effect of truncation on heavy-tailed models ect of truncation on heavy-tailed models arijit...

Effect ofTruncation onHeavy-tailed

Models

Arijit ChakrabartyCornell University

Introduction

Central LimitTheorem

Large Deviation

Comparison

Effect of Truncation on Heavy-tailed Models


Joint work with Gennady Samorodnitsky

June 23, 2009


Models


Introduction


Large Deviation

Comparison

Introduction

There are situations where heavy-tail has proven to be agood fit for the underlying distribution and at the same timethere is a natural upper bound for possible values.

I How to resolve this apparent paradox?Natural model: Truncated Heavy-tail.

I How does the upper bound effect the asymptotics?

I How to decide if the upper bound is “large enough”?


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

The setup

I B separable Banach space.

I ρ α-stable probability measure on B, α ∈ (0, 2).

I (H,H1,H2, . . .) are i.i.d. B-valued random variables inthe domain of attraction of ρ.

I (L, L1, L2, . . .) are i.i.d. [0,∞) valued random variableswith EL2 <∞.

I The families (H,H1,H2, . . .) and (L, L1, L2, . . .) areindependent.

I Xnj = Hj1{‖Hj‖≤Mn} +Hj

‖Hj‖(Mn + Lj)1{‖Hj‖>Mn}where Mn is a sequence of positive numbers going to∞.

I Sn :=∑n

j=1 Xnj .


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

Notations

I µ denotes the Levy measure of ρ.

I σ denotes the normalized spectral measure of ρ.


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

Soft Truncation

Theorem (C.-Samorodnitsky)

If Mn grows fast enough so that

P(‖H‖ > Mn)� n−1

thenb−1n (Sn − an) =⇒ ρ

as n −→∞ . Here {an} and {bn} are such that

b−1n

n∑j=1

Hj − an

=⇒ ρ .


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

Hard Truncation

Mn satisfiesP(‖H‖ > Mn)� n−1

andMn � 1 .


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

A one-dimensional result


For every f ∈ B ′

B−1n (f (Sn)− Ef (Sn)) =⇒ N

(0,

2

2− α

∫S

f 2(s)σ(ds)

),

whereBn := [nM2

nP(‖H‖ > Mn)]1/2 .

Corollary

If B = Rd ,B−1

n (Sn − ESn) =⇒ Nd(0,Σ)

where

Σij =2

2− α

∫S

si sjσ(ds) .


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

The genaral small ball criterion

Theorem (Ledoux-Talagrand)

X ∈ L2(B) satisfies CLT if and only if for every ε > 0,

(small ball criterion) lim infn→∞

P(‖Sn − ESn‖ < ε√

n) > 0,

where

Sn :=n∑

i=1

Xi

and X1,X2, . . . are i.i.d. copies of X .


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

The small ball criterion


There is a Gaussian measure γ on B such that

B−1n (Sn − ESn) =⇒ γ

if and only if the following hold:

1. (small ball criterion) For every ε > 0

lim infn→∞

P(B−1n ‖Sn − ESn‖ < ε) > 0 ,

2. supn≥1 B−1n E‖Sn − ESn‖ <∞ .

In this case, the characteristic function of γ is given by

γ(f ) = exp

(− 2

2− α

∫S

f 2(s)σ(ds)

), f ∈ B ′ .


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

Type 2 spaces

DefinitionB is said to be of type p if there is Cp ∈ (0,∞) so that foridependent zero mean X1, . . . ,XN ,

E

∥∥∥∥∥∥N∑

j=1

Xj

∥∥∥∥∥∥p

≤ Cp

N∑j=1

E‖Xj‖p .

Theorem (Reference: Araujo-Gine)

Suppose B is a Banach space of type 2. Then everyX ∈ L2(B) satisfies CLT, ie, there is a Gaussian measure γsuch that

n−1/2n∑

i=1

[Xi − E (X )] =⇒ γ .

Conversely, if B is a Banach space where every X ∈ L2(B)satisfies the CLT, then B is of type 2.


Models


Introduction


Soft Truncation

Hard Truncation

Large Deviation

Comparison

Type 2 spaces (contd.)


If B is of type 2, then there is a Gaussian measure γ on Bsuch that

B−1n (Sn − ESn) =⇒ γ .

The characteristic function of γ is given by

γ(f ) = exp

(− 2

2− α

∫S

f 2(s)σ(ds)

), f ∈ B ′ .


