Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Some mixing properties of conditionally independent
processes
Manel Kacem, Stéphane Loisel and Véronique Maume-Deschamps
Séminaire Lyon-Le Mans, 3 mai 2012
Université Lyon 1, ISFA, Laboratoire SAF
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Motivation
The classical insurance risk model
Rn = u + c − S n ,
Sn =
n
X
i =1
Xi .
In risk theory, some models treating the structure of dependence
between individual risks Xi are factor models.
In actuarial theory these models are known by models with common
shock where dependence between r.v's Xi may come from a
time-varying common factor.
Many researchers in actuariat give attention to risk models in
Markovian environment (Asmussen(1989) and Cossette et
al.(2004)).
An important property of factor models is: all risks have the
property to be dependent but conditionally independent given the
common factor.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Literature VS our Model
Let Vi be the random factor observed at time i.
All i ∈ N, r.v's Xi are independent when conditioned to a factor
V n = (V1 , . . . , Vn ).
Cossette et al. (2004) proposed a compound binomial model modulated
by a Markovian environment. In their model, r.v's Xi are conditionally
independent given r.v's Vi , roughly speaking
Z
iu
Y
f (xi1 , . . . , xiu )
Nj (dxj , Vj )
E(f (Xi1 , . . . , Xiu )|V n ) =
xi1 ,...,xiu
j =i1
In our framework, r.v.'s (Xn )n∈N are considered to be conditionally
independent given the entire trajectory of the factor:
Z
iu
Y
E(f (Xi1 , . . . , Xiu )|V n ) =
f (xi1 , . . . , xiu )
Nj (dxj , V j ),
xi1 ,...,xiu
j =i1
where Nj (dxj , Vj ) and Nj (dxj , V j ) respectively denote the Kernel
transition of Xj given Vj and given V j .
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Goals
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Our Goals are
To provide some mixing properties of the process (Xi )i ∈N , under
necessary conditions on the factor process (Vi )i ∈N and on the
conditional mixing structure (see Denition 1).
2 To give a Self normalized Central Limit Theorem (SNCLT) for the
P
sum Sn = ni=1 Xi .
Our framework is illustrated by some example in insurance.
1
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Conditional independence
Our denition for mixing sequences
Conditionally mixing sequences
Conditional independence
We shall make the following standing assumptions.
H1: For all i ∈ N, r.v's Xi are conditionally independent given
V i = (V1 , . . . , Vi ).
H2: The conditional law of Xi |V n is the same as the one of Xi |V i ,
for all n ≥ i.
Assumptions H1 and H2 are equivalent to:
For any multi index i1 ≤ i2 . . . ≤ iu and for any n ≥ iu we have
E (f (Xi1 , . . . , Xiu |V n )) =
Z
xi1 ,...,xiu
f (xi1 , . . . , xiu )
u
Y
j =1
Nij (dxij , V ij ), (1.1)
where Nij (dxij , V ij ) denotes the kernel transition of Xij given V ij .
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Conditional independence
Our denition for mixing sequences
Conditionally mixing sequences
Conditional independence
Proposition 1
Let u, v and r be integers and let f and g be real valued bounded
functions. Consider the multi-indices
i1 < i2 . . . < iu < iu + r ≤ j1 < j2 . . . < jv . If H1 and H2 hold then
Cov(f (Xi1 , . . . , Xiu ), g (Xj1 , . . . , Xjv ))
= Cov(E(f (Xi1 , . . . , Xiu )|V iu ), E(g (Xj1 , . . . , Xjv )|V jv )), (1.2)
This means that the covariance of f (Xi1 , . . . , Xiu ) and g (Xj1 , . . . , Xjv ) is
equal to the covariance of their conditional expectation whenever
conditioning is done respectively with respect to V iu and V jv .
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Conditional independence
Our denition for mixing sequences
Conditionally mixing sequences
Denition 1
Let u, v be integers, a random process (X1 , . . . , Xn ) is said to be
ηu,v -mixing, if there exist a function r 7→ η(r ) decreasing to 0 as r goes
to innity and a constant C (u , v ) > 0 such that for any real valued
bounded functions f and g , for any integers u 0 ≤ u, v 0 ≤ v and for any
multi-indices satisfying the relation (?):
i1 < · · · < iu0 ≤ iu < iu + r ≤ j1 < · · · < jv 0 ≤ jv ,
(?)
we have
sup Cov f (Xi1 , . . . , Xiu0 ), g (Xj1 , . . . , Xjv 0 ) ≤ C (u , v )η(r )kf ka kg kb ,
(1.3)
where the supremum is taken over all the sequences (i1 , . . . , iu0 ) and
(j1 , . . . , jv 0 ) satisfying (2.4) and r ≥ j1 − iu is the gap of time between
past and future.
k ka and k kb are norms on bounded functions.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Conditional independence
Our denition for mixing sequences
Conditionally mixing sequences
With respect to these norms, we have various kind of mixing as referred
to the classical strongly mixing denitions.
