Theory of Probability Notes These are notes made in preparation for oral exams involving the following topics in probability : Random walks , Martingales , and Markov Chains . Textbook used :“Probability : Theory and Examples ," Durrett . Durrett Chapter 4 (Random Walks ). d Random Walk : Let X1,X2,q be iid taking values in R and let Sn : X1 +q+Xn. Sn is a random walk. This section is concerned with the properties of the sequence S1c,S2c,q Stopping Times P Z Stopping Time : Let E, FFF,F FFFn n0, be a filtered probability space. A stopping time T : E + W +K is a random variable such that T n 5 FFFn for all n 0, or equivalently, T : n 5 FFFn for all n 0. g The constant times (e.g., T è 10) are always stopping times. R g Let Xn n0 be an adapted process. Fix A 5 BBB . Then the first entry time into A, notated as TA :: inf n 0 : Xn 5 A, with the convention that inf 2 : +K is a stopping time. Associated with each stopping time N is a ]-field FFFN : information known at time N. FFFN is the collection of sets A that have A V N : n 5 FFFn, -n 9 K. When N : n, A must be measurable with respect to information known at time n. P Stopping Times Lemma : Let S,T,Tn n0 be stopping times on the filtered probability space E, FFF,F FFFn n0, . Then the following are stopping times: i S T T :: min S,T, ii S U T :: max S,T, iii S + T, iv lim inf n Tn and inf n Tn, v lim sup n Tn and sup n Tn. c c c Proof of i) S T T n : S n V T n 5 FFFn. n Notational Conventions for Measurable Space S, SSS : Assuming the random sequence S1c,S2c,q defined above, Let E :: c1,c2,q : ci 5 S, FFF :: SSS ≥ SSS ≥q, P :: S ≥ S ≥q where S is the distribution of Xi, and Xnc :: cn. Finite Permutation of N: A map Z from N onto N so that Zi é i for only finitely many i. Permutable Event A: An event A 5 FFF is permutable if Z?1A è c : Zc 5 A : A for any finite permutation Z. n Symmetric Function : A function f : R R is said to be symmetric if fx1,x2,q,xn : fxZ1,xZ2,q,xZn for each x1,q,xn 5 R and for each permutation Z of 1,2,q,n. n Exchangeable ]-field L: Let X1,X2,q be a sequence of r.v.s on E, FFF,F P. Let Fn :: f : R R symmetric m’ble , K and Ln :: ]Fn,Xn+1,Xn+2,q . The exchangeable ]-field L is defined as L :: Vn:1 Ln. The next results shows that for an iid sequence, there is no difference between L and TTT, they are both trivial. Hewitt Savage 0-1 Law (Gen. of Kolmogorov 0-1) D4.1.1: Let L be the exchangeable ]-field of iid X1,X2,q Then PA 5 0,1 for any A 5 L. Random Walk Possibilities (Consequence of the HSL) D4.1.2: For random walks on R, there are 4 possibilities, one of whichhasprobability1. i) Sn : 0 for all n, ii/iii) Sn çK, iv) ?K : lim inf Sn 9 lim sup Sn : K. r r Proof : Theorem D4.1.1 implies lim sup Sn is a constant c 5 ?K,K. Let Sn : Sn+1 ? X1. Since Sn has the same distribution as Sn, it follows that c : c ? X1. If c is finite, subtracting c from both sides we conclude X1 è 0 and i) occurs. Turning the last statement around, we see that if X1 is not equivalently zero, then c : ?K or K. The same analysis applies to the lim inf. Discarding the impossible combination lim sup Sn : ?K and lim inf Sn : +K, we have proved the results. n Exercises D4.1.1-7 Theorem D4.1.3: Let X1,X2,q be iid, FFFn : ]X1,q,Xn and T be a stopping time with PT 9 K 0. Conditional on T 9 K, XT+n, n 1 is independent of FFFT and has the same distribution as the original sequence. 