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

Convergence to stable laws in the space D cois Roueff 1 Philippe Soulier 2 Fran¸ Poitiers, November 9th, 2012 1 T´ el´ ecom Paris Tech 2 Universit´ e Paris Ouest

Motivation: non linear functionals of shot-noise processes Let X be a shot-noise type process: � X ( t ) = W k 1 { Γ k ≤ t < Γ k + η k } k ∈ Z where ◮ { Γ k } are the points of a point process on R with intensity λ > 0; ◮ { ( W k , η k ) } 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 Γ k are the points of a homogenous Poisson point process; W k ≥ 0 is the transmission rate of the k -th connection and η k is the duration of the k -th connection. ◮ The renewal-reward process: Γ k +1 − Γ k = η k . ◮ The error duration process (Parke 1999): Γ k = k . Used to model macro-economic quantities such as employment rate.

Long memory If E [ η k ] < ∞ , the number of active sources at any time is a.s. finite and these models have stationary versions. If E [ W 2 η ] < ∞ , then the process X is also weakly stationary and can exhibit second order long memory. For the infinite source Poisson process, cov ( X (0) , X ( t )) = λ E [ 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 ) t 2 H − 2 . with H = 3 − α . 2

Weak convergence Under an assumption of asymptotic independence on the joint distribution of ( W , η ), � Tt a − 1 { X ( s ) − E [ X (0)] } d s ⇒ Λ( t ) T 0 with P ( η > a T ) ∼ 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 M 1 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, N ( T ) � T � X ( s ) d s = W k η k + edge effects . 0 k =1 The discrete sum, suitably centered and normalized converges to a stable law. Also � T · � a − 1 d M 1 { X ( s ) − E [ X ( s )] } d s , T 0  N ( T · ) a − 1 �  → 0 . { W k η k − E [ W k η k ] } N k =1

Bounded non linear functionals Let φ be a bounded function and consider � T S ( φ ) = φ ( X ( s )) d s . 0 Example: the empirical distribution function: � T E T ( x ) = 1 { X ( s ) ≤ x } d s . 0

Yet another stable limit! Theorem Roueff et al. (2012) Consider the infinite source Poisson process. � Tt a − 1 { φ ( X ( s )) − E [ φ ( X (0))] } d s ⇒ Λ φ ( t ) T 0 where Λ φ is a L´ evy stable process and the convergence is in the M 1 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 N ( Tt ) � C j +1 � φ ( X ( s )) d s = φ ( X ( s )) d s + edge effects . 0 C j j =1 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 M 1 topology but not in the J 1 topology. The discrete sum converges in the J 1 topology.

The empirical process We can apply the previous result to obtain the fi.di. convergence of � T a − 1 T { E T ( x ) − F X ( x ) } = a − 1 { 1 { X ( s ) ≤ x } − F X ( x ) } d s T 0 to a stable process Λ( x ).

Tightness � C j +1 Define W i ( x ) = 1 { X ( s ) ≤ x } d s . Then, as previously, C j N ( T ) � T a − 1 { 1 { X ( s ) ≤ x } − F X ( x ) } ∼ a − 1 � { W i ( x ) − E [ W i ( x )] } . T T 0 i =1 We must prove the J 1 tightness of the sequences of random elements of D ( R ): n S n = a − 1 � { W i − E [ W i ] } , n i =1 where { W i } is an i.i.d. sequence of D ( R )-valued random variables. But how?

IDEA!? For finite dimensional random variables the convergence of S n is equivalent to the regular variation of the distribution of W 1 . 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 J 1 -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 X ∈ A ) = x − α ν ( A ) n →∞ n P ( � X � ∞ > a n x , lim ( ∗ ) � X � ∞ 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 { X i } be i.i.d. random elements of D with the same distribution as X . Let a n be such that P ( � X � ∞ > a n ) ∼ n − 1 and define S n = a − 1 � n i =1 { X i − E [ X i ] } . n The regular variation of the finite dimensional distributions with index α ∈ (1 , 2) is necessary and sufficient for the finite dimensional distributions of S n 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 J 1 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, n →∞ P ( � S <ǫ η → 0 lim sup lim n � ∞ > η ) = 0 ( ∗∗ ) with S <ǫ = a − 1 � n i =1 { X i 1 {� X � ∞ ≤ a n ǫ } − E [ X i 1 {� X � ∞ ≤ a n ǫ } ] } . n n

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, n →∞ P ( � S <ǫ η → 0 lim sup lim n � ∞ > η ) = 0 ( ∗∗ ) with S <ǫ = a − 1 � n i =1 { X i 1 {� X � ∞ ≤ a n ǫ } − E [ X i 1 {� X � ∞ ≤ a n ǫ } ] } . n n Then S n converges weakly in ( D , J 1 ) to the α -stable process ∞ � { Γ − 1 /α W i − E [Γ − 1 /α Z = ] E [ W i ] } i i i =1 where Γ i are the points of a unit rate Poisson process, { W i } is an i.i.d. sequence with distribution ν and the convergence of the series is almost sure in ( D , J 1 ).

Proof The proof follows the usual path.

Proof The proof follows the usual path. (1) Define N n = � n i =1 δ � Xi �∞ . The weak convergence of N n to a an Poisson point process N on D with mean measure x − α ν ( · ) is equivalent to the regular variation condition ( ∗ ). de Haan and Lin (2001).

Proof The proof follows the usual path. (1) Define N n = � n i =1 δ � Xi �∞ . The weak convergence of N n to a an Poisson point process N on D with mean measure x − α ν ( · ) 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, � ∞ � S ywN n ( d y , d w ) converges weakly in D to ǫ � ∞ i =1 Γ − 1 /α S ywN ( d y , d w ) = � ∞ � Z ǫ = ≤ ǫ } W i . 1 { Γ − 1 /α ǫ i i

Proof The proof follows the usual path. (1) Define N n = � n i =1 δ � Xi �∞ . The weak convergence of N n to a an Poisson point process N on D with mean measure x − α ν ( · ) 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, � ∞ � S ywN n ( d y , d w ) converges weakly in D to ǫ � ∞ i =1 Γ − 1 /α S ywN ( d y , d w ) = � ∞ � Z ǫ = ≤ ǫ } W i . 1 { Γ − 1 /α ǫ i i (3) Z ǫ − E [ Z ǫ ] converges in D to the stable process Z . The proof of this fact uses the negligibility assumption ( ∗∗ )

Proof continued (4) Define n � S >ǫ = { X i 1 {� X i � ∞ > a n ǫ } − E [ X i 1 {� X i � ∞ > a n ǫ } ] } n i =1 � ∞ �� ∞ � � � = ywN n ( d y , d w ) − E ywN n ( d y , d w ) . ǫ S ǫ S Then S >ǫ converges to Z ǫ − E [ Z ǫ ]. n (5) Finally, since S n = S >ǫ + S <ǫ n , the negligibility assumption n ( ∗∗ ) yields that S n converges to Z .