1 If k ka = k kb = k k∞ , we shall say that the process is α
(u ,v ) mixing
and we shall write α(r ) instead of η(r ). If the process is α(u,v ) for all
u , v and if supu,v ∈N C (u , v ) ≤ C < ∞ then the process is α mixing
in the sense of Rosenblatt (1956).
2 If k ka = k kb = k k1 : the process is Ψ
(u ,v ) mixing.
Note that
η(r ) is decreasing in r ,
for the
, C (u , v ) is uniform in u and v ,
Ψ(u,v ) =⇒ α(u,v ) mixing.
classical mixing coecients
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Conditional independence
Our denition for mixing sequences
Conditionally mixing sequences
Conditionally mixing sequences
Denition 2
Let u, v and r be integers and let (i1 , . . . , iu ) and (j1 , . . . , jv ) satisfying
(2.4). The sequence of r.v's (Xn )n∈N is called conditionally ηu,v mixing
with respect to V n if there exist a positive sequence η(r ) → 0 as r → ∞
and a constant C (u , v ) > 0 such that the following inequality holds for
any real and bounded functions f : Ru → R and g : Rv → R,
sup Cov(E(f (Xi1 , . . . , Xiu0 )|V iu0 ), E(g (Xj1 , . . . , Xjv 0 )|V jv 0 ))
≤ C (u , v )η(r ) kf ka kg kb ,
(1.4)
where the supremum is taken over all sequences (i1 , . . . , iu0 ) and
(j1 , . . . , jv 0 ) satisfying (2.4).
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Some notations
Some notations
Assumptions
Theorem
Proof
Our main result emphasizes the relation between the conditional mixing
properties of the process (Xi )i ∈N , the mixing properties of the process
(Vi )i ∈N , and the regularity of the transition kernels. Recall that
u
Y
i
,...,iu
1
N
(·, V iu ) =
N ik (·, V ik )
k =1
where N i1 ,...,iu (·, V iu ) denote the kernel transition of the random vector
(Xi1 , . . . , Xiu ) given V iu = (V1 , . . . , Viu ) and denote by
D (V jv , Veiu ,jv ,` ) = N j1 ,...,jv (·, V jv ) − N j1 ,...,jv (·, Veiu ,jv ,` ),
where
multi indices (i1 , . . . , iu ) and (j1 , . . . , jv ), satisfying (?),
` be an integer such that 0 < ` ≤ j1 − iu ,
Veiu ,jv ,` = (0, . . . , 0, Viu +`+1 , . . . , Vj1 , . . . , Vjv ) ∈ Rjv .
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Some notations
Assumptions
Theorem
Proof
Assumption 1
Kv and Kv0 denote positive constants depending on v , (κ(`))`∈N denotes
e j ,...,j (·) is a positive nite measure
a decreasing to zero sequence and N
1
v
on Rv . We denote k = j1 − iu − ` with 0 < ` ≤ j1 − iu . Let us introduce
the following assumptions satised for multi indices satifying (?).
sup D (V jv , Veiu ,jv ,` ) ≤ Kv kN j1 ,...,jv (·, V jv )k1 κ(k ).
V
(2.1)
e j ,...,j (·)κ(k ).
sup D (V jv , Veiu ,jv ,` ) ≤ N
1
v
V
(2.2)
Z
(2.3)
Z
ei ,j ,` ) ≤ κ(k )Kv .
D (V jv , V
u v
V xj1 ,...xjv
j
kN 1 ,...,jv (·, V jv )k∞ ≤ Kv0 kN j1 ,...,jv (·, V jv )k1 .
Z
kN j1 ,...,jv (·, V jv )k∞ < ∞.
xj1 ,...xjv
Séminaire Lyon-Le Mans, 3 mai 2012
(2.4)
(2.5)
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Some notations
Assumptions
Theorem
Proof
Theorem 1
Assume (Xi )i ∈N is conditionally independent with respect to (Vi )i ∈N .
The results are presented in Table 1.
