Central Limit Theorem
Etienne Pardoux
Aix–Marseille Université
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
1 / 11
Recall (sN (t), iN (t)) and (s(t), i(t)). We write
1
sN (t) = s(t) + √ UN (t),
N
1
iN (t) = i(t) + √ VN (t).
N
If we replace sN , iN by the above right–hand sides, exploit the
(s(t), i(t)) equation
√ to suppress the terms of order 1, and multiply the
resulting SDEs by N, we get
Etienne Pardoux (AMU)
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2 / 11
Recall (sN (t), iN (t)) and (s(t), i(t)). We write
1
sN (t) = s(t) + √ UN (t),
N
1
iN (t) = i(t) + √ VN (t).
N
If we replace sN , iN by the above right–hand sides, exploit the
(s(t), i(t)) equation
√ to suppress the terms of order 1, and multiply the
resulting SDEs by N, we get
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
2 / 11
We get
Z t
UN (r )VN (r )
√
dr
UN (t) = −β
s(r )VN (r ) + i(r )UN (r ) +
N
0
Z t
1
s(r )VN (r )+i(r )UN (r ) UN (r )VN (r )
√
− √ M1 βN
s(r )i(r )+
+
dr ,
N
N
N
0
Z t
UN (r )VN (r )
√
VN (t) = β
s(r )VN (r ) + i(r )UN (r ) +
dr
N
0 Z t
1
s(r )VN (r )+i(r )UN (r ) UN (r )VN (r )
√
+
+ √ M1 βN
s(r )i(r )+
N
N
N
0
Z t
Z t
VN (r )
1
i(r ) + √
dr .
−α
VN (r )dr − √ M2 αN
N
N
0
0
N
Let MN
1 (t) and M2 (t) be the two martingales in the above
equations.
Etienne Pardoux (AMU)
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3 / 11
We get
Z t
UN (r )VN (r )
√
dr
UN (t) = −β
s(r )VN (r ) + i(r )UN (r ) +
N
0
Z t
1
s(r )VN (r )+i(r )UN (r ) UN (r )VN (r )
√
− √ M1 βN
s(r )i(r )+
+
dr ,
N
N
N
0
Z t
UN (r )VN (r )
√
VN (t) = β
s(r )VN (r ) + i(r )UN (r ) +
dr
N
0 Z t
1
s(r )VN (r )+i(r )UN (r ) UN (r )VN (r )
√
+
+ √ M1 βN
s(r )i(r )+
N
N
N
0
Z t
Z t
VN (r )
1
i(r ) + √
dr .
−α
VN (r )dr − √ M2 αN
N
N
0
0
N
Let MN
1 (t) and M2 (t) be the two martingales in the above
equations.
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
3 / 11
Let
Z t
1
VN (r )
dr ,
=
= P2 αN
i(r ) + √
N
N
0
0≤s≤t
Z t
VN (r )
N
hM2 it = α
i(r ) + √
dr ,
N
0
[MN
2 ]t
X
2
|∆MN
2 (s)|
N
and similarly for [MN
1 ]t , hM1 it .
2
N
N
2
N
We have that |MN
1 (t)| − hM1 it and |M2 (t)| − hM2 it are
martingales.
√
√
It is plain that |UN (t)| ≤ 2 N, |VN (t)| ≤ 2 N. Hence
2
N
2
E[(MN
1 (t)) ] ≤ 9βt, E[(M2 (t)) ] ≤ 3αt,
p
√
N
E(|MN
3αt.
1 (t)|) ≤ 3 βt, E(|M2 (t)|) ≤
Etienne Pardoux (AMU)
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4 / 11
Let
Z t
1
VN (r )
dr ,
=
= P2 αN
i(r ) + √
N
N
0
0≤s≤t
Z t
VN (r )
N
hM2 it = α
i(r ) + √
dr ,
N
0
[MN
2 ]t
X
2
|∆MN
2 (s)|
N
and similarly for [MN
1 ]t , hM1 it .
2
N
N
2
N
We have that |MN
1 (t)| − hM1 it and |M2 (t)| − hM2 it are
martingales.
√
√
It is plain that |UN (t)| ≤ 2 N, |VN (t)| ≤ 2 N. Hence
2
N
2
E[(MN
1 (t)) ] ≤ 9βt, E[(M2 (t)) ] ≤ 3αt,
p
√
N
E(|MN
3αt.
1 (t)|) ≤ 3 βt, E(|M2 (t)|) ≤
Etienne Pardoux (AMU)
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4 / 11
Let
Z t
1
VN (r )
dr ,
=
= P2 αN
i(r ) + √
N
N
0
0≤s≤t
Z t
VN (r )
N
hM2 it = α
i(r ) + √
dr ,
N
0
[MN
2 ]t
X
2
|∆MN
2 (s)|
N
and similarly for [MN
1 ]t , hM1 it .
