Stable Lévy motion with values in the Skorohod space€¦ · Multivariate Case Regular variation...

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Stable Levy motion with values in theSkorohod space

Raluca BalanUniversity of Ottawa

Joint work with Becem Saidani

Stable Levymotion with values

in the Skorohodspace

Stable FCLT

Multivariate Case

Regular variationin D

D-valued α-stableLevy motion

Functional limittheorem in D

Table of Contents

1 Stable FCLT

2 Multivariate Case

3 Regular variation in D

4 D-valued α-stable Levy motion

5 Functional limit theorem in D

Stable FCLT

Multivariate Case

1. Stable Functional CLT

Theorem 1 (Skorohod, 1957)

(Xi )i≥1 i.i.d. with regularly varying tails of index α ∈ (0, 2):

P(|X | > x) = x−αL(x), limx→∞

P(X > x)

P(|X | > x)= p ∈ [0, 1],

where L is slowly varying at ∞. If Sn(t) =∑[nt]

i=1 Xi , then1

an(Sn(t)− bn(t))

d→ Z (t)t≥0 in D([0,∞)),

where Z (t)t≥0 is an α-stable Levy motion, aαn ∼ nL(an),bn(t) = 0 if α < 1 and bn(t) = E [Sn(t)] if α > 1.

Stable FCLT

Multivariate Case

Properties of α-stable Levy motion:

(i) Z (t2)− Z (t1), . . . ,Z (tn)− Z (tn−1) are independent, forany t1 < t2 < . . . < tn

(ii) Z (t2)− Z (t1)d= Z (t2 − t1) for any t1 < t2

(iii) Z (t) ∼ Sα(t1/ασ, β, 0) with β = p − q, q = 1− p and

σα = Γ(2−α)1−α cos

(πα2

)=: C−1

α . We have:

E (e iuZ(t)) = exp

(e iuy − 1)να,p(dy)

, if α < 1

E (e iuZ(t)) = exp

(e iuy − 1− iuy)να,p(dy)

, if α > 1

να,p(y) =(pαy−α−11y∈(0,∞)+qα(−y)−α−11y∈(−∞,0)

Stable FCLT

Multivariate Case

Idea of proof α < 1 (Resnick, 1986)

1) Point process convergence:

Nn =n∑

δXi/and→ N =

∑i≥1

δJi = PRM(να,p)

follows from nP( Xan∈ ·) v→ να,p in R0 = [−∞,∞]− 0.

2) Continuity of truncated summation: P(N ∈ Λ) = 1∑i≥1

δxi 7→∑i≥1

xi1|xi |>ε continuous on Λ

3) S(ε)n = a−1

∑ni=1 Xi1|Xi |>εan

d→ Z (ε) =∑

i≥1 Ji1|Ji |>ε4) Let ε→ 0.

Stable FCLT

Multivariate Case

Idea of proof (functional case)1) Point process convergence:

Nn =∑i≥1

δ( in,Xian

d→ N =∑i≥1

δ(Ti ,Ji ) = PRM(Leb × να,p)

2) Continuity of truncated summation: the map

Qε :∑i≥1

δ(ti ,xi ) 7→

∑ti≤t

xi1|xi |>ε

t∈[0,T ]

∈ D([0,T ])

is continuous on a set Λ and P(N ∈ Λ) = 1.3) Define

S(ε)n = Qε(Nn) =

[nt]∑i=1

Xi1|Xi |>εan

t∈[0,T ]

Z (ε) = Qε(N) =

∑Ti≤t

Ji1|Ji |>ε

t∈[0,T ]

Stable FCLT

Multivariate Case

By Continuous Mapping Theorem, for any ε > 0,

S(ε)n (·) d→ Z (ε)(·) in D([0,T ])

4) Let ε→ 0:

limε→0

lim supn→∞

P( supt∈[0,T ]

|S (ε)n (t)− Sn(t)| > δ) = 0

supt≤T|Z (ε)(t)− Z (t)| → 0 a.s.

The conclusion follows by Theorem 4.2 of Billingsley (1968).

RemarkA similar argument works when α > 1, using centering.

