18.175: Lecture 14 Infinite divisibility and so forth

Transcription

1 Lecture : Lecture 14 Infinite divisibility and so forth Scott Sheffield MIT

2 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

3 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

4 Infinitely divisible laws Say a random variable X is infinitely divisible, for each n, there is a random variable Y such that X has the same law as the sum of n i.i.d. copies of Y.

5 Infinitely divisible laws Say a random variable X is infinitely divisible, for each n, there is a random variable Y such that X has the same law as the sum of n i.i.d. copies of Y. What random variables are infinitely divisible?

6 Infinitely divisible laws Say a random variable X is infinitely divisible, for each n, there is a random variable Y such that X has the same law as the sum of n i.i.d. copies of Y. What random variables are infinitely divisible? Poisson, Cauchy, normal, stable, etc.

7 Infinitely divisible laws Say a random variable X is infinitely divisible, for each n, there is a random variable Y such that X has the same law as the sum of n i.i.d. copies of Y. What random variables are infinitely divisible? Poisson, Cauchy, normal, stable, etc. Let s look at the characteristic functions of these objects. What about compound Poisson random variables (linear combinations of independent Poisson random variables)? What are their characteristic functions like?

8 Infinitely divisible laws Say a random variable X is infinitely divisible, for each n, there is a random variable Y such that X has the same law as the sum of n i.i.d. copies of Y. What random variables are infinitely divisible? Poisson, Cauchy, normal, stable, etc. Let s look at the characteristic functions of these objects. What about compound Poisson random variables (linear combinations of independent Poisson random variables)? What are their characteristic functions like? What if have a random variable X and then we choose a Poisson random variable N and add up N independent copies of X.

9 Infinitely divisible laws Say a random variable X is infinitely divisible, for each n, there is a random variable Y such that X has the same law as the sum of n i.i.d. copies of Y. What random variables are infinitely divisible? Poisson, Cauchy, normal, stable, etc. Let s look at the characteristic functions of these objects. What about compound Poisson random variables (linear combinations of independent Poisson random variables)? What are their characteristic functions like? What if have a random variable X and then we choose a Poisson random variable N and add up N independent copies of X. More general constructions are possible via Lévy Khintchine representation.

10 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

11 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

12 Higher dimensional limit theorems Much of the CLT story generalizes to higher dimensional random variables.

13 Lecture 14 Higher dimensional limit theorems Much of the CLT story generalizes to higher dimensional random variables. For example, given a random vector (X, Y, Z), we can define φ(a, b, c) = Ee i(ax +by +cz).

14 Lecture 14 Higher dimensional limit theorems Much of the CLT story generalizes to higher dimensional random variables. For example, given a random vector (X, Y, Z), we can define φ(a, b, c) = Ee i(ax +by +cz). This is just a higher dimensional Fourier transform of the density function.

15 Lecture 14 Higher dimensional limit theorems Much of the CLT story generalizes to higher dimensional random variables. For example, given a random vector (X, Y, Z), we can define φ(a, b, c) = Ee i(ax +by +cz). This is just a higher dimensional Fourier transform of the density function. The inversion theorems and continuity theorems that apply here are essentially the same as in the one-dimensional case.

16 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

17 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

18 Lecture 14 Exchangeable events Start with measure space (S, S, µ). Let Ω = {(ω 1, ω 2,...) : ω i S}, let F be product σ-algebra and P the product probability measure.

19 Lecture 14 Exchangeable events Start with measure space (S, S, µ). Let Ω = {(ω 1, ω 2,...) : ω i S}, let F be product σ-algebra and P the product probability measure. Finite permutation of N is one-to-one map from N to itself that fixes all but finitely many points.

20 Exchangeable events Start with measure space (S, S, µ). Let Ω = {(ω 1, ω 2,...) : ω i S}, let F be product σ-algebra and P the product probability measure. Finite permutation of N is one-to-one map from N to itself that fixes all but finitely many points. Event A F is permutable if it is invariant under any finite permutation of the ω i.

