Peer-reviewed Journal Articles

2014

A. Račkauskas and Ch. Suquet. Computing the distribution of sequential Hölder norms of the Brownian motion. Communications in Statistics – Theory and Methods, 2014. to appear.

In this paper we give explicit formulas and study practical computations for the distribution function of sequential Hölder norms of a Brownian motion and of a Brownian bridge. We also discuss some statistical applications in the detection of some short ``epidemic'' changes in a sample.

@article{Rackauskas-Suquet:2014a,

author       = {Ra{\v c}kauskas, A. and Suquet, Ch.},

title        = {Computing the distribution of sequential H{\''o}lder norms of the Brownian motion},

journal      = {Communications in Statistics -- Theory and Methods},

year         = {2014},

note         = {to appear},
}
J. Markevičiūtė, A. Račkauskas, and Ch. Suquet. Testing the epidemic change in nearly nonstationary process. Nonlinear Analysis: Modelling and Control, 19(1):67–82, 2014.

Some tests for an epidemic type change in a first order nearly nonstationary autoregressive process are investigated. Limit distributions of the tests are found under no change. Consistency is examined under short epidemics in the mean of innovations

@article{Markeviciute-Rackauskas-Suquet:2013,

author       = {Markevi\v{c}i\={u}t\.{e}, J. and Ra{č}kauskas, A. and Suquet, Ch.},

title        = {Testing the epidemic change in nearly nonstationary process},

journal      = {Nonlinear Analysis: Modelling and Control},

volume       = {19},

number       = {1},

pages        = {67--82},

year         = {2014},
}

2013

A. Račkauskas and Ch. Suquet. Functional laws of large numbers in Hölder spaces. ALEA Lat. Am. J. Probab. Math. Stat., 10(2):609–624, 2013.

Let $S_n=X_1+\dots+X_n$, $n\ge 1$, where $(X_i)_{i\ge 1}$ are random variables. Let $\mu$ be a constant and $I$ be the identity function on $[0,1]$. We study the almost sure convergence to $\mu I$ of the two polygonal line partial sums processes $\zeta_n$ and $\zeta_n^{\mathrm{ad}}$ with respective vertices $(k/n,S_k)$ and $(\tau_k,S_k)$, $0\le k\le n$, where $\tau_k=T_k/T_n$ and $T_k=\abs{X_1}+\dots+\abs{X_k}$. These convergences are considered in the space $C[0,1]$ or in the Hölder spaces $\mathcal{H}_\alpha^o[0,1]$, $0\le \alpha<1$. In $C[0,1]$, any strong law of large numbers satisfied by $S_n$ is inherited by $\zeta_n$. In $\mathcal{H}_\alpha^o[0,1]$, assuming moreover that the $X_i$'s are i.i.d., $n^{-1}\zeta_n$ converges almost surely to $\mu I$ if and only if $\E\abs{X_1}^{1/(1-\alpha)}<\infty$ and $\mu=\E X_1$. In contrast, the same convergence for $\zeta_n^{\mathrm{ad}}$ is equivalent to $\E\abs{X_1}<\infty$ and $\mu=\E X_1$.

@article{Rackauskas-Suquet:2013a,

author       = {Ra{č}kauskas, A. and Suquet, Ch.},

title        = {Functional Laws of Large Numbers in {H}\"{o}lder Spaces},

journal      = {ALEA Lat. Am. J. Probab. Math. Stat.},

volume       = {10},

number       = {2},

pages        = {609--624},

year         = {2013},
}

2012

J. Markevičiūtė, A. Račkauskas, and Ch. Suquet. Functional central limit theorems for sums of nearly nonstationary processes. Lithuanian Math. J., 52(3):282–296, July 2012.

We study some Hölderian functional central limit theorems for the polygonal line partial sum processes built on a first order autoregressive process $y_{n,k} = \phi_n y_{n,k-1} + \epsilon_k$ with $\phi_n$ converging to $1$ and i.i.d. centered square integrable innovations. In the case where $\phi_n = \eexp{\gamma/n}$ with a negative constant $\gamma$, we prove that the limiting process is an integrated Ornstein-Uhlenbeck one. In the case where $\phi_n = 1 - \gamma_n/n$, with $\gamma_n$ tending to infinity slower than $n$, the convergence to Brownian motion is established in Hölder space in terms of the rate of $\gamma_n$ and the integrability of the $\epsilon_k$'s.

@article{Markeviciute-Rackauskas-Suquet:2012,

author       = {Markevi\v{c}i\={u}t\.{e}, J. and Ra{č}kauskas, A. and Suquet, Ch.},

title        = {Functional central limit theorems for sums of nearly nonstationary processes},

journal      = {Lithuanian Math. J.},

volume       = {52},

number       = {3},

pages        = {282--296},

month        = jul,

year         = {2012},
}

2011

A. Račkauskas and Ch. Suquet. Functional central limit theorems for self-normalized partial sums of linear processes. Lithuanian Math. J., 51(2):251–259, April 2011.

We prove the invariance principle under self-normalization by blocks for linear processes with summable filters and {i.i.d.} innovations in the domain of attraction of the normal distribution.

@article{Rackauskas-Suquet:2011,

author       = {Ra{č}kauskas, A. and Suquet, Ch.},

title        = {Functional central limit theorems for self-normalized partial sums of linear processes},

journal      = {Lithuanian Math. J.},

volume       = {51},

number       = {2},

pages        = {251--259},

month        = apr,

year         = {2011},
}
A. Račkauskas and Ch. Suquet. Operator fractional Brownian motion as limit of polygonal lines processes in Hilbert space. Stoch. Dyn., 11(1):49–70, 2011.

In this paper we study long memory phenomenon of functional time series. We consider an operator fractional Brownian motion with values in a Hilbert space defined via operator valued Hurst coefficient. We prove that this process is a limiting one for polygonal lines constructed from partial sums of time series having space varying long memory.

