Convergence to stable laws in the space D cois Roueff 1 Philippe - - PowerPoint PPT Presentation

Convergence to stable laws in the space D

Fran¸ cois Roueff1 Philippe Soulier2 Poitiers, November 9th, 2012

1T´

el´ ecom Paris Tech

2Universit´

e Paris Ouest

Motivation: non linear functionals of shot-noise processes

Let X be a shot-noise type process: X(t) =

• k∈Z

Wk1{Γk≤t<Γk+ηk} where

◮ {Γk} are the points of a point process on R with intensity

λ > 0;

◮ {(Wk, ηk)} is an i.i.d. sequence of random vectors with value

in R × R+.

This type of process encompasses different particular cases:

◮ The infinite source Poisson process: the Γk are the points of a

homogenous Poisson point process; Wk ≥ 0 is the transmission rate of the k-th connection and ηk is the duration of the k-th connection.

◮ The renewal-reward process: Γk+1 − Γk = ηk. ◮ The error duration process (Parke 1999): Γk = k. Used to

model macro-economic quantities such as employment rate.

Long memory

If E[ηk] < ∞, the number of active sources at any time is a.s. finite and these models have stationary versions. If E[W 2η] < ∞, then the process X is also weakly stationary and can exhibit second

• rder long memory. For the infinite source Poisson process,

cov(X(0), X(t)) = λE[W 2(η − t)+] . If the function t → E[W 2(η − t)+] is regularly varying with index α ∈ (1, 2), then X has second order long memory: cov(X(0), X(t)) = L(t)t2H−2 . with H = 3 − α 2 .

Weak convergence

Under an assumption of asymptotic independence on the joint distribution of (W , η), a−1

T

Tt {X(s) − E[X(0)]}ds ⇒ Λ(t) with P(η > aT) ∼ T −1 and Λ is an α-stable L´ evy process. Taqqu and Levy (1986) (renewal reward) Mikosch et al. (2002) Shot noise and On-Off; Maulik et al. (2002); Mikosch and Samorodnitsky (2007), (λ → ∞). The convergence is in Skorohod’s M1 topology, since the limit is

• discontinuous. Resnick and Van den Berg (2000)

Why stable limits for finite variance processes?

Let N(T) denote then number of complete sessions before time T. Then, T X(s)ds =

N(T)

• k=1

Wkηk+ edge effects . The discrete sum, suitably centered and normalized converges to a stable law. Also dM1

• a−1

T

T· {X(s) − E[X(s)]}ds, a−1

N N(T·)

• k=1

{Wkηk − E[Wkηk]}   → 0 .

Bounded non linear functionals

Let φ be a bounded function and consider S(φ) = T φ(X(s))ds . Example: the empirical distribution function: ET(x) = T 1{X(s)≤x}ds .

Yet another stable limit!

Theorem Roueff et al. (2012) Consider the infinite source Poisson process. a−1

T

Tt {φ(X(s)) − E[φ(X(0))]}ds ⇒ Λφ(t) where Λφ is a L´ evy stable process and the convergence is in the M1 topology.

Why stable again?

The paths of the process X can be decomposed into busy and idle

• period. A cycle is comprised of an idle and the subsequent busy
• period. The cycles are i.i.d. The length of a cycle is heavy tailed

Resnick and Samorodnitsky (1997). Tt φ(X(s))ds =

N(Tt)

• j=1

Cj+1

Cj

φ(X(s))ds + edge effects . where N(T) is now the number of complete cycles before time T. Since φ is bounded, the integral over a cycle is big only if the length of a cycle is big. The integral and the discrete sum, suitably centered and normalized, are close in the M1 topology but not in the J1

• topology. The discrete sum converges in the J1 topology.

The empirical process

We can apply the previous result to obtain the fi.di. convergence of a−1

T {ET(x) − FX(x)} = a−1 T

T {1{X(s)≤x} − FX(x)}ds to a stable process Λ(x).