21 Exchangeable events Start with measure space (S, S, µ). Let Ω = {(ω 1, ω 2,...) : ω i S}, let F be product σ-algebra and P the product probability measure. Finite permutation of N is one-to-one map from N to itself that fixes all but finitely many points. Event A F is permutable if it is invariant under any finite permutation of the ω i. Let E be the σ-field of permutable events.

22 Exchangeable events Start with measure space (S, S, µ). Let Ω = {(ω 1, ω 2,...) : ω i S}, let F be product σ-algebra and P the product probability measure. Finite permutation of N is one-to-one map from N to itself that fixes all but finitely many points. Event A F is permutable if it is invariant under any finite permutation of the ω i. Let E be the σ-field of permutable events. This is related to the tail σ-algebra we introduced earlier in the course. Bigger or smaller?

23 Hewitt-Savage 0-1 law If X 1, X 2,... are i.i.d. and A A then P(A) {0, 1}.

24 Hewitt-Savage 0-1 law If X 1, X 2,... are i.i.d. and A A then P(A) {0, 1}. Idea of proof: Try to show A is independent of itself, i.e., that P(A) = P(A A) = P(A)P(A). Start with measure theoretic fact that we can approximate A by a set A n in σ-algebra generated by X 1,... X n, so that symmetric difference of A and A n has very small probability. Note that A n is independent of event A n that A n holds when X 1,..., X n and X n1,..., X 2n are swapped. Symmetric difference between A and A n is also small, so A is independent of itself up to this small error. Then make error arbitrarily small.

25 Application of Hewitt-Savage: If X i are i.i.d. in R n then S n = n i=1 X i is a random walk on R n.

26 Application of Hewitt-Savage: If X i are i.i.d. in R n then S n = n i=1 X i is a random walk on R n. Theorem: if S n is a random walk on R then one of the following occurs with probability one:

27 Application of Hewitt-Savage: If X i are i.i.d. in R n then S n = n i=1 X i is a random walk on R n. Theorem: if S n is a random walk on R then one of the following occurs with probability one: Sn = 0 for all n

28 Application of Hewitt-Savage: If X i are i.i.d. in R n then S n = n i=1 X i is a random walk on R n. Theorem: if S n is a random walk on R then one of the following occurs with probability one: Sn = 0 for all n Sn

29 Application of Hewitt-Savage: If X i are i.i.d. in R n then S n = n i=1 X i is a random walk on R n. Theorem: if S n is a random walk on R then one of the following occurs with probability one: Sn = 0 for all n Sn Sn

30 Application of Hewitt-Savage: If X i are i.i.d. in R n then S n = n i=1 X i is a random walk on R n. Theorem: if S n is a random walk on R then one of the following occurs with probability one: Sn = 0 for all n Sn Sn = lim inf S n < lim sup S n =

31 Application of Hewitt-Savage: If X i are i.i.d. in R n then S n = n i=1 X i is a random walk on R n. Theorem: if S n is a random walk on R then one of the following occurs with probability one: Sn = 0 for all n Sn Sn = lim inf S n < lim sup S n = Idea of proof: Hewitt-Savage implies the lim sup S n and lim inf S n are almost sure constants in [, ]. Note that if X 1 is not a.s. constant, then both values would depend on X 1 if they were not in ±

32 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

33 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

34 Stopping time definition Say that T is a stopping time if the event that T = n is in F n for i n.

35 Stopping time definition Say that T is a stopping time if the event that T = n is in F n for i n. In finance applications, T might be the time one sells a stock. Then this states that the decision to sell at time n depends only on prices up to time n, not on (as yet unknown) future prices.

36 Lecture 14 Stopping time examples Let A 1,... be i.i.d. random variables equal to 1 with probability.5 and 1 with probability.5 and let X 0 = 0 and X n = n i=1 A i for n 0.

