3.3 non-degenerate U-statistics with Markovian entries
The moderate deviation principle in Theorem 1.9 is stated for independent random variables. Catoni showed in [Cat03], that the estimation of the logarithm of the Laplace transform can be generalized
for Markov chains via a coupled process. In the following one can see, that analogously to the proof of Theorem 1.9 these results yield the moderate deviation principle.
In this section we use the notation introduced in [Cat03], Chapter 3. Let us assume that X
k k
∈N
is a Markov chain such that for X := X
1
, . . . , X
n
the following inequali- ties hold
P τ
i
i + k G
i
, X
i
≤ Aρ
k
∀k ∈ N a.s. 3.19
P τ
i
i + k F
n
,
i
Y
i
≤ Aρ
k
∀k ∈ N a.s. 3.20
for some positive constants A and ρ 1. Here
i
Y :=
i
Y
1
, . . . ,
i
Y
n
, i = 1, . . . , n, are n coupled stochastic processes satisfying for any i that
i
Y is equal in distribution to X . For the list of the properties of these coupled processes, see page 14 in [Cat03]. Moreover, the σ-algebra
G
i
in 3.19 is generated by
i
Y , the σ-algebra F
n
in 3.20 is generated by X
1
, . . . , X
n
. Finally the coupling stopping times τ
i
are defined as τ
i
= inf{k ≥ i|
i
Y
k
= X
k
}. Now we can state our result:
T
HEOREM
3.4. Let us assume that X
k k
∈N
is a Markov chain such that for X := X
1
, . . . , X
n
3.19 and
3.20 hold true. Let U
n
hX be a non-degenerate U-statistic with bounded kernel function h and lim
n →∞
V p
nU
n
hX ∞. Then for every sequence a
n
, where lim
n →∞
a
n
n = 0 and lim
n →∞
n a
2 n
= 0 , the sequence
n a
n
U
n
hX
n
satisfies the moderate deviation principle with speed s
n
=
a
2 n
n
and rate function I given by
I x := sup
λ∈R
¨ λx −
λ
2
2 lim
n →∞
V p
nU
n
hX «
. Proof.
As for the independent case we define f X := p
nU
n
hX
1
, . . . , X
n
. Corollary 3.1 of [Cat03] states, that in the above situation the inequality
log E
exp s f X − sEf X −
s
2
2 V
f X ≤
s
3
p n
BCA
3
1 − ρ
3
ρ log ρ
−1
2AB −
s p
n
−1 +
+ s
3
p n
B
3
A
3
31 − ρ
3
+ 4B
2
A
3
1 − ρ
3
ρ log ρ
−1
2AB −
s p
n
−1 +
2650
holds for some constants B and C. This is the situation of Theorem 1.9 except that in this case dn is defined by
1 p
n BCA
3
1 − ρ
3
ρ log ρ
−1
2AB −
s p
n
−1 +
+ 1
p n
B
3
A
3
31 − ρ
3
+ 4B
2
A
3
1 − ρ
3
ρ log ρ
−1
2AB −
s p
n
−1 +
. This expression depends on s. We apply the adapted Theorem 1.9 for s
n
=
a
2 n
n
, t
n
:=
a
n
p n
and s := λ
a
n
p n
as before. Because of
s p
n
= λ
a
n
n n
→∞
−→ 0, the assumptions of Theorem 1.9 are satisfied: 1.
s
2 n
t
3 n
dn = a
n
n BCA
3
1 − ρ
3
ρ log ρ
−1
2AB −
s p
n
−1 +
+ a
n
n B
3
A
3
31 − ρ
3
+ 4B
2
A
3
1 − ρ
3
ρ log ρ
−1
2AB −
s p
n
−1 +
n →∞
−→ 0 . 2.
s
n
t
2 n
V f X = V
p nU
n
hX ∞ as assumed.
Therefore we can use the Gärtner-Ellis theorem to prove the moderate deviation principle for
n a
n
U
n
hX
n
.
C
OROLLARY
3.5. Let X
Kategori