Stable FCLT

Multivariate Case

2. Multivariate Case

A random vector X in Rd is regularly varying (RV) if

an∈ ·)

v→ ν in Rd0 (1)

where ν is a Radon measure on Rd0 = [−∞,∞]d − 0 with

ν(Rd0 − Rd) = 0. In this case, there exists α > 0 s.t.

ν(cA) = c−αν(A), for any c > 0,A ∈ B(Rd).

Moreover, (1) is equivalent to:

(( |X |an,X

)∈ ·)

v→ cνα × Γ1 in (0,∞]× Sd ,

for some probability measure Γ1 on Sd = x ∈ Rd ; |x | = 1,c > 0 and να(r ,∞) = r−α for r > 0 and να∞ = 0.

Stable FCLT

Multivariate Case

Multivariate Stable FCLT

Theorem 2 (Resnick, 2007)

(Xi )i≥1 i.i.d. in Rd satisfying (1). If Sn(t) =∑[nt]

i=1 Xi , then

an(Sn(t)− bn(t))

d→ Z (t)t≥0 in D([0,∞);Rd),

where Z (t)t≥0 is an α-stable Levy motion in Rd :

E [e iuZ(t)] = exp

(e iu·x − 1)ν(dx)

, if α < 1;

E [e iuZ(t)] = exp

(e iu·x − 1− iu · x)ν(dx)

, if α > 1.

Stable FCLT

Multivariate Case

3. Regular variation in D

D = D([0, 1]) is the set of cadlag functions on [0, 1]Xi = Xi (s)s∈[0,1], i ≥ 1 i.i.d. random elements in D

Examples

1) Xi (s) =number of internet transactions on a website onday i at time s during the day2) Xi (s)=high tide water level on day i at location s on theshore of Netherlands (de Haan and Lin, 2001)

GoalStudy the asymptotic behaviour of the partial sum process:

Sn(t, s) =

[nt]∑i=1

Xi (s), t ≥ 0, s ∈ [0, 1]

Note: Sn(·) ∈ D([0,∞);D) where Sn(t) = Sn(t, s)s∈[0,1]

Stable FCLT

Multivariate Case

Basic properties of D (Billingsley 1968, 1999)

Uniform normD is a Banach space equipped with the uniform norm:

‖x‖ = sups∈[0,1]

|x(s)|

(D, ‖ · ‖) is not separable

Skorohod J1-distance

D is a metric space equipped with the Skorohod J1-distance:

dJ1(x , y) = infλ∈Λ‖λ− e‖ ∨ ‖x − y λ‖

The Borel σ-field on (D, J1) coincides with the σ-fieldgenerated by the projections πs : D→ R, πs(x) = x(s).

Stable FCLT

Multivariate Case

Polar coordinate transformationWe consider the map T : D0 → (0,∞)× SD given by:

T (x) =

(‖x‖, x

‖x‖

)where D0 = D− 0 and SD = x ∈ D; ‖x‖ = 1T is a homeomorphism (since ‖ · ‖ is J1-continuous)

The product space

D0 = (0,∞]× SD is a Polish space with distance:

((r , z), (r ′, z ′)

∣∣∣∣1r − 1

∣∣∣∣ ∧ d0J1

(z , z ′)

D0 is not locally compact

Stable FCLT

Multivariate Case

Definition (de Haan and Lin, 2001)

A random element X in D is regularly varying if ∃ an →∞,α > 0, c > 0 and a probability measure Γ1 on SD such that

((‖X‖an

‖X‖

)∈ ·)

w→ cνα × Γ1 in D0. (2)

(µnw→ µ if µn(A)→ µ(A) for any A bounded, µ(∂A) = 0)

Let ν be the measure on D0 given by

ν T−1 = cνα × Γ1 =: ν. (3)

Example

If X = X (s)s∈[0,1] is an α-stable Levy motion with samplepaths in D, then X is regularly varying in D.