37 Lecture 14 Stopping time examples Let A 1,... be i.i.d. random variables equal to 1 with probability.5 and 1 with probability.5 and let X 0 = 0 and X n = n i=1 A i for n 0. Which of the following is a stopping time? 1. The smallest T for which X T = The smallest T for which X T { 10, 100} 3. The smallest T for which X T = The T at which the X n sequence achieves the value 17 for the 9th time. 5. The value of T {0, 1, 2,..., 100} for which X T is largest. 6. The largest T {0, 1, 2,..., 100} for which X T = 0.

38 Lecture 14 Stopping time examples Let A 1,... be i.i.d. random variables equal to 1 with probability.5 and 1 with probability.5 and let X 0 = 0 and X n = n i=1 A i for n 0. Which of the following is a stopping time? 1. The smallest T for which X T = The smallest T for which X T { 10, 100} 3. The smallest T for which X T = The T at which the X n sequence achieves the value 17 for the 9th time. 5. The value of T {0, 1, 2,..., 100} for which X T is largest. 6. The largest T {0, 1, 2,..., 100} for which X T = 0.

39 Lecture 14 Stopping time examples Let A 1,... be i.i.d. random variables equal to 1 with probability.5 and 1 with probability.5 and let X 0 = 0 and X n = n i=1 A i for n 0. Which of the following is a stopping time? 1. The smallest T for which X T = The smallest T for which X T { 10, 100} 3. The smallest T for which X T = The T at which the X n sequence achieves the value 17 for the 9th time. 5. The value of T {0, 1, 2,..., 100} for which X T is largest. 6. The largest T {0, 1, 2,..., 100} for which X T = 0. Answer: first four, not last two.

40 Stopping time theorems Theorem: Let X 1, X 2,... be i.i.d. and N a stopping time with N <.

41 Stopping time theorems Theorem: Let X 1, X 2,... be i.i.d. and N a stopping time with N <. Conditioned on stopping time N <, conditional law of {X N+n, n 1} is independent of F n and has same law as original sequence.

42 Stopping time theorems Theorem: Let X 1, X 2,... be i.i.d. and N a stopping time with N <. Conditioned on stopping time N <, conditional law of {X N+n, n 1} is independent of F n and has same law as original sequence. Wald s equation: Let X i be i.i.d. with E X i <. If N is a stopping time with EN < then ES N = EX 1 EN.

43 Stopping time theorems Theorem: Let X 1, X 2,... be i.i.d. and N a stopping time with N <. Conditioned on stopping time N <, conditional law of {X N+n, n 1} is independent of F n and has same law as original sequence. Wald s equation: Let X i be i.i.d. with E X i <. If N is a stopping time with EN < then ES N = EX 1 EN. Wald s second equation: Let X i be i.i.d. with E X i = 0 and EXi 2 = σ 2 <. If N is a stopping time with EN < then ES N = σ 2 EN.

44 Lecture 14 Wald applications to SRW S 0 = a Z and at each time step S j independently changes by ±1 according to a fair coin toss. Fix A Z and let N = inf{k : S k {0, A}. What is ES N?

45 Lecture 14 Wald applications to SRW S 0 = a Z and at each time step S j independently changes by ±1 according to a fair coin toss. Fix A Z and let N = inf{k : S k {0, A}. What is ES N? What is EN?

46 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

47 Lecture 14 Outline Infinite divisibility Higher dimensional CFs and CLTs Random walks Stopping times Arcsin law, other SRW stories

48 Reflection principle How many walks from (0, x) to (n, y) that don t cross the horizontal axis?

49 Reflection principle How many walks from (0, x) to (n, y) that don t cross the horizontal axis? Try counting walks that do cross by giving bijection to walks from (0, x) to (n, y).

50 Lecture 14 Ballot Theorem Suppose that in election candidate A gets α votes and B gets β < α votes. What s probability that A is a head throughout the counting?

51 Lecture 14 Ballot Theorem Suppose that in election candidate A gets α votes and B gets β < α votes. What s probability that A is a head throughout the counting? Answer: (α β)/(α + β). Can be proved using reflection principle.

52 Arcsin theorem Theorem for last hitting time.

53 Arcsin theorem Theorem for last hitting time. Theorem for amount of positive positive time.