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Large Deviations in Hard Truncation

Assume:

I B = Rd .

I If α > 1, EH = 0. If α = 1, H is symmetric.

I EeεL <∞ for some ε > 0.

I Mn positive sequence with

lim Mn = ∞and lim nP(‖H‖ > Mn) = ∞ .


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Large Deviations


Sn/{nMnP(‖H‖ > Mn)} follows LDP with speednP(‖H‖ > Mn) and rate function Λ∗ where Λ is given by

Λ(λ) :=

∫{‖x‖≤1}

(e〈λ,x〉 − 1

)ν(dx)

if 0 < α < 1∫{‖x‖≤1}

(e〈λ,x〉 − 1− 〈λ, x〉

)ν(dx)

if α = 1∫{‖x‖≤1}

(e〈λ,x〉 − 1− 〈λ, x〉

)ν(dx)− α

α−1

∫S〈λ, s〉σ(ds)

if 1 < α < 2

and ν is defined by

ν(A) :=µ(A ∩ B1)

µ(Bc1 )

+ σ(A ∩ S) .


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Moderate Deviations


If n1/2MnP1/2(‖H‖ > Mn)� an � nMnP(‖H‖ > Mn), thena−1n (Sn − ESn) follows LDP with speed

a2n/{nM2

nP(‖H‖ > Mn)} and rate Λ∗ where

Λ(λ) :=1

2〈λ,Dλ〉

and D is the d × d matrix with

Dij :=α

2− α

∫S

si sjσ(ds) .

If, in addition, D is invertible, then Λ∗ is given by

Λ∗(x) =1

2〈x ,D−1x〉 .


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Large Deviations in Soft Truncation

I B = Rd .

I If α > 1, EH = 0. If α = 1, H is symmetric.

I Mn positive sequence with

lim nP(‖H‖ > Mn) = 0 .


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Large Deviations for the untruncated case

Theorem (Hult et al.)

Suppose X ,X1,X2, . . . are i.i.d. Rd -valued random variablesin the domain of attraction of some α-stable distribution(0 < α < 2) with Levy measure µ. Define

Sn :=n∑

i=1

Xi .

Then for any sequence (λn) increasing to ∞ such that

λ−1n Sn

P−→ 0,

[nP(‖X‖ > λn)]−1P(λ−1n Sn ∈ ·)

v−→ µ(·)µ(Bc

1 ).


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Large Deviations


If bn � xn � Mn, then

P(x−1n Sn ∈ ·)

nP(‖H‖ > xn)v−→ µ(·)

µ(Bc1 )

on Rd \ {0}.


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Large Deviations


Suppose k ≥ 1 and that P(L > x) = o(P(‖H‖ > x)k−1) asx −→∞. Then, as n −→∞,

P(M−1n Sn ∈ ·)

{nP(‖H‖ > Mn)}kv−→ 1

k!νk

on Rd \ Bk−1, where

νk(A) :=

∫· · ·∫

1

k∑j=1

xj ∈ A

ν(dx1) . . . ν(dxk)

and

ν(A) :=µ(A ∩ B1)

µ(Bc1 )

+ σ(A ∩ S) .


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Large Deviations


(The boundary case: k = 1) For σ-continuous A ⊂ S, asn −→∞,

P

(‖Sn‖ > Mn,

Sn

‖Sn‖∈ A

)∼

nP(‖H‖ > Mn)

∫A

P(〈x , ρ〉 > 0)σ(dx) .


Models


Introduction


Large Deviation

Hard Truncation

Soft Truncation

Comparison

Large Deviations


(The boundary case: k ≥ 2, when σ has atoms) Assumethat for every s with σ({s}) > 0,

limt→∞

P(‖H‖ > t, H

‖H‖ = s)

P(‖H‖ > t)= σ({s}) .

Suppose k ≥ 2 and P(L > x) = o(P(‖H‖ > x)k−1). Then,for σ-continuous A ⊂ S,

P

(‖Sn‖ > kMn,

Sn

‖Sn‖∈ A

)∼

{nP(‖H‖ > Mn)}k 1

k!

∑s∈A

P(〈s, ρ〉 ≥ 0)σ({s})k .


Models


Introduction


Large Deviation

Comparison

Comparison

limit law large deviation probability

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effect of truncation on heavy-tailed models ect of truncation on heavy-tailed models arijit...

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