On the central limit theorem along subsequences of sums of i.i.d. random variables

Deli Li (), Oleg Klesov () and George Stoica ()

Statistical Papers, 2014, vol. 55, issue 4, 1035-1045

Abstract: Let $$\mathbb{N }=\{1, 2, 3, \ldots \}$$ N = { 1 , 2 , 3 , … } . Let $$\{X, X_{n}; n \in \mathbb N \}$$ { X , X n ; n ∈ N } be a sequence of i.i.d. random variables, and let $$S_{n}=\sum _{i=1}^{n}X_{i}, n \in \mathbb N $$ S n = ∑ i = 1 n X i , n ∈ N . Then $$ S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})$$ S n / n ⇒ N ( 0 , σ 2 ) for some $$\sigma ^{2} > \infty $$ σ 2 > ∞ whenever, for a subsequence $$\{n_{k}; k \in \mathbb N \}$$ { n k ; k ∈ N } of $$\mathbb N $$ N , $$ S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})$$ S n k / n k ⇒ N ( 0 , σ 2 ) . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when $$\{\sqrt{n}; n \in \mathbb N \}$$ { n ; n ∈ N } is replaced by $$\{\sqrt{na_{n}};n \in \mathbb N \}$$ { n a n ; n ∈ N } with $$\lim _{n \rightarrow \infty } a_{n}=\infty $$ lim n → ∞ a n = ∞ . We show that, for given positive nondecreasing sequence $$\{a_{n}; n \in \mathbb N \}$$ { a n ; n ∈ N } with $$\lim _{n \rightarrow \infty } a_{n}=\infty $$ lim n → ∞ a n = ∞ and $$\lim _{n \rightarrow \infty } a_{n+1}/a_{n}=1$$ lim n → ∞ a n + 1 / a n = 1 and given nondecreasing function $$h(\cdot ): (0, \infty ) \rightarrow (0, \infty )$$ h ( · ) : ( 0 , ∞ ) → ( 0 , ∞ ) with $$\lim _{x \rightarrow \infty } h(x)=\infty $$ lim x → ∞ h ( x ) = ∞ , there exists a sequence $$\{X, X_{n}; n \in \mathbb N \}$$ { X , X n ; n ∈ N } of symmetric i.i.d. random variables such that $$\mathbb E h(|X|)=\infty $$ E h ( | X | ) = ∞ and, for some subsequence $$\{n_{k}; k \in \mathbb N \}$$ { n k ; k ∈ N } of $$\mathbb N $$ N , $$ S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)$$ S n k / n k a n k ⇒ N ( 0 , 1 ) . In particular, for given $$0 > p > 2$$ 0 > p > 2 and given nondecreasing function $$h(\cdot ): (0, \infty ) \rightarrow (0, \infty )$$ h ( · ) : ( 0 , ∞ ) → ( 0 , ∞ ) with $$\lim _{x \rightarrow \infty } h(x)=\infty $$ lim x → ∞ h ( x ) = ∞ , there exists a sequence $$\{X, X_{n}; n \in \mathbb N \}$$ { X , X n ; n ∈ N } of symmetric i.i.d. random variables such that $$\mathbb E h(|X|)=\infty $$ E h ( | X | ) = ∞ and, for some subsequence $$\{n_{k}; k \in \mathbb N \}$$ { n k ; k ∈ N } of $$\mathbb N $$ N , $$ S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)$$ S n k / n k 1 / p ⇒ N ( 0 , 1 ) . Copyright Springer-Verlag Berlin Heidelberg 2014

Keywords: Central limit theorem; Law of large numbers; Subsequences; Sums of i.i.d. random variables; Primary: 60F05; Secondary: 60G50 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s00362-013-0551-9

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