Table 1
Kind of mixing
for (Vi )i ∈N
Satised equation
of assumption 1
Result: type of mixing
for (Xi )i ∈N
Ψu,v
(2.1)
Ψu,v
αu,v
(2.1) and (2.4)
Ψu,v
(2.3) and (2.5)
αu,v
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Proof of the main result
Some notations
Assumptions
Theorem
Proof
Lemma 1
Assume that (Vi )i ∈N is η(u,v ) mixing in the sense of Denition 1. For any
multi indices i1 < · · · < iu < j1 < · · · < jv and 0 < ` ≤ j1 − iu and
bounded functions f and g , we have
Cov(E(f (Xi1 , . . . , Xiu )|V i ), E(g (Xj1 , . . . , Xjv )|V j ))
u
v
Z
ei ,j ,` )| +
≤ 2kf k1
|g | · sup |D (V jv , V
u v
xj1 ,...,xjv
Z
xi1 ,...,xiu ,xj1 ,...,xjv
V
ei ,j ,` )kb .
|f g | ηu,v (`)kN i1 ,...,iu (·, V iu )ka · kN j1 ,...,jv (·, V
u v
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Some notations
Assumptions
Theorem
Proof
Lemma 2
Assume that (Vi )i ∈N is η(u,v ) mixing in the sense of Denition 1. For any
multi indices i1 < · · · < iu < j1 < · · · < jv and 0 < ` ≤ j1 − iu and
bounded functions f and g , we have
Cov(E(f (Xi1 , . . . , Xiu )|V i ), E(g (Xj1 , . . . , Xjv )|V j ))
u
v
Z
Z
Z
ei ,j ,` )|
|D (V jv , V
≤2
|f | sup N i1 ,...,iu (·, V iu ) × kg k∞
u v
xi1 ,...,xiu
V
Z
+
xi1 ,...,xiu ,xj1 ,...,xjv
V xj1 ,...,xjv
ei ,j ,` )kb .
|f g | ηu,v (`)kN i1 ,...,iu (·, V iu )ka · kN j1 ,...,jv (·, V
u v
Note that when (2.1) or (2.2) are satised, we use Lemma 1 while when
(2.3) is fullled, we use Lemma 2.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Proof of the Lemmas
Some notations
Assumptions
Theorem
Proof
Cov(E(f (Xi1 , . . . , Xiu )|V i ), E(g (Xj1 , . . . , Xjv )|V j ))
u
v
Z
i
,...,iu
j
,...,jv
1
1
(·, V iu ), N
(·, V jv ) ,
= f g · CovV N
xi1 ,...,xiu ,xj1 ,...,xjv
and
CovV N i1 ,...,iu (·, V i ), N j1 ,...,jv (·, V j ) u
v
ei ,j ,` ) ≤ CovV N i1 ,...,iu (·, V iu ), D (V jv , V
u v
ei ,j ,` ) .
+ CovV N i1 ,...,iu (·, V iu ), N j1 ,...,jv (·, V
u v
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Moment inequalities
Self Normalised Central Limit Theorem
Moment inequalities
Lemma 3
Let (Xi )i ∈N be a stationary α(u , v ) mixing sequences. Let kf kp < ∞ and
kg kq < ∞, where 1 < p , q ≤ ∞ and p1 + q1 < 1, then
|Cov(f (Xi 1 , . . . , Xiu ), g (Xj 1 , . . . , Xjv ))|
≤ (C (u , v ) + 6)(E |f |p ) p (E |g |q ) q α(r )1− p − q
1
1
1
1
where r ≥ j1 − iu where r is a positive integer.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Moment inequalities
Self Normalised Central Limit Theorem
Moment inequalities
Corollary 1
Let Sn =
n
X
Xi where E (Xi ) = 0 and denote by E(Snp ) the moment of
i =1
order p of the partial sum Sn . Let kXn kr = E(|Xn |r )1/r , δ > 0 and
denote by
K2 = sup(C (1, 1) + 6) kXi k22+δ , K4 = sup(C (2, 2) + 6) kXi k44+2δ .
i
i
Then
E(Sn2 ) ≤ 2nK2
4
E(Sn ) ≤ 4!n
2
K22
n −1
X
r =0
n−1
X
r =0
!2
α(r )
δ
δ+2
Séminaire Lyon-Le Mans, 3 mai 2012
δ
α(r ) δ+2 ,
+ nK4
n −1
X
δ
(r + 1)2 α(r ) δ+2 .