4/22/2020 JodinMorey 1 Tn 1 Tn 1 Let T a stopping time, T0 :: 0, Tnc :: Tn?1c + TO ? c for n 1, and tnc :: TO ? c, we now extend D4.1.3: P Theorem D4.1.4: Suppose T 9 K : 1. Then the "random vectors" Vn : tn,XTn?1+1,q,XTn are independent and identically distributed. Proof : It is clear from theorem D4.1.3 that Vn and V1 have the same distribution. The independence also follows from theorem D4.1.3 and induction since V1,q,Vn?1 5 FFFTn?1 . n Exercises D4.1.8-11 Wald ’s Identity : Let U1,U2,q be iid with finite mean S :: EUn . Set U0 and let Sn : U1 +q+Un. Let T be a stopping time with ET 9 K. Then, EST : SET. Exercises D4.1.12-14 2 2 Theorem D4.1.6 (Wald ’s 2nd Equation ): Let U1,U2,q be iid with S :: EUn : 0, EUn :: ] 9 K, and 2 2 Sn : U1 + U2 +q. If T is a stopping time with ET 9 K, then EST : ] ET. 2 Theorem D4.1.7 (used in the proof of D4.1.6): Let X1,X2,q be iid with EXn : 0 and EXn : 1, and let 1 9 K for c 9 1, Tc : inf n 1 : |Sn | cn 2 , then ETc : K for c 1. E X2 Lemma D4.1.8: If T is a stopping time with ET : , then TTn 0. K ETTn Recurrence Recurrent Value : x 5 S is considered recurrent if, for every L 0, we have P |Sn ? x| 9 L i.o. : 1. Possible Value (of random walk ): x 5 S is a possible value if, for any L 0, 0n such that P|Sn ? x| 9 L 0. Let VVV be the set of all recurrent values, and UUU be the set of all possible values, then: Theorem D4.2.1: VVV is either 2 or a closed a subgroup of Rd. If VVV is a closed subset, then VVV : UUU. Transient /Recurrent : If VVV : 2, the random walk is said to be transient, otherwise it is called recurrent. Let 0 : 0 and n : inf m n?1 : Sm : 0 be the time of the nth return to 0. Theorem D4.2.2: For any random walk, the following are equivalent: P P K P i) 1 9 K : 1, ii) Sm : 0 i.o. : 1, and iii) >m:0 Sm : 0 : K. P P P K K Proof : If 1 9 K : 1, then n 9 K : 1 for all n and Sm : 0 i.o. : 1. Let V :: >m:0 1Sm:0 : >n:0 1 n9K be the number of visits to 0, counting the visit at time 0. Taking expected value and using Fubini’s theorem to put the expected K K K n 1 value inside the sum: EV : PSm : 0 : P n 9 K : P 1 9 K : . The 2nd equality shows >m:0 >n:0 >n:0 1?P 19K that ii “ iii and, in combinations with the last two, shows that if i) is false, then iii) is false (i.e.,iii “ i ). n d Theorem D4.2.3 (6.33): The Simple (lattice) Symmetric Random Walk (SSRW) Sn on Z is recurrent for d : 1,2 and is transient for d 3. Examples 6.27-6.32 2 Exercise 6.34: Consider SSRW on Z. Show that E0T0 : +K. Prove the same for SSRW on Z . Theorem D4.2.6: The convergence (divergence) of BnP|Sn | 9 L for any single value of L 0 is sufficient to determine the transience (recurrence) of Sn. The proof of this theorem uses the following lemmas: K P P K P Lemma D4.2.4 (Gen. of D4.2.2): If >n:1 |Sn | 9 L 9 K, then |Sn | 9 L i.o. : 0. If >n:1 |Sn | 9 L : K, then P |Sn | 9 2L i.o. : 1. K P d K P Lemma D4.2.5: Let m be an integer 2. Then, >n:0 |Sn | 9 mL 2m >n:0 |Sn | 9 L. Sn p Theorem D4.2.7 (Chung -Fuchs Theorem ): Suppose d : 1. If the weak law of large numbers holds in the form n 0, then Sn is recurrent. 4/22/2020 JodinMorey 2 R2 Sn Theorem D4.2.8: If Sn is a random walk in and 1 “ a non-degenerate normal distribution (non-degenerate means n 2 dim support lim n Sn/ n : 2), then Sn is recurrent. Remark : The conclusion is also true if the limit is degenerate, but in that case, the random walk is essentially one-(or zero)-dimensional, and the result follows from the Chung-Fuchs Theorem. Let f :: EeitX j be the characteristic function of one of the steps of the random walk, then we have: 1 Theorem D4.2.9: Let J 0. Sn is recurrent if and only if X d Re dy : K. ?J,J 1?fy 1 Theorem D4.2.10: Let J 0. Sn is recurrent if and only if sup r91 X d Re dy : K. ?J,J 1?rfy The next two lemmas are used in the proof of D4.2.10: Lemma D4.2.11 (Parseval Relation ): Let S and T be probability measures on Rd with characteristic functions f and b.