@article{Rackauskas-Suquet:2011b,

author       = {Ra{č}kauskas, A. and Suquet, Ch.},

title        = {Operator fractional {B}rownian motion as limit of polygonal lines processes in {H}ilbert space},

journal      = {Stoch. Dyn.},

volume       = {11},

number       = {1},

pages        = {49--70},

year         = {2011},
}

2010

A. Račkauskas and Ch. Suquet. On limit theorems for Banach space valued linear processes. Lithuanian Math. J., 50(1):71–87, 2010.

Let $(\epsilon_i)_{i\in\mathbb{Z}}$ be {i.i.d.} random elements in the separable Banach space $\mathbb{E}$ and $(a_i)_{i\in\mathbb{Z}}$ be continuous linear operators from $\mathbb{E}$ to the Banach space $\mathbb{F}$, such that $\sum_{i\in\mathbb{Z}}\|a_i\|$ is finite. We prove that the linear process $(X_n)_{n\in\mathbb{Z}}$ defined by $X_n:=\sum_{i\in\mathbb{Z}}a_i(\epsilon_{n-i})$ inherits from $(\epsilon_i)_{i\in\mathbb{Z}}$ the central limit theorem as well as functional central limit theorems in various Banach spaces of $\mathbb{F}$ valued functions, including Hölder spaces.

@article{Rackauskas-Suquet:2010a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {On limit theorems for {B}anach space valued linear processes},

journal      = {Lithuanian Math. J.},

volume       = {50},

number       = {1},

pages        = {71--87},

year         = {2010},
}

2009

M. Juodis, A. Račkauskas, and Ch. Suquet. Hölderian functional central limit theorems for linear processes. ALEA Lat. Am. J. Probab. Math. Stat., 5:1–18, 2009.

Let $(\xi_n)_{n\ge 1}$ be the polygonal partial sums processes built on the linear processes $X_n=\sum_{i\ge 0}a_i\epsilon_{n-i}$, where $(\epsilon_i)_{i\in\mathbb{Z}}$ are i.i.d., centered and square integrable random variables with $\sum_{i\ge 0}a_i^2<\infty$. We investigate functional central limit theorem for $\xi_n$ in the {H}ölder spaces $\mathrm{H}_\alpha^o[0,1]$ of functions $x:[0,1]\to\mathbb{R}$ such that $|x(t+h)-x(t)|=o(h^\alpha)$ uniformly in $t$. When $\sum_{i\ge 0}|a_i|<\infty$ (short memory case), we show that $n^{-1/2}\xi_n$ converges weakly in $\mathrm{H}_\alpha^o[0,1]$ to some Brownian motion under the optimal assumption that $P(|\epsilon_0|\ge t)=o(t^{-p})$, where $1/p=1/2-\alpha$. This extends the Lamperti invariance principle for i.i.d. $X_n$'s. When $a_i=\ell(i)i^{-\beta}$, $1/2<\beta<1$, with $\ell$ positive, increasing and slowly varying, $(X_n)_{n\ge 1}$ has long memory. The limiting process for $\xi_n$ is then the fractional Brownian motion $W^H$ with Hurst index $H=3/2-\beta$ and the normalizing constants are $ b_n=c_\beta n^H\ell(n)$. For $0 < \alpha < H-1/2$, the weak convergence of $b_n^{-1}\xi_n$ to $W^H$ in $\mathrm{H}_\alpha^o[0,1]$ is obtained under the mild assumption that $\E\epsilon_0^2 < \infty.$ For $H-1/2 < \alpha < H$, the same convergence is obtained under $P(|\epsilon_0|\ge t)=o(t^{-p})$, where $1/p=H-\alpha$.

@article{Juodis-Rackauskas-Suquet:2007,

author       = {Juodis, M. and Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {H\"{o}lderian functional central limit theorems for linear processes},

journal      = {ALEA Lat. Am. J. Probab. Math. Stat.},

volume       = {5},

pages        = {1--18},

year         = {2009},
}
Ch. Suquet. Reproducing kernel Hilbert spaces and random measures. In H.G.W. Begehr and F. Nicolosi, editors, Proceedings of 5th International ISAAC Congress, Catania, Italy, 25-30 July 2005, pages 143–152. World Scientific, 2009.

We present a constructive approach of Guilbart's embedding of signed measures in a reproducing kernel Hilbert space. This enables us to define signed random measures as Hilbertian random elements and exploit the nice probabilistic properties of Hilbert space when studying limit theorems for random measures. Strong law of large numbers, central limit theorems, functional central limit theorems are given. As a statistical illustration we propose a test procedure to detect an epidemic change in the distribution of random measures.

@inproceedings{Suquet:2005,

author       = {Suquet, Ch.},

editor       = {Begehr, H.G.W. and Nicolosi, F.},

title        = {Reproducing kernel {H}ilbert spaces and random measures},

booktitle    = {Proceedings of 5th International ISAAC Congress, Catania, Italy, 25-30 July 2005},

pages        = {143--152},

year         = {2009},

publisher    = {World Scientific},
}

2008

A. Račkauskas and Ch. Suquet. Hölderian invariance principle for Hilbertian linear processes. ESAIM Probab. Statist., 13:261–275, 2008.

Let $(\xi_n)_{n\ge 1}$ be the polygonal partial sums processes built on the linear processes $X_n=\sum_{i\ge 0}a_i(\epsilon_{n-i})$, $n\ge 1$, where $(\epsilon_i)_{i\in\mathbb{Z}}$ are i.i.d., centered random elements in some separable Hilbert space $\mathbb{H}$ and the $a_i$'s are bounded linear operators $\mathbb{H}\to \mathbb{H}$, with $\sum_{i\ge 0}\lVert a_i \rVert < \infty$. We investigate functional central limit theorem for $\xi_n$ in the Hölder spaces $\mathrm{H}^o_\rho(\mathbb{H})$ of functions $x:[0,1]\to\mathbb{H}$ such that $\lVert x(t+h)-x(t)\rVert =o(\rho(h))$ uniformly in $t$, where $\rho(h)=h^\alpha L(1/h)$, $0\leq h\leq 1$ with $0<\alpha\leq 1/2$ and $L$ slowly varying at infinity. We obtain the $\mathrm{H}^o_\rho(\mathbb{H})$ weak convergence of $\xi_n$ to some $\mathbb{H}$ valued Brownian motion under the optimal assumption that for any $c>0$, $tP(\lVert \epsilon_0 \rVert > ct^{1/2}\rho(1/t)) = o(1)$ when $t$ tends to infinity, subject to some mild restriction on $L$ in the boundary case $\alpha=1/2$. Our result holds in particular with the weight functions $\rho(h)=h^{1/2}\ln^\beta(1/h)$, $\beta>1/2$.