Stable FCLT

Multivariate Case

4. D-valued stable Levy motion

DefinitionLet ν be a measure on D s.t. ν0 = 0 and (3) holds.For any t > 0, let Z (t) be a random element in D.Z (t)t≥0 is a D-valued α stable Levy motion if Z (0) = 0,(i) Z (t2)− Z (t1), . . . ,Z (tn)− Z (tn−1) are independent

(ii) Z (t2)− Z (t1)d= Z (t2 − t1) for any t1 < t2

(iii) Z (t, s)s∈[0,1] is a cadlag α-stable process s.t. if α < 1,

E [e i∑m

j=1 ujZ(t,sj )] = exp

(e iu·y − 1)µs1,...,sm(dy)

and if α > 1,

E [e i∑m

j=1 ujZ(t,sj )] = exp

(e iu·y − 1− iu · y)µs1,...,sm(dy)

whereµs1,...,sm = ν π−1

s1,...,sm .

Stable FCLT

Multivariate Case

Example

Let L(t, s)t,s∈[0,1] be an α-stable Levy sheet withsample paths in D([0, 1]2). Then L(t)t∈[0,1] is aD-valued α-stable Levy motion.

Proof of (iii): L(t) = L(t, s)s∈[0,1] is a cadlag α-stableLevy motion. Then L(1) is RV in D with limiting measurecνα × Γ1 (Hult and Lindskog, 2007). Let ν be given by (3).(L(1, s1), . . . , L(1, sm)) is RV in Rm with limiting measureν π−1

s1,...,sm (Hult and Lindskog, 2005)But (L(1, s1), . . . , L(1, sm)) is α-stable and (if α < 1)

E [e i∑m

j=1 ujL(1,sj )] = exp

(e iu·y − 1)µs1,...,sm(dy)

(L(1, s1), . . . , L(1, sm)) is RV in Rm with limiting measureµs1,...,sm . Hence

µs1,...,sm = ν π−1s1,...,sm

Stable FCLT

Multivariate Case

Construction of D-valued stable Levy motion

Compound Poisson building blocks

Let N be a PRM on [0,∞)× D0 of intensity Leb × ν.Let εj ↓ 0 with ε0 = 1. For any t ≥ 0 and s ∈ [0, 1], define

Zj(t, s) =

∫[0,t]×Ij×SD

rz(s)N(du, dr , dz)

where Ij = (εj , εj−1] for j ≥ 1 and I0 = (1,∞). The processZj(t) = Zj(t, s)s∈[0,1] has all sample paths in D and

E (e iuZj (t,s)) = exp

∫Ij×SD

(e iurz(s) − 1)ν(dr , dz)

Stable FCLT

Multivariate Case

Let ϕ(s) =∫SD z(s)Γ1(dz) and ψ(s) =

∫SD |z(s)|2Γ1(dz).

E (Zj(t, s)) = tϕ(s)

rνα(dr), j ≥ 0

Var(Zj(t, s)) = tψ(s)

r2να(dr), j ≥ 0

Zj(t, s)j≥0 are independent. By Kolmogorov’s criterion,∑j≥1

(Zj(t, s)− E (Zj(t, s))

)converges a.s.

If α < 1, let Z (εk )(t, s) =∑k

j=0 Zj(t, s) and

Z (t, s) =∑j≥0

Zj(t, s).

If α > 1, let Z(εk )

(t, s) =∑k

(Zj(t, s)− E (Zj(t, s))

Z (t, s) =∑j≥0

(Zj(t, s)− E (Zj(t, s))

Stable FCLT

Multivariate Case

Lemma(Z (t, s1), . . . ,Z (t, sm)) has a stable distribution in Rm.

Proof: Assume α < 1. (The case α > 1 is similar.)

E (e iuZ(t,s)) = exp

(e iurz(s) − 1)ν(dr , dz)

(e iuy − 1)µs(dy)

where µs = ν π−1

s = ν π−1s . By the scaling property of ν,

µs(cA) = c−αµs(A) for any c > 0,A ⊂ R. Hence,

µs(dy) =(c+s y−α−11y>0 + c−s (−y)−α−11y<0

and Z (t, s) ∼ Sα(t1/ασs , βs , 0) for some σs > 0, βs ∈ R.

