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">$$\rho _n,\,0\le \rho _n1$$</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></mrow></math></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>

We consider the problem of testing the null hypothesis of no change against the alternative of multiple change points in a series of independent observations. We propose an ANOVA-type test statistic and obtain its asymptotic null distribution. We also give approximations of its limiting critical...

Let {χk(t),t≥0} be a stationary χ-process with k degrees of freedom being independent of some non-negative random variable T. In this paper we derive the exact asymptotics of P{supt∈[0,T]χk(t)u} as u→∞ when T has a regularly varying tail with index λ∈[0,1). Three other novel...

This paper presents some asymptotic results for statistics of Brownian semi-stationary (BSS) processes. More precisely, we consider power variations of BSS processes, which are based on high frequency (possibly higher order) differences of the BSS model. We review the limit theory discussed by...

Let ηt be a Poisson point process of intensity t≥1 on some state space Y and let f be a non-negative symmetric function on Yk for some k≥1. Applying f to all k-tuples of distinct points of ηt generates a point process ξt on the positive real half-axis. The scaling limit of ξt as t tends...

In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [0,T] when T→∞. We further exhibit the asymptotic behaviour of the covariation of the increments of the...

We consider generalized linear models for regression modeling of count time series. We give easily verifiable conditions for obtaining weak dependence for such models. These results enable the development of maximum likelihood inference under minimal conditions. Some examples which are useful to...

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the...

This paper presents some asymptotic results for statistics of Brownian semi-stationary (BSS) processes. More precisely, we consider power variations of BSS processes, which are based on high frequency (possibly higher order) differences of the BSS model. We review the limit theory discussed in...