@article{Rackauskas-Suquet:2008a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {H\"{o}lderian invariance principle for {H}ilbertian linear processes},

journal      = {ESAIM Probab. Statist.},

volume       = {13},

pages        = {261--275},

year         = {2008},
}

2007

A. Račkauskas and Ch. Suquet. Estimating a changed segment in a sample. Acta Appl. Math., 97:189–210, 2007.

In the paper we consider a changed segment model for sample distributions. We generalize Dümbgen's change point estimator and obtain optimal rates of convergence of estimators of the begining and the length of the changed segment.

@article{Rackauskas-Suquet:2007a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Estimating a changed segment in a sample},

journal      = {Acta Appl. Math.},

volume       = {97},

pages        = {189--210},

year         = {2007},
}
A. Račkauskas, Ch. Suquet, and V. Zemlys. Hölderian functional central limit theorem for multi-indexed summation process. Stochastic Processes Appl., 117(8):1137–1164, 2007.

Let $\{X_{\bm{j}}$; $\bm{j}\in\N^d$, $\bm{j}\geq \bm{1}\}$ be an i.i.d. random field of square integrable centered random elements in the separable Hilbert space $\mathbb{H}$ and $\xi_{\bm{n}}$, $\bm{n}\in\N^d$, be the summation processes based on the collection of sets $[0,t_1]\times\dots\times[0,t_d]$, $0\le t_i\le 1$, $i=1,\dots,d$. When $d\ge 2$, we characterize the weak convergence of $(n_1\dots n_d)^{-1/2}\xi_{\bm{n}}$ in the Hölder space $\mathrm{H}_\alpha^o(\mathbb{H})$ by the finiteness of the weak $p$ moment of $\Vert X_{\bm{1}}\Vert$ for $p=(1/2-\alpha)^{-1}$. This contrasts with the Hölderian FCLT for $d=1$ and $\mathbb{H}=\mathbb{R}$ (Račkauskas, Suquet, 2003) where the necessary and sufficient condition is $P(|X_1|>t)=o(t^{-p})$.

@article{Rackauskas-Suquet-Zemlys:2007,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch. and Zemlys, V.},

title        = {H\"{o}lderian functional central limit theorem for multi-indexed summation process},

journal      = {Stochastic Processes Appl.},

volume       = {117},

number       = {8},

pages        = {1137--1164},

year         = {2007},
}

2006

D. Hamadouche and Ch. Suquet. Optimal Hölderian functional central limit theorem for uniform empirical and quantile processes. Mathematical Methods of Statistics, 15(2):207–223, 2006.

Let $\mathrm{H}_\rho^o$ be the {H}ölder space of functions $x: [0,1]\to \mathbb{R}$ such that $\vert x(t+h) - x(t) \vert = o(\rho(h))$ uniformly in $t\in [0,1]$, where $\rho(h)=h^\alpha L(1/h)$ with $0<\alpha<1$ and $L$ is normalized slowly varying. Denote by $\xi_n^{\mathrm{pg}}$ the polygonal smoothing of the uniform empirical process. We prove that $\xi_n^{\mathrm{pg}}$ converges weakly in $\mathrm{H}_\rho^o$ to the Brownian bridge $B$ if and only if $h^{1/4}= o(\rho(h))$. We also prove that the polygonal smoothing $\chi_n^{\mathrm{pg}}$ of the uniform quantile process converges weakly in $\mathrm{H}_\rho^o$ to $B$ if and only if $h^{1/2}\ln(1/h)=o(\rho(h))$.

@article{Hamadouche-Suquet:2006,

author       = {Hamadouche, D. and Suquet, Ch.},

title        = {Optimal {H}{\"{o}}lderian functional central limit theorem for uniform empirical and quantile processes},

journal      = {Mathematical Methods of Statistics},

volume       = {15},

number       = {2},

pages        = {207--223},

year         = {2006},
}
A. Račkauskas and Ch. Suquet. Testing epidemic changes of infinite dimensional parameters. Statistical Inference for Stochastic Processes, 9:111–134, 2006.

To detect epidemic change in the mean of a sample of size $n$ of random elements in a Banach space, we introduce new test statistics DI based on weighted increments of partial sums. We obtain their limit distributions under the null hypothesis of no change in the mean. Under alternative hypothesis our statistics can detect very short epidemics of length $\log^\gamma n$, $\gamma>1$. We present applications to detect epidemic changes in distribution function or characteristic function of real valued observations as well as changes in covariance matrixes of random vectors.

@article{Rackauskas-Suquet:2006a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Testing epidemic changes of infinite dimensional parameters},

journal      = {Statistical Inference for Stochastic Processes},

volume       = {9},

pages        = {111--134},

year         = {2006},
}

2005

A. Ayache, P. Heinrich, L. Marsalle, and Ch. Suquet. Hölderian random functions. In J. Lévy Véhel and E. Lutton, editors, Fractals in Engineering. New Trends in Theory and Applications. Springer-Verlag, 2005.

Hölder regularity which plays a key rôle in fractal geometry raises an increasing interest in probability and statistics. In this paper we discuss various aspects of local and global regularity for stochastic processes and random fields. As a main result we show the invariability of the pointwise Hölder exponent of a continuous and nowhere differentiable random field which has stationary increments and satisfies a zero-one law. We also survey some recent uses of Hölder spaces in limit theorems for stochastic processes and statistics.