Tightness

Define Wi(x) = Cj+1

Cj

1{X(s)≤x}ds. Then, as previously, a−1

T

T {1{X(s)≤x} − FX(x)} ∼ a−1

T N(T)

• i=1

{Wi(x) − E[Wi(x)]} . We must prove the J1 tightness of the sequences of random elements of D(R): Sn = a−1

n n

• i=1

{Wi − E[Wi]} , where {Wi} is an i.i.d. sequence of D(R)-valued random variables. But how?

IDEA!?

For finite dimensional random variables the convergence of Sn is equivalent to the regular variation of the distribution of W1. Why not use the concept of regularly varying processes introduced by de Haan and Lin (2001) and Hult and Lindskog (2005)?

Regularly varying stochastic processes

Let D = D([0, 1]) be the set of c` adl` ag functions indexed by [0, 1], endowed with Skorohod’s J1-topology. Let S be the subset of functions f ∈ D such that f ∞ = 1. A D-valued random variable X is said to be regularly varying with index α if there exists a probability measure ν on S such that lim

n→∞ nP(X∞ > anx,

X X∞ ∈ A) = x−αν(A) (∗) for all Borel subset A of S. The measure ν is called the spectral measure of the process X.

◮ This definition implies that X∞ is regularly varying with

index α and that the finite dimensional distribution of X are multivariate regularly varying with the same index.

◮ Hult and Lindskog (2005) showed that the functional regular

variation condition (∗) is equivalent to the regular variation of the finite dimensional distributions of the process X and a tightness condition.

Convergence to stable processes

Let X be a regularly varying random variable with values in D and let {Xi} be i.i.d. random elements of D with the same distribution as X. Let an be such that P(X∞ > an) ∼ n−1 and define Sn = a−1

n

n

i=1{Xi − E[Xi]}.

The regular variation of the finite dimensional distributions with index α ∈ (1, 2) is necessary and sufficient for the finite dimensional distributions of Sn to converge to those of a stable process. What about convergence in D? The problem is that D is not a Banach space and not even a topological vector space, i.e. addition is not continuous with respect to the J1 topology.

Theorem

Let X be regularly varying in D with index α ∈ (1, 2) and spectral measure ν.

◮ For all t ∈ [0, 1], ν({w : |w(t) − w(t−)| > 0}) = 0 (no fixed

jump).

◮ For all ǫ > 0,

lim

η→0 lim sup n→∞ P(S<ǫ n ∞ > η) = 0 (∗∗)

with S<ǫ

n

= a−1

n

n

i=1{Xi1{X∞≤anǫ} − E[Xi1{X∞≤anǫ}]}.

Theorem

Let X be regularly varying in D with index α ∈ (1, 2) and spectral measure ν.

◮ For all t ∈ [0, 1], ν({w : |w(t) − w(t−)| > 0}) = 0 (no fixed

jump).

◮ For all ǫ > 0,

lim

η→0 lim sup n→∞ P(S<ǫ n ∞ > η) = 0 (∗∗)

with S<ǫ

n

= a−1

n

n

i=1{Xi1{X∞≤anǫ} − E[Xi1{X∞≤anǫ}]}.

Then Sn converges weakly in (D, J1) to the α-stable process Z =

∞

• i=1

{Γ−1/α

i

Wi − E[Γ−1/α

i

]E[Wi]} where Γi are the points of a unit rate Poisson process, {Wi} is an i.i.d. sequence with distribution ν and the convergence of the series is almost sure in (D, J1).

Proof

The proof follows the usual path.

Proof

The proof follows the usual path. (1) Define Nn = n

i=1 δ Xi ∞

an

. The weak convergence of Nn to a Poisson point process N on D with mean measure x−αν(·) is equivalent to the regular variation condition (∗). de Haan and Lin (2001).

Proof

The proof follows the usual path. (1) Define Nn = n

i=1 δ Xi ∞

an

. The weak convergence of Nn to a Poisson point process N on D with mean measure x−αν(·) is equivalent to the regular variation condition (∗). de Haan and Lin (2001). (2) The assumption of no fixed jump implies that for all ǫ > 0, ∞

ǫ

• S ywNn(dy, dw) converges weakly in D to

Zǫ = ∞

ǫ

• S ywN(dy, dw) = ∞

i=1 Γ−1/α i

1{Γ−1/α

i

≤ǫ}Wi.

