Convergence of the empirical twosample statistics with mixing data
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 CONVERGENCE OF THE EMPIRICAL TWOSAMPLE STATISTICS WITH
 Convergence of the twosample
(1 {g (X_{i}, X_{j}) 6 s} − P{g (X_{i}, X_{j}) 6 s}) CONVERGENCE OF THE EMPIRICAL TWOSAMPLE STATISTICS WITH MIXING DATA HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV Abstract. We consider the empirical twosample statistic with strictly mixing strictly stationary data and inverstigate its convergence in Skorohod spaces. We then provide an application of such convergence. 1. INTRODUCTION AND MAIN RESULTS Let a strictly stationary mixing sequence of real valued random variables. In this paper, we will study the asymptotic behavior of the empirical statistic defined by _{ (1,1)} where kj→ R is a measurable function and for a real number x, [x] denotes the unique integer k such that k 6 x < k + 1. The following is known: • In [2], the convergence of the 1sample Ustatistic defined by 1
has been investigated. • In [4], the convergence of the 2sample Ustatistic of kernel h defined by Tn (t) := 1 n3/2 [nt] Xi =1 nX j=[nt]+1 h (Xi, Xj) (1.3) has been investigated. In this paper, we focus on the case of mixing sequences. Let (Ω, F, P) be a probability space. The αmixing and βmixing coefficients between two subσalgebras A and B of F are defined defined respectively by
CONVERGENCE OF THE EMPIRICAL TWOSAMPLE STATISTICS WITH βMIXING DATA HEROLD DEHLING, DAVIDE GIRAUDO AND OLIMJON SHARIPOV Abstract. We consider the empirical twosample Ustatistic with strictly βmixing strictly stationary data and inverstigate its convergence in Skorohod spaces. We then provide an application of such convergence.
Let (X_{i})_{i}_{>1} a strictly stationary mixing sequence of real valued random variables. In this paper, we will study the asymptotic behavior of the empirical Ustatistic defined by
(1.1) where g : R^{2} → R is a measurable function and for a real number x, [x] denotes the unique integer k such that k 6 x < k + 1. The following is known:
1 X
(1.2) _{n}3/2 16i6=j6n has been investigated.
has been investigated. In this paper, we focus on the case of mixing sequences. Let (Ω, F, P) be a probability space. The αmixing and βmixing coeﬃcients between two subσalgebras A and B of F are defined defined respectively by
Date: April 3, 2020. Key words and phrases. Ustatistics, empirical process . 1
_{k}_{>1} k^{p}α (k) converges.
where the supremum runs over all the partitions (A_{i})^{I}_{i}_{=1} and (B_{j})^{J}_{j}_{=1} of Ω of elements of A and B respectively. Given a sequence (X_{i})_{i}_{>1}, we associate its sequences of α and βmixing coeﬃcients by letting
where F_{u}^{v}, 1 6 u 6 v 6 +∞ is the σalgebra generated by the random variables X_{i}, u 6 i 6 v (u 6 i for v = +∞). 1.1. Convergence of the twosample Ustatistic in Skorohod spaces D ([−R, R] × [0, 1]). Let us state one of the two main results of the paper. Theorem 1.1. Let (X_{i})_{i}_{∈}_{Z} be a strictly stationary sequence. Let e_{n} be the twosample Ustatistics empirical process with kernel g : R × R → R defined for n > 1, 0 6 t 6 1 and s ∈ R by
Suppose that the following four conditions holds. (A.1) For all u ∈ R, the random variable g (u, X_{1}) has a density f_{1}_{,u} +∞. (A.2) For all v ∈ R, the random variable g (X_{1}, v) has a density f_{2}_{,v} +∞. P
P (A.4) The series _{k}_{>1}_{ }kβ (k) converges. Then for all R, ^{and} ^{sup}x,u∈R ^{f}1,u ^{(}^{x}^{)} ^{<} ^{and} ^{sup}x,v∈R ^{f}2,v ^{(}^{x}^{)} ^{<}
where (W (s, t) , s ∈ R, t ∈ [0, 1]) is a centered Gaussian process, with covariance given for 0 6 t 6 t^{0} 6 1 and s, s^{0} ∈ R by the following formula:
Remark 1.2. We did not make a symmetry assumption on g. When g is symmetric, in the sense that g (u, v) = g (v, u) for all u and v ∈ R, the covariance of the limiting process W reads
3
^{n} ^{−1} ^{P}_{16}_{i}_{6}_{n} 1 {g (X_{i}, X_{j}) 6 s}. 2 Download 380.03 Kb. Do'stlaringiz bilan baham: 
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