@inproceedings{Ayache-Heinrich-Marsalle-Suquet:2005,

author       = {Ayache, A. and Heinrich, P. and Marsalle, L. and Suquet, Ch.},

editor       = {L\'{e}vy V\'{e}hel, J. and Lutton, E.},

title        = {{H}\"{o}lderian random functions},

booktitle    = {Fractals in Engineering. New Trends in Theory and Applications},

year         = {2005},

publisher    = {Springer-Verlag},
}
A. Račkauskas and Ch. Suquet. Estimation de points de rupture pour des paramètres infinidimensionnels dans les courtes épidémies. Lietuvos Matematikos Rinkinys, 45(4):1–20, 2005.

Nous considérons une rupture épidémique dans la loi d'un échantillon à valeurs réelles ou dans l'espérance d'éléments aléatoires banachiques. Pour ces deux modèles, nous proposons des procédures consistantes d'estimation de la longueur et de la localisation de la rupture épidémique.

@article{Rackauskas-Suquet:2005b,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Estimation de points de rupture pour des param\`{e}tres infinidimensionnels dans les courtes \'{e}pid\'{e}mies},

journal      = {Lietuvos Matematikos Rinkinys},

volume       = {45},

number       = {4},

pages        = {1--20},

year         = {2005},
}
A. Račkauskas and Ch. Suquet. Estimation of change points of infinite dimensional parameters in short epidemic. Lithuanian Mathematical Journal, 45(4):458–474, 2005.

@article{Rackauskas-Suquet:2005c,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Estimation of change points of infinite dimensional parameters in short epidemic},

journal      = {Lithuanian Mathematical Journal},

volume       = {45},

number       = {4},

pages        = {458--474},

year         = {2005},
}
A. Račkauskas and Ch. Suquet. Hölderian invariance principles and some applications for testing epidemic changes. In A. Kirman and G. Teyssière, editors, Long Memory in Economics, pages 129–156. Springer, 2005. ISBN: 3-540-22694-X.

@incollection{Rackauskas-Suquet:2005a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

editor       = {Kirman, A. and Teyssi\`{e}re, G.},

title        = {H{\"{o}}lderian invariance principles and some applications for testing epidemic changes},

booktitle    = {Long Memory in Economics},

pages        = {129-156},

year         = {2005},

publisher    = {Springer},

note         = {{ISBN}: 3-540-22694-{X}},
}

2004

A. Račkauskas and Ch. Suquet. Central limit theorems in Hölder topologies for Banach space valued random fields. Theory of Probability and its Applications, 49(1):109–125, 2004.

For rather general moduli of smoothness $\rho$, like $\rho(h)=h^\alpha \ln^\beta (c/h)$, we consider the Hölder spaces $\Hr(B)$ of functions $[0,1]^d \to B$ where $B$ is a separable Banach space. Using isomorphism between $\Hr(B)$ and some sequence Banach space we follow a very natural way to study in terms of second differences the central limit theorem for i.i.d. sequences of random elements in $\Hr(B)$.

@article{Rackauskas-Suquet:2004c,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Central limit theorems in {H}{\"{o}}lder topologies for {B}anach space valued random fields},

journal      = {Theory of Probability and its Applications},

volume       = {49},

number       = {1},

pages        = {109--125},

year         = {2004},
}
A. Račkauskas and Ch. Suquet. Hölder norm test statistics for epidemic change. Journal of Statistical Planning and Inference, 126(2):495–520, 2004.

To detect epidemic change in the mean of a sample of size $n$, we introduce new test statistics UI and DI based on weighted increments of partial sums. We obtain their limit distributions under the null hypothesis of no change in the mean. Under alternative hypothesis our statistics can detect very short epidemics of length $\log^\gamma n$, $\gamma>1$. Using self-normalization and adaptiveness to modify UI and DI, allows us to prove the same results under very relaxed moment assumptions. Trimmed versions of UI and DI are also studied.

@article{Rackauskas-Suquet:2004b,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {{H}{\"{o}}lder norm test statistics for epidemic change},

journal      = {Journal of Statistical Planning and Inference},

volume       = {126},

number       = {2},

pages        = {495--520},

year         = {2004},
}
A. Račkauskas and Ch. Suquet. Necessary and sufficient condition for the Hölderian functional central limit theorem. Journal of Theoretical Probability, 17(1):221–243, 2004.

Let $(X_i)_{i\geq 1}$ be an i.i.d. sequence of random elements in the Banach space $B$, $S_n:= X_1+\cdots + X_n$ and $\xi_n$ be the random polygonal line with vertices $(k/n,S_k)$, $k=0,1,\dots,n$. Put $\rho(h)=h^\alpha L(1/h)$, $0\leq h\leq 1$ with $0<\alpha\leq 1/2$ and $L$ slowly varying at infinity. Let $\mathrm{H}^o_\rho(B)$ be the Hölder space of functions $x:[0,1]\mapsto B$, such that $||x(t+h)-x(t)||=o(\rho(h))$, uniformly in $t$. We characterize the weak convergence in $\mathrm{H}^o_\rho(B)$ of $n^{-1/2}\xi_n$ to a Brownian motion. In the special case where $B=\R$ and $\rho(h)=h^\alpha$, our necessary and sufficient conditions for such convergence are $\mathbf{E} X_1=0$ and $\mathbf{P}(\vert X_1\vert>t)=o(t^{-p(\alpha)})$ where $p(\alpha)=1/(1/2-\alpha)$. This completes Lamperti (1962) invariance principle.

@article{Rackauskas-Suquet:2004a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Necessary and sufficient condition for the {H}{\"{o}}lderian functional central limit theorem},

journal      = {Journal of Theoretical Probability},

volume       = {17},

number       = {1},

pages        = {221--243},

year         = {2004},
}

2003

A. Račkauskas and Ch. Suquet. Hölderian invariance principle for triangular arrays of random variables. Lithuanian Mathematical Journal, 43(4):423–438, 2003.