Proof

The proof follows the usual path. (1) Define Nn = n

i=1 δ Xi ∞

an

. The weak convergence of Nn to a Poisson point process N on D with mean measure x−αν(·) is equivalent to the regular variation condition (∗). de Haan and Lin (2001). (2) The assumption of no fixed jump implies that for all ǫ > 0, ∞

ǫ

• S ywNn(dy, dw) converges weakly in D to

Zǫ = ∞

ǫ

• S ywN(dy, dw) = ∞

i=1 Γ−1/α i

1{Γ−1/α

i

≤ǫ}Wi.

(3) Zǫ − E[Zǫ] converges in D to the stable process Z. The proof of this fact uses the negligibility assumption (∗∗)

Proof continued

(4) Define S>ǫ

n

=

n

• i=1

{Xi1{Xi∞>anǫ} − E[Xi1{Xi∞>anǫ}]} = ∞

ǫ

• S

ywNn(dy, dw) − E ∞

ǫ

• S

ywNn(dy, dw)

• .

Then S>ǫ

n

converges to Zǫ − E[Zǫ]. (5) Finally, since Sn = S>ǫ

n

+ S<ǫ

n , the negligibility assumption

(∗∗) yields that Sn converges to Z.

What’s new here?

◮ If Xn converges to X in (D, J1), then E[Xn] does not

necessarily converge to E[X]. The no-fixed-jump condition guarantees this convergence.

◮ The weak and almost sure uniform convergence in D of

Zǫ − E[Zǫ] to a stable process is not trivial. The proof of this fact should only use the measure ν, but we know ν only through X. The negligibility condition (∗) is an implicit condition on ν. More explicit conditions are possible.

Related Results

Let {Wi, i ≥ 1} be sequence of i.i.d. random elements in D. Assume that

◮ E[|Wi(t)|α] < ∞ for all t. ◮ There exist a continuous, increasing function F, p1 > α,

p2 > α/2 and γ > 1/2 such that, for s < t < u, E[|W (t) − W (s)|p1] ≤ {F(t) − F(s)}γ , E[|W (t) − W (s)|p2|W (u) − W (t)|p2] ≤ {F(t) − F(s)}2γ .

Related Results

Let {Wi, i ≥ 1} be sequence of i.i.d. random elements in D. Assume that

◮ E[|Wi(t)|α] < ∞ for all t. ◮ There exist a continuous, increasing function F, p1 > α,

p2 > α/2 and γ > 1/2 such that, for s < t < u, E[|W (t) − W (s)|p1] ≤ {F(t) − F(s)}γ , E[|W (t) − W (s)|p2|W (u) − W (t)|p2] ≤ {F(t) − F(s)}2γ . Then the series ∞

i=1{Γ−1/α i

Wi − E[Γ−1/α

i

Wi]} is almost surely uniformly convergent.

◮ Basse-O’Connor and Rosi´

nsi (2011), (to appear in AoP), for symmetric stable processes.

◮ Davydov and Dombry (2012), (Stat. Prob. Letters), under a

slightly different series representation and for p = 2.

(They also assume E[W α

∞] < ∞ which actually is a consequence of the

• ther assumptions).

Questions

If {Wi, i ≥ 1} are i.i.d. random elements in S (i.e. Wi∞ = 1), with distribution ν, then the series ∞

i=1{Γ−1/α i

Wi − E[Γ−1/α

i

Wi]} converge almost surely pointwise. (i) Is the sum in D? (ii) Is the convergence almost surely uniform? (iii) If the answer is positive, is the spectral measure (in the sense

• f regular variation) equal to ν?

Questions

If {Wi, i ≥ 1} are i.i.d. random elements in S (i.e. Wi∞ = 1), with distribution ν, then the series ∞

i=1{Γ−1/α i

Wi − E[Γ−1/α

i

Wi]} converge almost surely pointwise. (i) Is the sum in D? (ii) Is the convergence almost surely uniform? (iii) If the answer is positive, is the spectral measure (in the sense

• f regular variation) equal to ν?

It follows from Theorem 2.1 of Basse-O’Connor and Rosi´ nsi (2011) that a positive answer to (i) implies a positive answer to (ii). The answer to (iii) is conjectured to be positive in that case.