r =0
(3.1)
(3.2)
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Moment inequalities
Self Normalised Central Limit Theorem
Moment inequalities
Theorem 2
Let 2 < p < r ≤ ∞, 2 < v ≤ r and Xn , n ≥ 1 be an α− mixing sequence
of random variables with E Xn = 0. Assume that for some C > 0 and
θ>0
α(n) ≤ Cn−θ ,
(3.3)
where θ > (2(rpr−p)) . Then, there exists A = A(ε, r , p , θ, C ) < ∞ for any
ε > 0, such that
E |Sn |p ≤ Anp/2 (K2 Dn,2,2 )p/2 + max kXi kpr ,
i ≤n
Séminaire Lyon-Le Mans, 3 mai 2012
(3.4)
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Moment inequalities
Self Normalised Central Limit Theorem
Self Normalised Central Limit Theorem
We introduce the following estimator which is a modied version of some
estimators introduced in Shao et al.(1995)(write ` = `n ).
2
n−` X
Sj (`) − X `
√
,
Bn,2 =
n − ` + 1 j =0
`
1
2
with
Sj (`) =
j +`
X
n−`
X
1
Xk and X ` =
S (`).
n − ` + 1 j =0 j
k =j +1
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Moment inequalities
Self Normalised Central Limit Theorem
Self Normalised Central Limit Theorem
Theorem 3
Let {Xn , n ∈ N} be a stationary α(1,1) mixing process with E(X1 ) = µ.
Assume that for some δ > 0,
E |X1 |
2+δ
< ∞,
Var (Sn )
→ σ 2 , as n → ∞,
n
∞
X
r =1
δ
α(r ) 2+δ < ∞,
`n = ` , `n → ∞ , ` = o (n) , n → ∞
(3.5)
(3.6)
(3.7)
(3.8)
If 0 < σ < ∞, then Bn,2 is a consistent estimator of σ in the L2 -norm,
Sn − nµ d .
p
→ N (0, 1).
nBn,2
Séminaire Lyon-Le Mans, 3 mai 2012
(3.9)
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Moment inequalities
Self Normalised Central Limit Theorem
Self Normalised Central Limit Theorem
Theorem 4
Let {Xn , n ∈ N} be a stationary α(2,2) mixing process satisfying mixing
condition with E(X1 ) = µ. Assume that
(
Sn − n µ
√
n
4+2δ
)
,n ≥ 1 ,
(3.10)
is uniformly integrable with δ > 0 . Assume that `n = `, `n → ∞,
` = o (n) as n → ∞. If in addition 3.7 and for 0 < σ < ∞, 3.6 holds then

2 ! 12
n
S (`) − `µ σ2
d.
→
N (0, ).
Bn,2 − E √
`
3
`
√

Séminaire Lyon-Le Mans, 3 mai 2012
(3.11)
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Example in a discrete case
Consider the process (Xi )(i ∈N ) such that Xi = Ii × Bi , where
the Ii 's are Bernouilli r.v's, conditionally independent with respect to
V i,
the claim amounts Bi 's are considered independent and independent
of the Ii 's and of V i .
(Vi )(i ∈N ) is mixing sequence of Bernouilli random variables taking
value one in time of crisis and value zero in a stable period.
We shall consider that the conditional Law of Ii has the following
structure:
i
X
h(Vj )
P (Ii = 1|V i ) =
i −j ,
j =1 2
where h is a measurable, non negative and bounded function:
0 < ρ ≤ h(v ) ≤ κ2 , with κ < 1 to ensure that P (Ii = 1|V i ) < 1.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Lemma 3
The transition kernel N i1 ,...,iu (·|V iu ) satises conditions (2.1) and (2.4) of
Assumption 1.
where we recall
sup D (V jv , Veiu ,jv ,` ) ≤ Kv kN j1 ,...,jv (·, V jv )k1 κ(k ),
V
(2.1)
kN j1 ,...,jv (·, V jv )k∞ ≤ Kv0 kN j1 ,...,jv (·, V jv )k1 .
(2.4)
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Estimation of the variance in the discrete example
Let Sn =
Pn
i =1 Ii .