This paper extends Hölderian invariance principle by Lamperti (1962) to the case of partial sums processes based on a triangular array of row-wise independent random variables. As an application, necessary and sufficient conditions are obtained for the almost sure (resp. in probability) weak Hölder convergence of partial sums processes based on bootstrapped samples.

@article{Rackauskas-Suquet:2003d,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {H{\"{o}}lderian invariance principle for triangular arrays of random variables},

journal      = {Lithuanian Mathematical Journal},

volume       = {43},

number       = {4},

pages        = {423--438},

year         = {2003},
}
A. Račkauskas and Ch. Suquet. Invariance principle under self-normalization for nonidentically distributed random variables. Acta Applicandae Mathematicae, 79(1-2):83–103, 2003.

Let $V_n^{-1}\zeta_n$ be the adaptive process of self-normalized partial sums $S_k$ of independent random variables $X_i$, defined by linear interpolation between the points $(V_k^2/V_n^2, S_k/V_n)$, $k\leq n$, where $V_k^2=\sum_{ i\leq k}X_i^2$. We prove that if the $X_k$'s are symmetric, $V_n^{-1}\zeta_n$ converges weakly to the Brownian motion $W$ in each Hölder space supporting $W$ if and only if $V_n^{-1}\max_{k\le n}\abs{X_k}=o_P(1)$. We give some partial extension to the non symmetric case.

@article{Rackauskas-Suquet:2003b,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Invariance principle under self-normalization for nonidentically distributed random variables},

journal      = {Acta Applicandae Mathematicae},

volume       = {79},

number       = {1-2},

pages        = {83--103},

year         = {2003},
}
A. Račkauskas and Ch. Suquet. Necessary and sufficient condition for the Lamperti invariance principle. Theory of Probability and Mathematical Statistics, 68:115–124, 2003.

Let $(X_i)_{i\geq 1}$ be an i.i.d. sequence of random variables with null expectation and variance one, $S_n:= X_1+\cdots + X_n$ and $\xi_n$ the random polygonal line with vertices $(k/n,S_k)$, $k=0,1,\dots,n$. By Lamperti (1962) theorem, if $X_1$ has a moment of order $p>2$, $n^{-1/2}\xi_n$ weakly converges to the standard Brownian motion $W$ in the Hölder space $H_\alpha$ for $\alpha<1/2-1/p$. We prove here that a necessary and sufficient condition for the $H_\alpha$-weak convergence of this process to $W$ is that $P(\vert X_1\vert>t)=o(t^{-p(\alpha)})$ where $p(\alpha)=1/(1/2-\alpha)$.

@article{Rackauskas-Suquet:2003a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Necessary and sufficient condition for the {L}amperti invariance principle},

journal      = {Theory of Probability and Mathematical Statistics},

volume       = {68},

pages        = {115--124},

year         = {2003},
}
A. Račkauskas and Ch. Suquet. Principe d'invariance hölderien pour des tableaux triangulaires de variables aléatoires. Lietuvos Matematikos Rinkinys, 43(4):513–532, 2003.

@article{Rackauskas-Suquet:2003c,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Principe d'invariance h{\"{o}}lderien pour des tableaux triangulaires de variables al\'{e}atoires},

journal      = {Lietuvos Matematikos Rinkinys},

volume       = {43},

number       = {4},

pages        = {513--532},

year         = {2003},
}

2002

B. Morel and Ch. Suquet. Hilbertian invariance principles for the empirical process under association. Math. Methods Statist., 11(2):203–220, 2002.

We prove that a necessary and sufficient condition for the weak $\mathrm{L}^2[0,1]$ convergence of the empirical process of a stationary associated sequence $(X_k)$ of uniform random variables is simply $\sum (2/3-\E\max(X_0,X_k))<\infty$. This condition is more natural in this setting than the classical covariance summability conditions. In the same spirit, we discuss the weak convergence in the Besov spaces $\mathrm{B}_2^{s,2}$ ($s<1/2$).

@article{Morel-Suquet:2002,

author       = {Morel, B. and Suquet, Ch.},

title        = {Hilbertian invariance principles for the empirical process under association},

journal      = {Math. Methods Statist.},

volume       = {11},

number       = {2},

pages        = {203--220},

year         = {2002},
}
A. Račkauskas and Ch. Suquet. Hölder convergences of multivariate empirical characteristic functions. Mathematical Methods of Statistics, 11(3):341–357, 2002.

We study the asymptotic behavior of the multivariate empirical characteristic function in various Hölder topologies: strong laws of large numbers and central limit theorem for the empirical characteristic process. A rather precise connection is established between the topological framework and the integrability of the marginal distribution of the sample. Some application to test statistics are proposed.

@article{Rackauskas-Suquet:2002b,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {{H}{\"{o}}lder convergences of multivariate empirical characteristic functions},

journal      = {Mathematical Methods of Statistics},

volume       = {11},

number       = {3},

pages        = {341--357},

year         = {2002},
}
A. Račkauskas and Ch. Suquet. On the Hölderian functional central limit theorem for i.i.d. random elements in Banach space. In I. Berkes, E. Cs\'aki, and M. Csörgő, editors, Limit Theorems in Probability and Statistics Balatonlelle 1999, volume 2, pages 485–498, Budapest, 2002. J\'anos Bolyai Mathematical Society.

We investigate the weak Hölder convergence of polygonal partial sums processes of independent copies of a Banach valued random variable $X$. We establish a close connection between the relevant Hölder modulus of smoothness and the tail behaviour of $X$ and of the normalized partial sums.

