Lecture 18: Poisson Random Variables and Stable Laws, Slides of Mathematics

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18.175: Lecture 18

Poisson random variables

Scott Sheffield

MIT

18.175 Lecture 18

Outline

Extend CLT idea to stable random variables

18.175 Lecture 18

Recall continuity theorem

� (^) Strong continuity theorem: If μ n =⇒^ μ∞ then

φn(t) → φ∞(t) for all t. Conversely, if φn(t) converges to a

limit that is continuous at 0, then the associated sequence of

distributions μn is tight and converges weakly to a measure μ

with characteristic function φ.

18.175 Lecture 18

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Recall CLT idea

Let X be a random variable.

The characteristic function of X is defined by

itX φ(t) = φX (t) := E [e ].

(m) And if X has an mth moment then E [X

m ] = i

m φ (0). X

In particular, if E [X ] = 0 and E [X

2 ] = 1 then φX (0) = 1 and

φ

x (0) = 0 and φ

xx (0) = −1. X X

Write LX := − log φX. Then LX (0) = 0 and

L

x (0) = −φ

x (0)/φX (0) = 0 and X X

L

xx = −(φ

xx (0)φX (0) − φ

x (0)

2 )/ φX (0)

2 = 1. X X X

− 1 / 2 n If Vn = n i=

Xi where Xi are i.i.d. with law of X , then

LV

n

(t) = nLX (n

− 1 / 2 t).

When we zoom in on a twice differentiable function near zero √

(scaling vertically by n and horizontally by n) the picture

looks increasingly like a parabola.

18.175 Lecture 18

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Stable laws

Example: Suppose that P(X 1 > x) = P(X 1 < −x) = x

−α / 2

for 0 < α < 2. This is a random variable with a “power law

tail”.

Compute 1 − φ(t) ≈ C |t|

α when |t| is large.

If X 1 , X 2 ,... have same law as X 1 then we have l

E exp(itSn/n

1 /α ) = φ(t/n

α )

n = 1 − (1 − φ(t/n

1 /α )). As

n → ∞, this converges pointwise to exp(−C |t|

α ).

1 /α Conclude by continuity theorems that Xn/n =⇒ Y where

Y is a random variable with φY (t) = exp(−C |t|

α )

Let’s look up stable distributions. Up to affine

transformations, this is just a two-parameter family with

characteristic functions exp[−|t|

α (1 − iβsgn(t)Φ)] where

Φ = tan(πα/2) where β ∈ [− 1 , 1] and α ∈ (0, 2].

18.175 Lecture 18

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Stable-Poisson connection

Let’s think some more about this example, where

P(X 1 > x) = P(X 1 < −x) = x

−α / 2 for 0 < α < 2 and

X 1 , X 2 ,... are i.i.d.

1 /α 1 −α − b

−α )n

− 1 Now P(an < X 1 < bn

1 α = (a. 2

So {m ≤ n : Xm/n

1 /α ∈ (a, b)} converges to a Poisson

distribution with mean (a

−α − b

−α )/2.

More generally {m ≤ n : Xm/n

1 /α ∈ (a, b)} converges in law n α to Poisson with mean 2 |x| A α+1^ dx^ <^ ∞.

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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 Poisson random variables)? What are their

characteristic functions like?

More general constructions are possible via L´evy Khintchine

representation.

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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.

18.175 Lecture 18

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