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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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))
t≥0
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
t
∫R
(e iuy − 1)να,p(dy)
, if α < 1
E (e iuZ(t)) = exp
t
∫R
(e iuy − 1− iuy)να,p(dy)
, if α > 1
να,p(y) =(pαy−α−11y∈(0,∞)+qα(−y)−α−11y∈(−∞,0)
)dy
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
Idea of proof α < 1 (Resnick, 1986)
1) Point process convergence:
Nn =n∑
i=1
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
n
∑ni=1 Xi1|Xi |>εan
d→ Z (ε) =∑
i≥1 Ji1|Ji |>ε4) Let ε→ 0.
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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) =
1
an
[nt]∑i=1
Xi1|Xi |>εan
t∈[0,T ]
Z (ε) = Qε(N) =
∑Ti≤t
Ji1|Ji |>ε
t∈[0,T ]
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
2. Multivariate Case
A random vector X in Rd is regularly varying (RV) if
nP
(X
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:
nP
(( |X |an,X
|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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
1
an(Sn(t)− bn(t))
t≥0
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
t
∫Rd
(e iu·x − 1)ν(dx)
, if α < 1;
E [e iuZ(t)] = exp
t
∫Rd
(e iu·x − 1− iu · x)ν(dx)
, if α > 1.
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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:
dD0
((r , z), (r ′, z ′)
)=
∣∣∣∣1r − 1
r ′
∣∣∣∣ ∧ d0J1
(z , z ′)
D0 is not locally compact
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
nP
((‖X‖an
,X
‖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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
t
∫Rm
(e iu·y − 1)µs1,...,sm(dy)
,
and if α > 1,
E [e i∑m
j=1 ujZ(t,sj )] = exp
t
∫Rm
(e iu·y − 1− iu · y)µs1,...,sm(dy)
,
whereµs1,...,sm = ν π−1
s1,...,sm .
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
∫Rm
(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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
t
∫Ij×SD
(e iurz(s) − 1)ν(dr , dz)
.
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
Let ϕ(s) =∫SD z(s)Γ1(dz) and ψ(s) =
∫SD |z(s)|2Γ1(dz).
E (Zj(t, s)) = tϕ(s)
∫Ij
rνα(dr), j ≥ 0
Var(Zj(t, s)) = tψ(s)
∫Ij
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
j=0
(Zj(t, s)− E (Zj(t, s))
)and
Z (t, s) =∑j≥0
(Zj(t, s)− E (Zj(t, s))
)
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
t
∫D0
(e iurz(s) − 1)ν(dr , dz)
= exp
t
∫R
(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
)dy
and Z (t, s) ∼ Sα(t1/ασs , βs , 0) for some σs > 0, βs ∈ R.
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
d0J1
(x(t), y(t)) ≤ ‖x − y‖D.
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
5. Functional limit theorem (B-Saidani, 2019)
Theorem 3.Let (Xi )i≥1 be i.i.d. RV in D:
nP
((‖X‖an
,X
‖X‖
)∈ ·)
w→ cνα × Γ1 in D0.
Let Sn(t, s) =∑[nt]
i=1 Xi (s). If α > 1, assume an additionaltechnical condition. Then
1
an(Sn(t)− bn(t))
t≥0
d→ Z (t)t≥0 in D([0,∞);D),
where Z (t)t≥0 is the process of Theorems 1-2.
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
Proof (case α < 1)1) Point process convergence:
Nn =∑i≥1
δ( in,‖Xi‖an
,Xi
‖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) =
1
an
[nt]∑i=1
Xi1‖Xi‖>εan
t∈[0,T ]
Z (ε) = Qε(N) =
∑Ti≤t
RiWi1Ri>ε
t∈[0,T ]
Stable Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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
1
an
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
• 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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D
• 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 Levymotion with values
in the Skorohodspace
Stable FCLT
Multivariate Case
Regular variationin D
D-valued α-stableLevy motion
Functional limittheorem in D Thank you!