@inproceedings{Rackauskas-Suquet:2002a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

editor       = {Berkes, I. and Cs\'aki, E. and Cs{\"{o}}rg\H{o}, M.},

title        = {On the {H}{\"{o}}lderian functional central limit theorem for i.i.d. random elements in {B}anach space},

booktitle    = {Limit Theorems in Probability and Statistics Balatonlelle 1999},

volume       = {2},

pages        = {485--498},

year         = {2002},

publisher    = {J\'anos Bolyai Mathematical Society},

address      = {Budapest},
}

2001

A.V. Bulinski and Ch Suquet. Normal approximation for quasi associated random fields. Statist. Probab. Lett., 54:215–226, 2001.

For quasi associated random fields (comprising negatively and positively dependent fields) on $\mathbb{Z}^d$ we use Stein's method to establish the rate of normal approximation for partial sums taken over arbitrary finite subsets $\mathcal{U}$ of $\mathbb{Z}^d$.

@article{,

author       = {Bulinski, A.V. and Suquet, Ch},

title        = {Normal approximation for quasi associated random fields},

journal      = {Statist. Probab. Lett.},

volume       = {54},

pages        = {215--226},

year         = {2001},
}
A. Račkauskas and Ch. Suquet. Hölder versions of Banach spaces valued random fields. Georgian Mathematical Journal, 8(2):347–362, 2001.

For rather general moduli of smoothness $\rho$, like $\rho(h)=h^\alpha \ln^\beta (c/h)$, we consider the Hölder spaces $\Hr(B)$ of functions $[0,1]^d \to B$ where $B$ is a separable Banach space. We establish isomorphism between $\Hr(B)$ and some sequence Banach space. With this analytical tool, we follow a very natural way to study in terms of second differences the existence of a version in $\Hr(B)$ for a given random field.

@article{Rackauskas-Suquet:2001b,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {{H}{\"{o}}lder versions of {B}anach spaces valued random fields},

journal      = {Georgian Mathematical Journal},

volume       = {8},

number       = {2},

pages        = {347--362},

year         = {2001},
}
A. Račkauskas and Ch. Suquet. Invariance principles for adaptive self-normalized partial sums processes. Stoch. Process. Appl., 95:63–81, 2001.

Let $\zeta^{\mathrm{se}}_n$ be the adaptive polygonal process of self-normalized partial sums $S_k=\sum_{1\leq i\leq k}X_i$ of i.i.d. random variables defined by linear interpolation between the points $(V_k^2/V_n^2, S_k/V_n)$, $k\leq n$, where $V_k^2=\sum_{ i\leq k}X_i^2$. We investigate the weak Hölder convergence of $\zeta^{\mathrm{se}}_n$ to the Brownian motion $W$. We prove particularly that when $X_1$ is symmetric, $\zeta^{\mathrm{se}}_n$ converges to $W$ in each Hölder space supporting $W$ if and only if $X_1$ belongs to the domain of attraction of the normal distribution. This contrasts strongly with Lamperti's FCLT where a moment of $X_1$ of order $p>2$ is requested for some Hölder weak convergence of the classical partial sums process. We also present some partial extension to the non symmetric case.

@article{Rackauskas-Suquet:2001a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Invariance principles for adaptive self-normalized partial sums processes},

journal      = {Stoch. Process. Appl.},

volume       = {95},

pages        = {63--81},

year         = {2001},
}

2000

A. Račkauskas and Ch. Suquet. Convergence of self-normalized partial sums processes in $C[0,1]$ and $D[0,1]$. Publications IRMA de Lille, 53-VI, 2000.

Let $(X_i)_{i\geq 1}$ be an i.i.d. sequence of mean zero random variables, $S_n:= X_1+\cdots + X_n$ and $V_n^2:=X_1^2+\cdots +X_n^2$. We consider four sequences of partial sums processes: the broken lines with vertices at the points $(k/n,S_k/V_n)$ or $(V_k^2/V_n^2,S_k/V_n)$ and the corresponding random step functions. We prove that each of them converges weakly in $C[0,1]$ or $D[0,1]$ to the Brownian motion if and only if $X_1$ belongs to the domain of attraction of the normal distribution. These results contrast with the classical Donsker Prohorov invariance principles where the N.S.C. for such convergences is $\E X_1^2 < \infty$.

@article{Rackauskas-Suquet:2000,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Convergence of self-normalized partial sums processes in ${C}[0,1]$ and ${D}[0,1]$},

journal      = {Publications {IRMA} de Lille},

volume       = {53-{VI}},

year         = {2000},
}

1999

D. Hamadouche and Ch. Suquet. Weak Hölder convergence of stochastic processes with application to the perturbed empirical process. Applicationes Math., 26:63–83, 1999.

We consider stochastic processes as random elements in some spaces of Hölder functions vanishing at infinity. The corresponding scale of spaces $C_0^{\alpha,o}$ is shown to be isomorphic to some scale of Banach sequences spaces. This enables us to obtain some tightness criterion in these spaces. As an application, we prove the weak Hölder convergence of the convolution-smoothed empirical process of an i.i.d. sample $(X_1,\ldots,X_n)$ under a natural assumptions about the regularity of the marginal distribution function $F$ of the sample. In particular when $F$ is Lipschitz, the best possible bound $\alpha<1/2$ for the weak $\alpha$-Hölder convergence of such processes is achieved.

@article{Hamadouche-Suquet:1999,

author       = {Hamadouche, D. and Suquet, Ch.},

title        = {Weak {H}{\"{o}}lder convergence of stochastic processes with application to the perturbed empirical process},

journal      = {Applicationes Math.},

volume       = {26},

pages        = {63--83},

year         = {1999},
}
A. Račkauskas and Ch. Suquet. Central limit theorem in Hölder spaces. Probab. Math. Stat., 19:155–174, 1999.

Stochastic processes are considered within the framework of Hölder spaces $H_\alpha^0$ as paths spaces. Exploiting Ciesielski's isomorphisms between $H_\alpha^0$ and sequences spaces via the Faber Schauder triangular functions allows us to express our basic assumptions in terms of the second differences of the processes, giving more flexibility. We obtain general conditions for the existence of a version with paths in $H_\alpha^0$ and for tightness of sequences of random elements in these spaces. Central limit theorems in $H_\alpha^0$ are established and convergence rates are given with respect to Prohorov and bounded Lipschitz metrics. As an application, we study the weak Hölder convergence of the characteristic empirical process.

