Limit laws for the maxima of stationary chi-processes under random index

Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\{\chi _{k}(t), t\ge 0\}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="italic">χ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> be a stationary <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\chi $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">χ</mi> </math> </EquationSource> </InlineEquation>-process with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$k$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>k</mi> </math> </EquationSource> </InlineEquation> degrees of freedom. In this paper, we consider the maxima <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$M_{k}(T)= \max \{\chi _{k}(t), \forall t\in [0,T]\}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>M</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">max</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="italic">χ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>∀</mo> <mi>t</mi> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">}</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> with random index <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\mathcal {T}_{T}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="script">T</mi> <mi>T</mi> </msub> </math> </EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\mathcal {T}_{T}/T$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi mathvariant="script">T</mi> <mi>T</mi> </msub> <mo stretchy="false">/</mo> <mi>T</mi> </mrow> </math> </EquationSource> </InlineEquation> converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$M_{k}(\mathcal {T}_{T})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>M</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">T</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> exists under some additional conditions. Copyright Sociedad de Estadística e Investigación Operativa 2014