Stable FCLT

Multivariate Case

Case α < 1

Theorem 1.(a) (B-Saidani, 2019)

For any t > 0, there exists a cadlag modificationZ (t) = Z (t, s)s∈[0,1] of Z (t) = Z (t, s)s∈[0,1] such that

limk→∞

‖Z (εk )(t)− Z (t)‖ → 0 a.s.

Z (t)t≥0 is a D-valued α-stable Levy motion.

Proof: ‖Zj(t)‖ ≤∫

[0,t]×Ij×SD rN(du, dr , dz) and

E∑j≥1

‖Zj(t)‖ ≤ t

∫(0,1]×SD

rν(dr , dz) <∞

It follows that∑

j≥1 ‖Zj(t)‖ <∞ a.s. and Z (εk )(t)k isCauchy in (D, ‖ · ‖) a.s. (with limit Z (t)).

Stable FCLT

Multivariate Case

Theorem 1.(b) (B-Saidani, 2019)

There exists a collection Z (t)t≥0 of random elements

in D such that P(Z (t) = Z (t)) = 1 for any t > 0 and

supt≤T‖Z (εk )(t)− Z (t)‖ → 0 ∀T > 0.

Moreover, the map t 7→ Z (t) is in Du([0,∞);D) a.s.

NotationDu([0,∞);D) is the set of cadlag functions x : [0,∞)→ Dwith respect to the uniform norm ‖ · ‖ on D.

RemarkDu([0, 1];D) is a Banach space with respect to thesuper-uniform norm:

‖x‖D = supt≤1‖x(t)‖.

Stable FCLT

Multivariate Case

Case α > 1

Theorem 2.(a) (B-Saidani, 2019)

For any t > 0, there exists a cadlag modificationZ (t) = Z (t, s)s∈[0,1] of Z (t) = Z (t, s)s∈[0,1] such that

limk→∞

‖Z (εk )(t)− Z (t)‖ → 0 a.s. (4)

Z (t)t≥0 is a D-valued α-stable Levy motion.

Proof: To prove (4), we use a version of Ito-Nisio Theorem(Basse O’Connor and Rosinski, 2013):

• Z (εk )(t) =

∑kj=0

(Zj(t)− E (Zj(t))

)is a sum of zero-mean

independent random elements in D• Z (εk )

(t)k is tight in (D, J1) (Roueff and Soulier, 2015)

• Z (εk )(t)

d→ Z (t) in D and Z (t, s)s∈[0,1] is u.i.

Stable FCLT

Multivariate Case

Theorem 2.(b) (B-Saidani, 2019)

There exists a collection Z (t)t≥0 of random elements

in D such that P(Z (t) = Z (t)) = 1 for any t > 0, themap t 7→ Z (t) is in D([0,∞);D) and

Z(εk )

(·) d→ Z (·) in D([0,∞);D)

as k →∞, k ∈ N ′ for a subsequence N ′. D([0,∞);D) is theset of cadlag functions x : [0,∞)→ D with respect to J1.

J1-distance

Here D([0, 1];D) is equipped with J1-distance: (Whitt, 1980)

dD(x , y) = infλ∈Λ‖λ− e‖ ∨ ρD(x , y λ) ,

where ρD is the uniform distance with respect to d0J1

ρD(x , y) = supt≤1

(x(t), y(t)) ≤ ‖x − y‖D.

Stable FCLT

Multivariate Case

Proof:

• Similarly to Roueff and Soulier (2015), it can be proved

that (Z(εk )

)k is tight in D([0,∞);D). There exists aprocess Y (t)t≥0 with sample paths in D([0,∞);D)(defined on another space (Ω′,F ′,P ′)) such that:

Z(εk )

(·) d→ Y (·), in D([0,∞);D)

as k →∞, k ∈ N ′.

• (Z (t1), . . . ,Z (tn))d= (Y (t1), . . . ,Y (tn)) for all

t1, . . . , tn > 0

• By Lemma 3.24 (Kallenberg, 2002), it follows thatZ (t)t≥0 has a modification Z (t)t≥0 with samplepaths in D([0,∞);D).