P (Ii = 1|V i ) = K
i
X
1 + Vj
i −j ,
j =1 2
The exact value of the limit variance is derived equal to
lim
n→∞
Var (Sn ) 8 2
= K q (1 − q ) + 2K (1 + q − 2K − 4Kq − 2K 2 q 2 )
n
3
Numerical application: Let K = 0.24 and q = 0.6 then the exact value of
the squared error σ is equal to 0.4637. We also derive easly the squared
error when the conditionning is with respect to only the last value of the
factor, and the result is equal to σ = 0.4864.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Simulations
For simulation we choose ` =
following table
√
n and results for Bn,2 are gived in the
Table 2
n
m
Bn,`
IC −
IC +
EQMA
EQMR
200000
50
0, 4607
0, 4360
0, 4854
0, 000148
0, 000689
100
0, 4621
0, 4374
0, 4869
0
300
0, 4615
0, 4266
0, 4964
0, 000261
0, 001213
400
0, 4624
0, 4274
0, 4974
0, 000258
0, 001198
50000
Séminaire Lyon-Le Mans, 3 mai 2012
,000135 0,000627
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Example in continuous case
The conditional kernel transition is denoted by Ni (dxi , V i ) = fVi i (·)dxi .
Example 1
Consider that the conditional law Xi |V i is Pareto (α, θi ) where α > 2 is
the shape parameter and θi > 0 is the scale parameter: the conditional
density of Xi , i ∈ N, has the form
α
θi
fVi i (xi ; α, θi ) = α × α+
for xi ≥ θi ,
xi 1
where the scale parameter θi depends on V i with the following structure:
α
θi =
i
X
h(Vj )
i −j ,
j =1 2
(4.1)
and where the function h satises 0 < h(v ) ≤ τ2 .
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Proposition 2
If 0 < τ1 ≤ h(v ) ≤ τ2 for any v then the process of Example 1 satises
(2.3) and (2.5) of Assumption 1.
Remark
Note that the case of an Exponential conditional law (instead of a Pareto
conditional law) could also be considered. In particular if we consider an
Exponential law with parameter (1/θi ) where θi is dened as in (4.1)
with α = 1 and 0 < τ1 ≤ h(v ) ≤ τ2 for any v , then (2.3) and (2.5) of
Assumption 1 are satised.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Models for applications of the absolutely continuous case
Coinsider a hedging strategie in an insurance compagnie.
Assume that (Zi )i ∈N is a sequence of i.i.d. r.v's representing random
transaction costs of the hedging startegie that are observed in a steady
environment.
Assume that the (Vj )j ≥1 are independent of Zi and are r.v's which
contribute to create liquidity shocks and market conditions during some
time.
Consider for each i ≥ 1, Xi = θi Zi where
i
X
h(Vj ) 1/α
θi = (
i −j )
j =1 2
is a multiplicative factor, the function h is as in the previous example.
If Zi is Pareto (α, 1) distributed, then Xi is Pareto (α, θi ) distributed as
in the previous Example.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Conclusion
1
2
3
We have proved some mixing properties for process (Xi )i ∈N when
r.v's are considred to be conditionally independent given an
unbounded memory of a factor.
P
We proved a Self Normalised Central Limit Theorem for ni=1 Xi
adapted with mixing properties satised by our models, namely ηu,v
mixing properties.
We give some illustrated examples and we compare the aggregate
risk of an insurance portfolio of conditionally independent risks when
the conditioning is done with respect to a bounded memory of the
factor or with respect to unbounded memory and hence obtain
additional information about the magnitude of risk.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Cossette, H., Landriault, D., Marceau, E. (2004)
model in a Markovian environment.
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Economics. 35, 425-443.
Bradley, R.C. (2005)
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survey and some open questions.
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Dedecker,J. Doukhan,P., Lang, G., León R., JR., Louhichi,S. Prieur, C.
(2007)
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Lect. Notes
Stat. 190.
Doukhan, P. Louhichi, S. (1999)
A new weak dependence condition and
applications to moment inequalities.
Stochastic Process. Appl. 84. 2,
313-342.
Peligrad, M. Shao, Q. (1995)
for
ρ−
Estimation of the variance of Partial Sums
mixing random variables.
Journal of Multivariate Analysis. no. 52,
140-157.
H. Yu, Q. Shao,Weak
convergence for weighted empirical processes of
dependent sequences.
(1996) The annal of probability. Vol.24 no. 4,
2098-2127.
Strongway Shi,
Estimation of the variance for strongly mixing sequences.
Applied. Math.J.Chinese Univ.Ser.B (2000), no. 15(1), 45-54.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes
Mixing sequences and Conditional independence
Mixing properties of the process (Xi )i ∈N
Self Normalised Central Limit Theorem
Examples and some applications
Example in a discrete case
Example in continuous case
Thank you for your attention.
Séminaire Lyon-Le Mans, 3 mai 2012
Some mixing properties of conditionally independent processes