2
N
N
2
N
We have that |MN
1 (t)| − hM1 it and |M2 (t)| − hM2 it are
martingales.
√
√
It is plain that |UN (t)| ≤ 2 N, |VN (t)| ≤ 2 N. Hence
2
N
2
E[(MN
1 (t)) ] ≤ 9βt, E[(M2 (t)) ] ≤ 3αt,
p
√
N
E(|MN
3αt.
1 (t)|) ≤ 3 βt, E(|M2 (t)|) ≤
Etienne Pardoux (AMU)
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We deduce from this and the above equations that
sup
N≥1, 0≤t≤T
sup
N≥1, 0≤t≤T
E (|UN (t)| + |VN (t)|) ≤ C1 (α, β, T ),
E |UN (t)|2 + |VN (t)|2 ≤ C2 (α, β, T ).
Exploiting Doob’s inequality, we deduce that for all T > 0,
!
2
2
sup E
sup |UN (t)| + |VN (t)|
< ∞.
0≤t≤T
N≥1
We want to take the limit in
Z t
UN (r )VN (r )
√
UN (t) = −β
dr −MN
s(r )VN (r ) + i(r )UN (r ) +
1 (t),
N
0
Z t
UN (r )VN (r )
√
VN (t) = β
s(r )VN (r ) + i(r )UN (r ) +
dr
N
0
Z t
N
−α
VN (r )dr + MN
1 (t) − M2 (t),
0
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
5 / 11
We deduce from this and the above equations that
sup
N≥1, 0≤t≤T
sup
N≥1, 0≤t≤T
E (|UN (t)| + |VN (t)|) ≤ C1 (α, β, T ),
E |UN (t)|2 + |VN (t)|2 ≤ C2 (α, β, T ).
Exploiting Doob’s inequality, we deduce that for all T > 0,
!
2
2
sup E
sup |UN (t)| + |VN (t)|
< ∞.
0≤t≤T
N≥1
We want to take the limit in
Z t
UN (r )VN (r )
√
UN (t) = −β
dr −MN
s(r )VN (r ) + i(r )UN (r ) +
1 (t),
N
0
Z t
UN (r )VN (r )
√
VN (t) = β
s(r )VN (r ) + i(r )UN (r ) +
dr
N
0
Z t
N
−α
VN (r )dr + MN
1 (t) − M2 (t),
0
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
5 / 11
We deduce from this and the above equations that
sup
N≥1, 0≤t≤T
sup
N≥1, 0≤t≤T
E (|UN (t)| + |VN (t)|) ≤ C1 (α, β, T ),
E |UN (t)|2 + |VN (t)|2 ≤ C2 (α, β, T ).
Exploiting Doob’s inequality, we deduce that for all T > 0,
!
2
2
sup E
sup |UN (t)| + |VN (t)|
< ∞.
0≤t≤T
N≥1
We want to take the limit in
Z t
UN (r )VN (r )
√
UN (t) = −β
dr −MN
s(r )VN (r ) + i(r )UN (r ) +
1 (t),
N
0
Z t
UN (r )VN (r )
√
VN (t) = β
s(r )VN (r ) + i(r )UN (r ) +
dr
N
0
Z t
N
−α
VN (r )dr + MN
1 (t) − M2 (t),
0
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
5 / 11
√
N
−1/2 M(Nt + Nt ),
where both MN
N
1 (t) and M2 (t) are of the form N
where tN ’s are random variables, with N −1/2 E[tN ] → 0 as N → ∞.
We want to show
Proposition
Under the above assumptions,
(
)
√
M(Nt + NtN )
√
, t ≥ 0 ⇒ {B(t), t ≥ 0},
N
where B(t) is a standard Brownian motion.
Let us first prove the result in case tN is a deterministic sequence s.t.
N −1/2 tN → 0 as N → ∞.
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6 / 11
√
N
−1/2 M(Nt + Nt ),
where both MN
N
1 (t) and M2 (t) are of the form N
where tN ’s are random variables, with N −1/2 E[tN ] → 0 as N → ∞.
We want to show
Proposition
Under the above assumptions,
(
)
√
M(Nt + NtN )
√
, t ≥ 0 ⇒ {B(t), t ≥ 0},
N
where B(t) is a standard Brownian motion.
Let us first prove the result in case tN is a deterministic sequence s.t.
N −1/2 tN → 0 as N → ∞.