Stable FCLT

Multivariate Case

5. Functional limit theorem (B-Saidani, 2019)

Theorem 3.Let (Xi )i≥1 be i.i.d. RV in D:

((‖X‖an

‖X‖

)∈ ·)

w→ cνα × Γ1 in D0.

Let Sn(t, s) =∑[nt]

i=1 Xi (s). If α > 1, assume an additionaltechnical condition. Then

an(Sn(t)− bn(t))

d→ Z (t)t≥0 in D([0,∞);D),

where Z (t)t≥0 is the process of Theorems 1-2.

Stable FCLT

Multivariate Case

Proof (case α < 1)1) Point process convergence:

Nn =∑i≥1

δ( in,‖Xi‖an

‖Xi‖

) d→ N =∑i≥1

δ(Ti ,Ri ,Wi ) = PRM(Leb×ν)

2) Continuity of truncated summation: the map

Qε :∑i≥1

δ(ti ,ri ,zi ) 7→

∑ti≤t

rizi1ri>ε

t∈[0,T ]

∈ D([0,T ];D)

is continuous on a set Λ and P(N ∈ Λ) = 1.3) Define

S(ε)n = Qε(Nn) =

[nt]∑i=1

Xi1‖Xi‖>εan

t∈[0,T ]

Z (ε) = Qε(N) =

∑Ti≤t

RiWi1Ri>ε

t∈[0,T ]

Stable FCLT

Multivariate Case

By Continuous Mapping Theorem, for any ε > 0,

S(ε)n (·) d→ Z (ε)(·) in D([0,T ];D)

4) Let ε→ 0:

limε→0

lim supn→∞

P( supt∈[0,T ]

‖S (ε)n (t)− Sn(t)‖ > δ) = 0

supt≤T‖Z (εk )(t)− Z (t)‖ → 0 a.s. (Theorem 1.(b))

The conclusion follows by Theorem 4.2 of Billingsley (1968).

RemarkA similar argument works when α > 1, using centering.

Stable FCLT

Multivariate Case

RemarkTheorem 3 extends a result of Roueff and Soulier (2015) tofunctional convergence. Their result says that if α > 1 and(Xi )i≥1 are i.i.d. regularly varying in D, then

n∑i=1

(Xi − E (Xi )

) d→ Y in D,

where Y = Y (s)s∈[0,1] is an α-stable process with samplepaths in D. Note that

Y (s)s∈[0,1]d= Z (1, s)s∈[0,1].

Stable FCLT

Multivariate Case

References

• Balan, R.M. and Saidani, B. (2019). Stable Levymotion with values in the Skorohod space: constructionand approximation. To appear in J. Theor. Probab.Preprint arXiv:1809.02103

• Basse-O’Connor, A. and Rosinski, J. (2013). On theuniform convergence of random series in Skorohodspace and representation of cadlag infinitely divisibleprocesses. Ann. Probab. 41, 4317-4341.

• Billingsley, P. (1968, 1999). Convergence ofProbability Measures. John Wiley.

Stable FCLT

Multivariate Case

• de Haan, L. and Lin, T. (2001). On convergencetowards an extreme value distribution in C [0, 1]. Ann.Probab. 29, 467-483.

• Hult, H. and Lindskog, F. (2005). Extremal behaviourof regularly varying stochastic processes. Stoch. Proc.Appl. 115, 249-274

• Hult, H. and Lindskog, F. (2007). Extremal behaviourof stochastic integrals driven by regularly varying Levyprocesses. Ann. Probab. 35, 309-339.

• Kallenberg, O. (2002). Foundations of ModernProbability. Second edition. Springer.

• Resnick, S.I. (2007). Heavy-Tail Phenomena:probabilistic and statistical modelling. Springer.

Stable FCLT

Multivariate Case

• Roueff, F. and Soulier, P. (2015). Convergence tostable laws in the space D. J. Appl. Probab. 52, 1-17.

• Skorohod, A. V. (1957). Limit theorems for stochasticprocesses with independent increments. Th. Probab.Appl. 2, 138-171.

• Whitt, W. (1980). Some useful functions for functionallimit theorems. Math. Oper. Res. 5, 67-85.

Stable FCLT

Multivariate Case

Functional limittheorem in D Thank you!

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