@article{Rackauskas-Suquet:1999a,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

title        = {Central limit theorem in {H}{\"{o}}lder spaces},

journal      = {Probab. Math. Stat.},

volume       = {19},

pages        = {155--174},

year         = {1999},
}
A. Račkauskas and Ch. Suquet. Random fields and central limit theorem in some generalized Hölder spaces. In B. Grigelionis et al., editor, Prob. Theory and Math. Statist., Proceedings of the 7th Vilnius Conference (1998), pages 599–616. TEV Vilnius and VSP Utrecht, 1999.

For rather general moduli of smoothness $\rho$ (like e.g. $\rho(h)=h^\alpha \ln^\beta (c/h)$ ) the Hölder spaces $\Hr([0,1]^d)$, are characterized by the rate of coefficients in the skew pyramidal basis. With this analytical tool, we study in terms of second differences the existence of a version in $\Hr$ for a given random field. In the same spirit, central limit theorems are obtained both for i.i.d. and martingale differences sequences of random elements in $\Hr$.

@inproceedings{Rackauskas-Suquet:1999b,

author       = {Ra\v{c}kauskas, A. and Suquet, Ch.},

editor       = {B. Grigelionis et al.},

title        = {Random fields and central limit theorem in some generalized {H}{\"{o}}lder spaces},

booktitle    = {Prob. Theory and Math. Statist., Proceedings of the 7th Vilnius Conference (1998)},

pages        = {599--616},

year         = {1999},

publisher    = {TEV Vilnius and VSP Utrecht},
}
Ch. Suquet. Tightness in Schauder decomposable Banach spaces. In Amer. Math. Soc. Transl., volume 193, pages 201–224. AMS, 1999.

@incollection{Suquet:1999,

author       = {Suquet, Ch.},

title        = {Tightness in {S}chauder decomposable {B}anach spaces},

booktitle    = {Amer. Math. Soc. Transl.},

volume       = {193},

pages        = {201--224},

year         = {1999},

publisher    = {AMS},
}

1998

P.E. Oliveira and Ch. Suquet. Weak convergence in $L^p[0,1]$ of the uniform empirical process under dependence. Statist. Probab. Lett., 39:363–370, 1998.

The weak convergence of the empirical process of strong mixing or associated random variables is studied in $L^p(0,1)$. We find minimal rates of convergence to zero of the mixing coefficients or the covariances, in either case, supposing stationarity of the underlying variables. The rates obtained improve, for $p$ not too large, the corresponding results in the classical $D(0,1)$ framework.

@article{Oliveira-Suquet:1998,

author       = {Oliveira, P.E. and Suquet, Ch.},

title        = {Weak convergence in ${L}^p[0,1]$ of the uniform empirical process under dependence},

journal      = {Statist. Probab. Lett.},

volume       = {39},

pages        = {363--370},

year         = {1998},
}
Ch. Suquet. Plotnost' mer v banahovih prostranstvah s razlozeniem Shaudera. Trudy Sankt Petersburgskogo Matematicheskogo Obschestva, 5:289–327, 1998. in Russian.

@article{Suquet:1998,

author       = {Suquet, Ch.},

title        = {Plotnost' mer v banahovih prostranstvah s razlozeniem {S}haudera},

journal      = {Trudy Sankt Petersburgskogo Matematicheskogo Obschestva},

volume       = {5},

pages        = {289--327},

year         = {1998},

note         = {in {R}ussian},
}
Ch. Suquet and M.-C. Viano. Change point detection in dependent sequences: invariance principles for some quadratic statistic. Math. Methods Statist., 7:157–191, 1998.

We consider the change point detection problem for the marginal distribution $F$ of a dependent process. We focus on three Cramér-von Mises type test statistics. Under the hypothesis that no change occurs, we prove their convergence in distribution for either an $\alpha$-mixing or an associated sequence of observations. Thanks to the quadratic context, the space $D[0,1]^2$ can be replaced by $L^2[0,1]^2$ in the preliminary topological tools and the convergence results hold for a significantly larger class of dependent sequences than for the classical Kolmogorov-Smirnov statistics.

@article{Suquet-Viano:1998,

author       = {Suquet, Ch. and Viano, M.-C.},

title        = {Change point detection in dependent sequences: invariance principles for some quadratic statistic},

journal      = {Math. Methods Statist.},

volume       = {7},

pages        = {157--191},

year         = {1998},
}

1997

P. Jacob and Ch. Suquet. Regression and asymptotical location of a multivariate sample. Statist. Probab. Lett., 35:173–179, 1997.

A broad class of multidimensional probability distributions is shown to have large samples which can be almost surely encompassed by a sequence of deterministic close fitting surfaces. Based on polar regression, a general method is proposed to estimate these surfaces.

@article{Jacob-Suquet:1997,

author       = {Jacob, P. and Suquet, Ch.},

title        = {Regression and asymptotical location of a multivariate sample},

journal      = {Statist. Probab. Lett.},

volume       = {35},

pages        = {173--179},

year         = {1997},
}

1996

P. Jacob and Ch. Suquet. Regression and edge estimation. Statist. Probab. Lett., 27:11–15, 1996.

This short paper points out the fact that, for a large class of multidimensional probability distributions with bounded support, every estimate of the regression can be modified in order to give an estimate of the edge of the support.

@article{Jacob-Suquet:1996,

author       = {Jacob, P. and Suquet, Ch.},

title        = {Regression and edge estimation},

journal      = {Statist. Probab. Lett.},

volume       = {27},

pages        = {11--15},

year         = {1996},
}

1995

P. Jacob and Ch. Suquet. Estimating the edge of a Poisson process by orthogonal series. J. Statist. Plann. Inference, 46:215–234, 1995.

