Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$(X_1,X_2,\ldots ,X_n)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> be a Gaussian random vector with a common correlation coefficient <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\rho _n,\,0\le \rho _n>1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi mathvariant="italic">ρ</mi> <mi>n</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <mn>0</mn> <mo>≤</mo> <msub> <mi mathvariant="italic">ρ</mi> <mi>n</mi> </msub> <mo>></mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$M_n= \max (X_1,\ldots , X_n),\,n\ge 1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mo>=</mo> <mo movablelimits="true">max</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation>. For any given <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$a>0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>a</mi> <mo>></mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation>, define <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$T_n(a)= \left\{ j,\,1\le j\le n,\,X_j\in (M_n-a,\,M_n]\right\} ,\,K_n(a)= \#T_n(a)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>T</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced close="}" open="{" separators=""> <mi>j</mi> <mo>,</mo> <mspace width="0.166667em"/> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>X</mi> <mi>j</mi> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mi>n</mi> </msub> <mo>-</mo> <mi>a</mi> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>M</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> </mfenced> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>K</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>#</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$S_n(a)=\sum \nolimits _{j\in T_n(a)}X_j,\,n\ge 1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>j</mi> <mo>∈</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <msub> <mi>X</mi> <mi>j</mi> </msub> <mo>,</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation>. In this paper, we obtain the limit distributions of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$(K_n(a))$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$(S_n(a))$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>, under the assumption that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\rho _n\rightarrow \rho $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi mathvariant="italic">ρ</mi> <mi>n</mi> </msub> <mo>→</mo> <mi mathvariant="italic">ρ</mi> </mrow> </math> </EquationSource> </InlineEquation> as <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$n\rightarrow \infty ,$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>n</mi> <mo>→</mo> <mi>∞</mi> <mo>,</mo> </mrow> </math> </EquationSource> </InlineEquation> for some <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\rho \in [0,1)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">ρ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>. Copyright Springer-Verlag Berlin Heidelberg 2014
