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18.175: Lecture 13

Infinite divisibility and L´evy processes

Scott Sheffield

MIT

18.175 Lecture 13

Outline

Poisson random variable convergence

Extend CLT idea to stable random variables

Infinite divisibility

Higher dimensional CFs and CLTs

Poisson random variables: motivating questions

I (^) How many raindrops hit a given square inch of sidewalk during a ten minute period?

18.175 Lecture 13

Poisson random variables: motivating questions

I (^) How many raindrops hit a given square inch of sidewalk during a ten minute period? I (^) How many people fall down the stairs in a major city on a given day?

Poisson random variables: motivating questions

I (^) How many raindrops hit a given square inch of sidewalk during a ten minute period? I (^) How many people fall down the stairs in a major city on a given day? I (^) How many plane crashes in a given year? I (^) How many radioactive particles emitted during a time period in which the expected number emitted is 5?

Poisson random variables: motivating questions

I (^) How many raindrops hit a given square inch of sidewalk during a ten minute period? I (^) How many people fall down the stairs in a major city on a given day? I (^) How many plane crashes in a given year? I (^) How many radioactive particles emitted during a time period in which the expected number emitted is 5? I (^) How many calls to call center during a given minute?

Poisson random variables: motivating questions

I (^) How many raindrops hit a given square inch of sidewalk during a ten minute period? I (^) How many people fall down the stairs in a major city on a given day? I (^) How many plane crashes in a given year? I (^) How many radioactive particles emitted during a time period in which the expected number emitted is 5? I (^) How many calls to call center during a given minute? I (^) How many goals scored during a 90 minute soccer game? I (^) How many notable gaffes during 90 minute debate?

Poisson random variables: motivating questions

I (^) How many raindrops hit a given square inch of sidewalk during a ten minute period? I (^) How many people fall down the stairs in a major city on a given day? I (^) How many plane crashes in a given year? I (^) How many radioactive particles emitted during a time period in which the expected number emitted is 5? I (^) How many calls to call center during a given minute? I (^) How many goals scored during a 90 minute soccer game? I (^) How many notable gaffes during 90 minute debate? I (^) Key idea for all these examples: Divide time into large number of small increments. Assume that during each increment, there is some small probability of thing happening (independently of other increments).

Bernoulli random variable with n large and np = λ

I (^) Let λ be some moderate-sized number. Say λ = 2 or λ = 3. Let n be a huge number, say n = 10^6. I (^) Suppose I have a coin that comes up heads with probability λ/n and I toss it n times.

Bernoulli random variable with n large and np = λ

I (^) Let λ be some moderate-sized number. Say λ = 2 or λ = 3. Let n be a huge number, say n = 10^6. I (^) Suppose I have a coin that comes up heads with probability λ/n and I toss it n times. I (^) How many heads do I expect to see?

18.175 Lecture 13

Bernoulli random variable with n large and np = λ

I (^) Let λ be some moderate-sized number. Say λ = 2 or λ = 3. Let n be a huge number, say n = 10^6. I (^) Suppose I have a coin that comes up heads with probability λ/n and I toss it n times. I (^) How many heads do I expect to see? I (^) Answer: np = λ. I (^) Let k be some moderate sized number (say k = 4). What is the probability that I see exactly k heads?

18.175 Lecture 13

Bernoulli random variable with n large and np = λ

I (^) Let λ be some moderate-sized number. Say λ = 2 or λ = 3. Let n be a huge number, say n = 10^6. I (^) Suppose I have a coin that comes up heads with probability λ/n and I toss it n times. I (^) How many heads do I expect to see? I (^) Answer: np = λ. I (^) Let k be some moderate sized number (say k = 4). What is the probability that I see exactly k heads? I (^) Binomial formula:( n k

pk^ (1 − p)n−k^ = n(n−1)(n− k2)!... (n−k+1)pk^ (1 − p)n−k^.

18.175 Lecture 13

Bernoulli random variable with n large and np = λ

I (^) Let λ be some moderate-sized number. Say λ = 2 or λ = 3. Let n be a huge number, say n = 10^6. I (^) Suppose I have a coin that comes up heads with probability λ/n and I toss it n times. I (^) How many heads do I expect to see? I (^) Answer: np = λ. I (^) Let k be some moderate sized number (say k = 4). What is the probability that I see exactly k heads? I (^) Binomial formula:( n k

pk^ (1 − p)n−k^ = n(n−1)(n− k2)!... (n−k+1)pk^ (1 − p)n−k^. I (^) This is approximately λ kk! (1 − p)n−k^ ≈ λ kk! e−λ. I (^) A Poisson random variable X with parameter λ satisfies P{X = k} = λ k k! e

−λ (^) for integer k ≥ 0.

18.175 Lecture 13

Probabilities sum to one

I (^) A Poisson random variable X with parameter λ satisfies p(k) = P{X = k} = λ

k k! e −λ (^) for integer k ≥ 0.