H IDDEN L ARGE D EVIATIONS
IN REGULARLY VARYING L ÉVY PROCESSES
B IKRAMJIT DAS
S INGAPORE U NIVERSITY
OF T ECHNOLOGY AND D ESIGN
ANZAPW 2015
JOINT WORK WITH
PARTHANIL R OY (ISI, KOLKATA )
Regular variation
• X ∼ F has a regularly varying tail (or heavy-tail) with tail index α > 0 if
F (x) := P(X > x) = x −α L(x),
where L(tx)/L(t) → 1 as t → ∞. Or, for x > 0,
lim
t→∞
1 − F (tx)
= x −α =: να (x, ∞).
1 − F (t)
• Pareto distribution: F (x) = x −α , x ≥ 1.
Hidden Large Deviations for regularly varying Lévy processes
9th April, 2015
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Multivariate Regular Variation (MRV)
Suppose {0} ∈ C0 ⊂ C ⊂ [0, ∞)d . Then Z ∈ C is regularly varying on C \ C0 , if there exist a
scaling function b(t) ∈ RV1/α , α > 0 and a non-null limit measure ν ∈ M(C \ C0 ) such that as
t →∞
h
i
Z
tP
∈ · → ν(·) in M(C \ C0 ).
b(t)
Write Z ∈ MRV (α, b, ν, C \ C0 ).
• Think d = 2: C = [0, ∞)2 , C0 = {0}. Then E = C \ C0 .
• Asymptotic independence ⇒ ν((0, ∞)2 ) = 0.
• E.g.: (X , Y ) has Pareto(1) margins + Gaussian Copula with correlation ρ < 1. Then
ν [0, (x, y)]c =
1 1
+ .
x y
Hidden Large Deviations for regularly varying Lévy processes
9th April, 2015
Bikramjit Das
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Hidden Regular Variation (HRV)
Let E = [0, ∞)2 \ {0} and E0 = [0, ∞)2 \ [axes] = (0, ∞)2 .
D EFINITION : Z is MRV on E with HRV on E0 = (0, ∞)2 if for 0 < α ≤ α0 ,
• Z ∈ MRV (α, b, ν, E) and Z ∈ MRV (α0 , b0 , ν0 , E0 ) with
•
lim b(t)
t→∞ b0 (t)
→ ∞ as t → ∞.
Maulik and Resnick [2005], Mitra and Resnick [2011]
Hidden Large Deviations for regularly varying Lévy processes
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800
30000
600
density
20000
400
density
10000
200
0.4
0.40
0.35
0.3
0.30
0
0.2
0.25
0.15
y
y
0
0.1
0.20
0.20
0.2
0.25
0.1
x
x
0.3
0.30
0.15
0.35
0.4
0.40
800
60000
600
40000
density
density
400
20000
0.4
200
0.40
0.35
0.3
0.30
0
0.2
0.25
0.15
y
y
0
0.1
0.20
0.20
0.2
0.25
0.1
x
x
0.3
0.30
0.15
0.35
0.4
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Stable subordinators
1. Suppose {Xt }t≥0 is a Lévy process.
• X0 = 0 and it has stationary independent increments.
• It has a version with almost surely right continuous paths with left limits (cádlág).
2. A Levy subordinator is a Levy process with non-decreasing sample paths.
3. A Levy subordinator is called a Stable subordinator if moreover,
d
Xt = t 1/α X1 ,
∀t > 0.
Here α ∈ (0, 1) and is called the exponent of the stable subordinator.
- the Lévy measure is given by
να ((x, ∞)) = x −α ,
x > 0.
Hidden Large Deviations for regularly varying Lévy processes
9th April, 2015
Bikramjit Das
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Large deviations of Stable subordinator
• Let X := {Xt }t≥0 be a stable subordinator. Define X (n) := {Xtn }t∈[0,1] as
Xtn = Xnt
n ≥ 1.
• It is easy to check that for any λn n1/α ,
X (n)
→0
λn
as D-valued random variables under Skorohod metric where D := D([0, 1], R).
• Hence for a set F ⊂ D[0, 1] bounded away from 0, then
X (n)
Qn (F ) := P
∈F
λn
→ 0.