Applications

Let ν be a probability measure on S. Let {Wi} be a sequence of i.i.d. D-valued random variables with distribution ν. Let {Ri} be a sequence of i.i.d. Pareto random variables with index α ∈ (1, 2). Define Sn = n−1/α

n

• i=1

{RiWi − E[Ri]E[Wi]} . The process RW is regularly varying in D since RW ∞ = R and its spectral measure is ν. Does Sn converge to a stable process in D?

A condition on ν is needed to ensure the negligibility in D of the small jumps. If for s < t < u, E[|W (t) − W (s)|p1] ≤ {F(t) − F(s)}γ , E[|W (t) − W (s)|p2|W (u) − W (t)|p2] ≤ {F(t) − F(s)}2γ . where F is continuous and increasing and γ > 1/2, then the negligibility condition (∗∗) holds and Sn converges to a stable process with spectral measure ν in (D, J1).

A condition on ν is needed to ensure the negligibility in D of the small jumps. If for s < t < u, E[|W (t) − W (s)|p1] ≤ {F(t) − F(s)}γ , E[|W (t) − W (s)|p2|W (u) − W (t)|p2] ≤ {F(t) − F(s)}2γ . where F is continuous and increasing and γ > 1/2, then the negligibility condition (∗∗) holds and Sn converges to a stable process with spectral measure ν in (D, J1). Is this condition optimal? What is the weakest possible condition? Is any condition needed at all?

Empirical process of the renewal-reward process

Consider the renewal-reward process X(t) =

• k≥0

Wk+11{Tk≤t<Tk+1} with T0 = 0, Tk = Y1 + · · · + Yk and {(Yk, Wk), k ≥ 1} is an i.i.d. sequence such that lim

n→∞ nP(W ≤ w; Y > anx) = x−αG ∗(W ) ,

for some α ∈ (1, 2) and G ∗ a probability distribution on (−∞, ∞). This is asymptotic independence in the sense of Maulik et al. (2002). This model can be seen as a simplification of the infinite source Poisson process.

The empirical process of X can be expressed as T 1{X(s)≤x}ds =

N(T)

• k=1

Yk1{Wk≤x} + edge effects. If G ∗ is continuous, then the process Y 1{G ∗(W )≤·} is regularly varying in D([0, 1]) with spectral measure ν defined by ν(A) = P(1[U∗,1] ∈ A) where U∗ is uniform in [0, 1]. This is the spectral measure of a L´ evy process Hult and Lindskog (2005).

The empirical process of X can be expressed as T 1{X(s)≤x}ds =

N(T)

• k=1

Yk1{Wk≤x} + edge effects. If G ∗ is continuous, then the process Y 1{G ∗(W )≤·} is regularly varying in D([0, 1]) with spectral measure ν defined by ν(A) = P(1[U∗,1] ∈ A) where U∗ is uniform in [0, 1]. This is the spectral measure of a L´ evy process Hult and Lindskog (2005). Appying our Theorem, we obtain that a−1

T

T {1{X(s)≤x} − F(x)} ⇒ Z ◦ G ∗(x) − G ∗(x)Z(1) . where Z is a L´ evy process on [0, 1].

Questions

◮ Converse: does the convergence in D of

a−1

n

n

i=1{Xi − E[Xi]}, Xi i.i.d., to a stable process imply that

X1 is regularly varying in D?

◮ Is a stable process with c´

adl´ ad path regularly varying?

◮ Empirical process of the infinite source Poisson process.

References

◮ R. Roueff, G. Samorodnitsky and Ph. Soulier,

Function-indexed empirical processes based on an infinite source Poisson transmission stream, Bernoulli, 2012, 18(3), 783-802.

◮ Yu. Davydov and C. Dombry, On the convergence of LePage

series in Skorokhod space, Statistics and Probability Letters, 2012, 82(1), 145-150.

◮ A. Basse-O’Connor and J. Rosi´

nski, On the uniform convergence of random series in Skorohod space and representations of c` adl` ag infinitely divisible processes, 2011, to appear in Annals of Probability. arxiv:1111.1682.