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
6 / 11
√
N
−1/2 M(Nt + Nt ),
where both MN
N
1 (t) and M2 (t) are of the form N
where tN ’s are random variables, with N −1/2 E[tN ] → 0 as N → ∞.
We want to show
Proposition
Under the above assumptions,
(
)
√
M(Nt + NtN )
√
, t ≥ 0 ⇒ {B(t), t ≥ 0},
N
where B(t) is a standard Brownian motion.
Let us first prove the result in case tN is a deterministic sequence s.t.
N −1/2 tN → 0 as N → ∞.
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
6 / 11
Proof of Proposition
We have to prove two things. As N → ∞,
(1) N −1/2 M(Nt) ⇒√B(t),
(2) N −1/2 [M(Nt + NtN ) − M(Nt)] → 0 in probability.
Proof of (1). It follows from the usual CLT. Indeed
[Nt]
M(Nt)
1 X
M(Nt) − M([Nt])
p
p
=p
[M(i) − M(i − 1)] +
,
[Nt]
[Nt] i=1
[Nt]
The r.v.’s M(i) − M(i − 1) are i.i.d. centered with variance 1, and the
last term above converges in probability to 0 as N → ∞, hence
M(Nt)
p
⇒ N (0, 1),
[Nt]
p
[Nt] M(Nt)
M(Nt)
√
= √
× p
N
N
[Nt]
⇒ B(t),
where B(t) ' N (0, t).
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
7 / 11
Proof of Proposition
We have to prove two things. As N → ∞,
(1) N −1/2 M(Nt) ⇒√B(t),
(2) N −1/2 [M(Nt + NtN ) − M(Nt)] → 0 in probability.
Proof of (1). It follows from the usual CLT. Indeed
[Nt]
M(Nt)
1 X
M(Nt) − M([Nt])
p
p
=p
[M(i) − M(i − 1)] +
,
[Nt]
[Nt] i=1
[Nt]
The r.v.’s M(i) − M(i − 1) are i.i.d. centered with variance 1, and the
last term above converges in probability to 0 as N → ∞, hence
M(Nt)
p
⇒ N (0, 1),
[Nt]
p
[Nt] M(Nt)
M(Nt)
√
= √
× p
N
N
[Nt]
⇒ B(t),
where B(t) ' N (0, t).
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
7 / 11
Proof of Proposition
We have to prove two things. As N → ∞,
(1) N −1/2 M(Nt) ⇒√B(t),
(2) N −1/2 [M(Nt + NtN ) − M(Nt)] → 0 in probability.
Proof of (1). It follows from the usual CLT. Indeed
[Nt]
M(Nt)
1 X
M(Nt) − M([Nt])
p
p
=p
[M(i) − M(i − 1)] +
,
[Nt]
[Nt] i=1
[Nt]
The r.v.’s M(i) − M(i − 1) are i.i.d. centered with variance 1, and the
last term above converges in probability to 0 as N → ∞, hence
M(Nt)
p
⇒ N (0, 1),
[Nt]
p
[Nt] M(Nt)
M(Nt)
√
= √
× p
N
N
[Nt]
⇒ B(t),
where B(t) ' N (0, t).
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
7 / 11
Proof of (2)
!
M(Nt + √Nt ) − M(Nt) N
√
P >ε
N
√
1
≤
Var
M(Nt
+
Nt
)
−
M(Nt)
N
2
Nε
√
N |tN |
=
Nε2
→ 0,
provided N −1/2 tN → 0 as N → ∞.
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CIMPA, Ziguinchor
8 / 11
The case tN random
Note that
P
We split the event
(
|t |
√N > η
N
|M(Nt +
√
≤
1 E|tN |
√ .
η N
)
NtN ) − M(Nt)|
√
>ε
N
into three pieces, nintersecting withothen three events (which
o constitute
√
√
a partition of Ω) 0 ≤ tN ≤ η N , −η N ≤ tN ≤ 0 and
n
o
|t |
√N > η .
N
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
9 / 11
The case tN random
Note that
P
We split the event
(
|t |
√N > η
N
|M(Nt +
√
≤
1 E|tN |
√ .
η N
)
NtN ) − M(Nt)|
√
>ε
N
into three pieces, nintersecting withothen three events (which
o constitute
√
√
a partition of Ω) 0 ≤ tN ≤ η N , −η N ≤ tN ≤ 0 and
n
o
|t |
√N > η .
N
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
9 / 11
The probability of the first event is dominated by
!
|M(Nt + s) − M(Nt)|
√
P
sup
>ε
N
0≤s≤Nη
4η
4
2
E
|M(N(t
+
η))
−
M(Nt)|
≤
≤ 2.