Let a homogeneous Poisson process with support delimited by the axis and the graph of a function $f$ defined on $[0,1]$. We study the asymptotical properties of a class of estimators of $f$ obtained by the method of orthogonal functions. We give sufficient conditions for uniform convergences, we prove the asymptotical normality of the multivariate distributions and we find confidence intervals.

@article{Jacob-Suquet:1995,

author       = {Jacob, P. and Suquet, Ch.},

title        = {Estimating the edge of a {P}oisson process by orthogonal series},

journal      = {J. Statist. Plann. Inference},

volume       = {46},

pages        = {215--234},

year         = {1995},
}
P.E. Oliveira and Ch. Suquet. An invariance principle in $L^2(0,1)$ for non stationary $\varphi$-mixing sequences. Comment. Math. Univ. Carolinae, 36(2):293–302, 1995.

Invariance principle in $L^2[0,1]$ is studied using signed random measures. This approach to the problem uses an explicit isometry between $L^2[0,1]$ and a reproducing kernel Hilbert space giving a very convenient setting for the study of compactness and convergence of the sequence of Donsker functions. As an application, we prove a $L^2[0,1]$ version of the invariance principle in the case of $\varphi$-mixing random variables. Our result is not available in the $D(0,1)$-setting.

@article{Oliveira-Suquet:1995a,

author       = {Oliveira, P.E. and Suquet, Ch.},

title        = {An invariance principle in ${L}^2(0,1)$ for non stationary $\varphi$-mixing sequences},

journal      = {Comment. Math. Univ. Carolinae},

volume       = {36},

number       = {2},

pages        = {293--302},

year         = {1995},
}
P.E. Oliveira and Ch. Suquet. An $L^2(0,1)$ invariance principle for LPQD random variables. Portugaliae Mathematica, 53:367–379, 1995.

Using an explicit isometry between Hilbert spaces and an embedding of the space of signed measures we prove an invariance principle with weak convergence in $L^2[0,1]$ for random variables which are linearly positive quadrant dependent under a Lindeberg type condition and some regularity on the covariance structure.

@article{Oliveira-Suquet:1995d,

author       = {Oliveira, P.E. and Suquet, Ch.},

title        = {An ${L}^2(0,1)$ invariance principle for {LPQD} random variables},

journal      = {Portugaliae Mathematica},

volume       = {53},

pages        = {367--379},

year         = {1995},
}
P.E. Oliveira and Ch. Suquet. $L^2(0,1)$ weak convergence of the empirical process for dependent variables. In A. Antoniadis and G. Oppenheim, editors, Wavelets and Statistics, volume 103 of Lecture Notes in Statistics, pages 331–344. 1995.

We consider the empirical process induced by dependent variables as a random element in $L^2[0,1]$. Using some special properties of the Haar basis, we obtain a general tightness condition. In the strong mixing case, this allows us to improve on the well known result of Yoshihara (of course for the $L^2$ continuous functionals). In the same spirit, we give also an application to associated variables which improves a recent result of Yu. Some statistical applications are presented.

@incollection{Oliveira-Suquet:1995c,

author       = {Oliveira, P.E. and Suquet, Ch.},

editor       = {Antoniadis, A. and Oppenheim, G.},

title        = {${L}^2(0,1)$ weak convergence of the empirical process for dependent variables},

booktitle    = {Wavelets and Statistics},

volume       = {103},

pages        = {331--344},

series       = {Lecture Notes in Statistics},

year         = {1995},
}
P.E. Oliveira and Ch. Suquet. Processus empirique sous dépendance dans $L^2[0,1]$. C. R. Acad. Sci. Paris Sér. I Math., 321:939–944, 1995.

We consider the empirical process induced by dependent variables as a random element in $L^2[0,1]$ . We obtain a general tightness condition. In the strong mixing case, this allows us to improve (in the $L^2$ setting) on the well known result of Yoshihara . We also give an application to associated variables which improves in the same way a recent result of Yu.

@article{Oliveira-Suquet:1995b,

author       = {Oliveira, P.E. and Suquet, Ch.},

title        = {Processus empirique sous dépendance dans ${L}^2[0,1]$},

journal      = {C. R. Acad. Sci. Paris S{\'{e}}r. I Math.},

volume       = {321},

pages        = {939--944},

year         = {1995},
}
Ch. Suquet. Distances euclidiennes sur les mesures signées et application à des théorèmes de Berry-Esséen. Bull. Belg. Math. Soc., 2:161–181, 1995.

@article{Suquet:1995,

author       = {Suquet, Ch.},

title        = {Distances euclidiennes sur les mesures sign\'{e}es et application \`{a} des th\'{e}or\`{e}mes de {B}erry-{E}ss\'{e}en},

journal      = {Bull. Belg. Math. Soc.},

volume       = {2},

pages        = {161--181},

year         = {1995},
}

1993

Ch. Suquet. Convergences stochastiques de suites de mesures aléatoires à signe considérées comme variables aléatoires hilbertiennes. Publ. Inst. Statist. Univ. Paris, 37:71–99, 1993.

@article{Suquet:1993,

author       = {Suquet, Ch.},

title        = {Convergences stochastiques de suites de mesures al\'{e}atoires \`{a} signe consid\'{e}r\'{e}es comme variables al\'{e}atoires hilbertiennes},

journal      = {Publ. Inst. Statist. Univ. Paris},

volume       = {37},

pages        = {71--99},

year         = {1993},
}

1990

Ch. Suquet. Une topologie pré-hilbertienne sur l'espace des mesures à signe bornées. Publ. Inst. Statist. Univ. Paris, 35:51–77, 1990.

@article{Suquet:1990,

author       = {Suquet, Ch.},

title        = {Une topologie pr\'{e}-hilbertienne sur l'espace des mesures \`{a} signe born\'{e}es},

journal      = {Publ. Inst. Statist. Univ. Paris},

volume       = {35},

pages        = {51--77},

year         = {1990},
}