• Clearly this is a rare event. At what rate does it happen?
Hidden Large Deviations for regularly varying Lévy processes
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• From Hult et al. [2005], we have that there exists a measure µ(0) on D \ {0} such that
λα
X (n)
n
P
∈F
n
λn
→ µ(0) (F )
(1)
for all F ⊂ D bounded away from D=0 = {constant functions} satisfying µ(0) (∂F ) = 0.
• Moreover, µ(0) is supported on
D=1 = {f ∈ D : f has only one jump}.
• This result is well-known for regularly varying Lévy processes, random walks with regularly varying (sub-exponential) step-sizes. See Hult et al. [2005].
• It supports the idea of one large jump principle for heavy-tails.
Hidden Large Deviations for regularly varying Lévy processes
9th April, 2015
Bikramjit Das
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Hidden large deviations
• Can we find rates of a different kind of rare event?
• What if we look for X (n) belonging to a set F bounded away from
D≤1 = D=0 ∪ D=1 .
• According to Hult et al. [2005] we get
X (n)
P
∈F
λn
=o
n
λα
n
.
• What about the exact rate?
• We restrict to stable subordinators for the talk, but we have shown the result for regularly
varying Lévy measures.
Hidden Large Deviations for regularly varying Lévy processes
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• We work under the notion of M-convergence; for details see Lindskog et al. [2014], Hult
and Lindskog [2006].
• We can show that there exists a measure µ(1) on D \ D≤1 such that
λα
n
n
2 X (n)
P
∈F
λn
→ µ(1) (F )
for all F ⊂ D bounded away from D≤1 = {at most one jump functions} satisfying
µ(1) (∂F ) = 0.
• Additionally, µ(1) is supported on
D=2 = {f ∈ D : f has exactly two jumps}.
• Hence P
h
X (n)
λn
∈F
i
≈
n
λα
n
2
µ(1) (F ).
• This is what we call Hidden Large Deviation at level 1: HLDP(1).
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Hidden large deviations: HLDP(j)
Proposition: For any j ≥ 1, there exists a measure µ(j) supported on Dj+1 such that
λα
n
n
j+1 P
X (n)
∈F
λn
→ µ(j) (F )
for all F ⊂ D bounded away from D≤j = {at most j jump functions} satisfying µ(j) (∂F ) = 0.
Here Dj+1 = {f ∈ D : f has exactly j + 1 jumps}.
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Idea of Proof: For level 0:
with N ∼ PRM(να × Leb) where να (x, ∞) = x −α , write
n
ZZ
0
Xt = Xnt = a nt +
x N(ds, dx)
[0,nt]×R+
ZZ
= ant +
[0,nt]×{x≤1}
ZZ
x [N(ds, dx) − ds να (dx)] +
ZZ
x N(ds, dx) +
1
[0,nt]×{x≤n α }
x N(ds, dx)
1
[0,nt]×{x>n α }
We can show that the red part contributes to the limit measure and the rest go to 0.
• Ideas are from Lindskog et al. [2014], Hult et al. [2005].
• We can show this for regularly varying Lévy measures under some conditions on λn .
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References
H ULT, H. AND L INDSKOG , F. (2006), Regular variation for measures on metric spaces, Publications de l’Institut Mathématique,
Nouvelle Série 80(94), 121 – 140
H ULT, H., L INDSKOG , F., M IKOSCH , T., AND S AMORODNITSKY, G. (2005), Functional large deviations for multivariate regularly
varying random walks, Ann. Appl. Probab. 15(4), 2651 – 2680
L INDSKOG , F., R ESNICK , S., AND R OY, J. (2014), Regularly varying measures on metric spaces: hidden regular variation and
hidden jumps Probability Surveys 11, 270 – 314
M AULIK , K. AND R ESNICK , S. (2005), Characterizations and examples of hidden regular variation Extremes 7(1), 31 – 67
M ITRA , A. AND R ESNICK , S. (2011), Hidden regular variation and detection of hidden risks, Stochastic Models 27(4), 591 –
614
DAS , B. AND R OY, P., Hidden large deviations under redulary varying Lévy measures, under preparation
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