Convergence of the empirical two-sample -statistics with -mixing data

1 {u > k + 1} h_2,s` (X_i) .

(2.10)

Defining for i ∈ In,u, 1 6 u 6 d + 1 the random variable Yn,i by

(Xi) ,

d

nt

(2.11)

Yn,i := k,`=1 ak,` ^n ^−n^[^ntk^] 1 {u 6 k} h1,s` (Xi) + ^[ n^k^] 1 {u > k + 1} h2,s`

X

it follows that

d

n

k,`=1 ak,`Wn (s`, tk) = n^−1^/^2

i=1 Yn,i. Observe also that E[Yn,i] = 0. We

want to check the conditions of Theorem A.1. The first condition follows from

d

P

P

d

|ak,`| (1 {u 6 k} |h1,s` (Xi)| + 1 {u > k + 1} |h2,s` (Xi)|) 6 2

(2.12)

|Yn,i| 6

|ak,`| .

k,`=1

k,`=1

X

X

For the second condition, we first observe that if 1 6 u < u⁰ 6 d + 1, then

1	Cov		∈^X	^Yn,i^,	∈^X		^Yn,i		→ 0.	(2.13)
						0
n
			^i In,u	i^0 In,u

Indeed, by Proposition A.2, for i ∈ I_n,u and i⁰ ∈ I_n,u0,

d

2

Cov (Yn,i, Yn,i0)

α (i^0

i)

X

^ak,`

=: α (i^0

i) K

(2.14)

| 6

−

2

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−

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