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

Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathbb{N }=\{1, 2, 3, \ldots \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="double-struck">N</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\{X, X_{n}; n \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>;</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> be a sequence of i.i.d. random variables, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$S_{n}=\sum _{i=1}^{n}X_{i}, n \in \mathbb N $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math> </EquationSource> </InlineEquation>. Then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$ S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo stretchy="false">/</mo> <msqrt> <mi>n</mi> </msqrt> <mo stretchy="false">⇒</mo> <mi>N</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi mathvariant="italic">σ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> for some <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\sigma ^{2} > \infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msup> <mi mathvariant="italic">σ</mi> <mn>2</mn> </msup> <mo>></mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation> whenever, for a subsequence <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\{n_{k}; k \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <msub> <mi>n</mi> <mi>k</mi> </msub> <mo>;</mo> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\mathbb N $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="double-struck">N</mi> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$ S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>S</mi> <msub> <mi>n</mi> <mi>k</mi> </msub> </msub> <mo stretchy="false">/</mo> <msqrt> <msub> <mi>n</mi> <mi>k</mi> </msub> </msqrt> <mo stretchy="false">⇒</mo> <mi>N</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi mathvariant="italic">σ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation>. Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\{\sqrt{n}; n \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <msqrt> <mi>n</mi> </msqrt> <mo>;</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> is replaced by <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$\{\sqrt{na_{n}};n \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <msqrt> <mrow> <mi>n</mi> <msub> <mi>a</mi> <mi>n</mi> </msub> </mrow> </msqrt> <mo>;</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> with <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\lim _{n \rightarrow \infty } a_{n}=\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo>→</mo> <mi>∞</mi> </mrow> </msub> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo>=</mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation>. We show that, for given positive nondecreasing sequence <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$\{a_{n}; n \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo>;</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> with <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$\lim _{n \rightarrow \infty } a_{n}=\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo>→</mo> <mi>∞</mi> </mrow> </msub> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo>=</mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$$\lim _{n \rightarrow \infty } a_{n+1}/a_{n}=1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo>→</mo> <mi>∞</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation> and given nondecreasing function <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$$h(\cdot ): (0, \infty ) \rightarrow (0, \infty )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> <mo>:</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo>→</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> with <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$$\lim _{x \rightarrow \infty } h(x)=\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>x</mi> <mo>→</mo> <mi>∞</mi> </mrow> </msub> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation>, there exists a sequence <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$$\{X, X_{n}; n \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>;</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> of symmetric i.i.d. random variables such that <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$$\mathbb E h(|X|)=\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="double-struck">E</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation> and, for some subsequence <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$$\{n_{k}; k \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <msub> <mi>n</mi> <mi>k</mi> </msub> <mo>;</mo> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> of <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$$\mathbb N $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="double-struck">N</mi> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$$ S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>S</mi> <msub> <mi>n</mi> <mi>k</mi> </msub> </msub> <mo stretchy="false">/</mo> <msqrt> <mrow> <msub> <mi>n</mi> <mi>k</mi> </msub> <msub> <mi>a</mi> <msub> <mi>n</mi> <mi>k</mi> </msub> </msub> </mrow> </msqrt> <mo stretchy="false">⇒</mo> <mi>N</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation>. In particular, for given <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$$0 > p > 2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>0</mn> <mo>></mo> <mi>p</mi> <mo>></mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation> and given nondecreasing function <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$$h(\cdot ): (0, \infty ) \rightarrow (0, \infty )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> <mo>:</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo>→</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> with <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$$\lim _{x \rightarrow \infty } h(x)=\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>x</mi> <mo>→</mo> <mi>∞</mi> </mrow> </msub> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation>, there exists a sequence <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$$\{X, X_{n}; n \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>;</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> of symmetric i.i.d. random variables such that <InlineEquation ID="IEq26"> <EquationSource Format="TEX">$$\mathbb E h(|X|)=\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="double-struck">E</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation> and, for some subsequence <InlineEquation ID="IEq27"> <EquationSource Format="TEX">$$\{n_{k}; k \in \mathbb N \}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <msub> <mi>n</mi> <mi>k</mi> </msub> <mo>;</mo> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> of <InlineEquation ID="IEq28"> <EquationSource Format="TEX">$$\mathbb N $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="double-struck">N</mi> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq29"> <EquationSource Format="TEX">$$ S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>S</mi> <msub> <mi>n</mi> <mi>k</mi> </msub> </msub> <mo stretchy="false">/</mo> <msubsup> <mi>n</mi> <mrow> <mi>k</mi> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> </msubsup> <mo stretchy="false">⇒</mo> <mi>N</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation>. 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