Nε2
ε
The probability of the second event is estimated analogously. As for
|
√N .
the third event, its proabbility is dominated by η1 E|t
N
It remains to choose η = ε3 /8 to deduce that
lim sup P
N
Etienne Pardoux (AMU)
|M(Nt +
√
!
NtN ) − M(Nt)|
√
> ε ≤ ε.
N
CIMPA, Ziguinchor
10 / 11
The probability of the first event is dominated by
!
|M(Nt + s) − M(Nt)|
√
P
sup
>ε
N
0≤s≤Nη
4η
4
2
E
|M(N(t
+
η))
−
M(Nt)|
≤
≤ 2.
Nε2
ε
The probability of the second event is estimated analogously. As for
|
√N .
the third event, its proabbility is dominated by η1 E|t
N
It remains to choose η = ε3 /8 to deduce that
lim sup P
N
Etienne Pardoux (AMU)
|M(Nt +
√
!
NtN ) − M(Nt)|
√
> ε ≤ ε.
N
CIMPA, Ziguinchor
10 / 11
The probability of the first event is dominated by
!
|M(Nt + s) − M(Nt)|
√
P
sup
>ε
N
0≤s≤Nη
4η
4
2
E
|M(N(t
+
η))
−
M(Nt)|
≤
≤ 2.
Nε2
ε
The probability of the second event is estimated analogously. As for
|
√N .
the third event, its proabbility is dominated by η1 E|t
N
It remains to choose η = ε3 /8 to deduce that
lim sup P
N
Etienne Pardoux (AMU)
|M(Nt +
√
!
NtN ) − M(Nt)|
√
> ε ≤ ε.
N
CIMPA, Ziguinchor
10 / 11
Weak convergence
Moreover one can rather easily show that the sequence
{(UN (t), VN (t)), t ≥ 0} is tight as a process whose trajectories
belong to C ([0, +∞); R2 ). Hence along a subsequence
{(UN (t), VN (t)), t ≥ 0} ⇒ {(U(t), V (t)), t ≥ 0},
where the limit satisfies
Z t
p Z tp
U(t) = −β
[s(r )V (r ) + i(r )U(r )] dr + β
s(r )i(r )dB1 (r ),
0
0
Z t
p Z tp
s(r )i(r )dB1 (r )
V (t) = β
[s(r )V (r ) + i(r )U(r )] dr − β
0
0
Z t
Z tp
√
−α
V (r )dr + α
i(r )dB2 (r ).
0
0
The process {(U(t), V (t)), t ≥ 0} is a Gaussian process of the
Ornstein–Uhlenbeck type.
The law of the limit is uniquely determined. Hence the whole sequence
converges.
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
11 / 11
Weak convergence
Moreover one can rather easily show that the sequence
{(UN (t), VN (t)), t ≥ 0} is tight as a process whose trajectories
belong to C ([0, +∞); R2 ). Hence along a subsequence
{(UN (t), VN (t)), t ≥ 0} ⇒ {(U(t), V (t)), t ≥ 0},
where the limit satisfies
Z t
p Z tp
U(t) = −β
[s(r )V (r ) + i(r )U(r )] dr + β
s(r )i(r )dB1 (r ),
0
0
Z t
p Z tp
s(r )i(r )dB1 (r )
V (t) = β
[s(r )V (r ) + i(r )U(r )] dr − β
0
0
Z t
Z tp
√
−α
V (r )dr + α
i(r )dB2 (r ).
0
0
The process {(U(t), V (t)), t ≥ 0} is a Gaussian process of the
Ornstein–Uhlenbeck type.
The law of the limit is uniquely determined. Hence the whole sequence
converges.
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
11 / 11
Weak convergence
Moreover one can rather easily show that the sequence
{(UN (t), VN (t)), t ≥ 0} is tight as a process whose trajectories
belong to C ([0, +∞); R2 ). Hence along a subsequence
{(UN (t), VN (t)), t ≥ 0} ⇒ {(U(t), V (t)), t ≥ 0},
where the limit satisfies
Z t
p Z tp
U(t) = −β
[s(r )V (r ) + i(r )U(r )] dr + β
s(r )i(r )dB1 (r ),
0
0
Z t
p Z tp
s(r )i(r )dB1 (r )
V (t) = β
[s(r )V (r ) + i(r )U(r )] dr − β
0
0
Z t
Z tp
√
−α
V (r )dr + α
i(r )dB2 (r ).
0
0
The process {(U(t), V (t)), t ≥ 0} is a Gaussian process of the
Ornstein–Uhlenbeck type.
The law of the limit is uniquely determined. Hence the whole sequence
converges.
Etienne Pardoux (AMU)
CIMPA, Ziguinchor
11 / 11