2008 – p. 6/7Probability of Extinction: Critical Case•Suppose m = 1 and IP (N = 1) < 1•Then {Zn} is a martingale and as before there exists arandom variable Z∞such thatZn→ Z∞a.s. as n → ∞•Proof of Proposition 4.6.5 shows that Z∞= 0 a.s. andpex= 1•Note that since IE(Zn) = 1 for all n, Zndoes not converge inL1.•Thus, {Zn} is not uniformly integrableMATH136/STAT219 Lecture 22, November 14, 2008 – p.