The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform
Yoichi Nishiyama - One of the best experts on this subject based on the ideXlab platform.
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From a stochastic Maximal Inequality to infinite-dimensional martingales.
arXiv: Probability, 2020Co-Authors: Yoichi NishiyamaAbstract:As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic Maximal Inequality derived by using the Taylor expansion, is presented. The Inequality dealing with finite-dimensional discrete-time martingales is pulled up to infinite-dimensional ones by using the monotone convergence arguments. The main results are some weak convergence theorems for sequences of separable random fields of discrete-time martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d. random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained.
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A stochastic Maximal Inequality, strict countability, and infinite-dimensional martingales
arXiv: Probability, 2017Co-Authors: Yoichi NishiyamaAbstract:As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic Maximal Inequality} derived by using the formula for integration by parts and on a new concept named {\em strict countability}, is presented. The main results are some weakconvergence theorems for sequences of separable random fields of discrete-time martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d.\ random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained.
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A stochastic Maximal Inequality, strict countability, and related topics
arXiv: Probability, 2013Co-Authors: Yoichi NishiyamaAbstract:As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic Maximal Inequality derived by using Ito's formula and on a new concept named strict countability, is presented. The main results are some weak convergence theorems for sequences of separable random fields of locally square-integrable martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d. random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained. An application to statistical estimation in semiparametric models is presented with an illustration to construct adaptive estimators in Cox's regression model.
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A Maximal Inequality for continuous martingales and $M$-estimation in a Gaussian white noise model
The Annals of Statistics, 1999Co-Authors: Yoichi NishiyamaAbstract:Some sufficient conditions to establish the rate of convergence of certain M-estimators in a Gaussian white noise model are presented They are applied to some concrete problems including jump point estimation and nonparametric maximum likelihood estimation for the regression function The results are shown by means of a Maximal Inequality for continuous martingales and some techniques developed recently in the context of empirical processes
Adam Osękowski - One of the best experts on this subject based on the ideXlab platform.
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A sharp Maximal Inequality for continuous martingales and their differential subordinates
Czechoslovak Mathematical Journal, 2013Co-Authors: Adam OsękowskiAbstract:Assume that X, Y are continuous-path martingales taking values in ℝν, ν ⩾ 1, such that Y is differentially subordinate to X. The paper contains the proof of the Maximal Inequality $$\left\| {\mathop {\sup }\limits_{t \geqslant 0} \left| {Y_t } \right|} \right\|_1 \leqslant 2\left\| {\mathop {\sup }\limits_{t \geqslant 0} \left| {X_t } \right|} \right\|_1 .$$ The constant 2 is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.
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Sharp Maximal Inequality for nonnegative martingales
Statistics & Probability Letters, 2011Co-Authors: Adam OsękowskiAbstract:Let X be a nonnegative martingale, let H be a predictable process taking values in [−1,1] and let Y be an Ito integral of H with respect to X. We establish the bound ‖supt≥0|Yt|‖1≤3‖supt≥0Xt‖1 and show that the constant 3 is the best possible.
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A Maximal Inequality for nonnegative sub- and supermartingales
Mathematical Inequalities & Applications, 2011Co-Authors: Adam OsękowskiAbstract:Let X = (Xt)t≥0 be a nonnegative semimartingale and H = (Ht)t≥0 be a predictable process taking values in [−1, 1]. Let Y denote the stochastic integral of H with respect to X. We show that (i) If X is a supermartingale, then || sup t≥0 Yt||1 ≤ 3|| sup t≥0 Xt||1 and the constant 3 is the best possible. (ii) If X is a submartingale satisfying ||X||∞ ≤ 1, then || sup t≥0 Yt||p ≤ 2Γ(p + 1), 1 ≤ p
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Sharp Maximal Inequality for stochastic integrals
Proceedings of the American Mathematical Society, 2008Co-Authors: Adam OsękowskiAbstract:Let X = (Xt)t 0 be a nonnegative supermartingale and H = (Ht)t 0 be a predictable process with values in ( 1,1). Let Y denote the stochastic integral of H with respect to X. The paper contains the proof of the sharp Inequality sup t 0 ||Yt||1 0||sup t 0 Xt||1,
Magda Peligrad - One of the best experts on this subject based on the ideXlab platform.
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Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples
Annals of Probability, 2013Co-Authors: Florence Merlevede, Magda PeligradAbstract:The aim of this paper is to propose new Rosenthal-type inequalities for moments of order p larger than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations. As in the recent results by Peligrad et al. (2007) and Rio (2009), the estimates of the moments are expressed in terms of the norms of projections of partial sums. The proofs of the results are essentially based on a new Maximal Inequality generalizing the Doob's Maximal Inequality for martingales and dyadic induction. Various applications are also provided.
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A Maximal _{}-Inequality for stationary sequences and its applications
Proceedings of the American Mathematical Society, 2006Co-Authors: Magda Peligrad, Sergey UtevAbstract:The paper aims to establish a new sharp Burkholder-type Maximal Inequality in Lp for a class of stationary sequences that includes martingale sequences, mixingales and other dependent structures. The case when the variables are bounded is also addressed, leading to an exponential Inequality for a maximum of partial sums. As an application we present an invariance principle for partial sums of certain maps of Bernoulli shifts processes.
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a Maximal _ Inequality for stationary sequences and its applications
Proceedings of the American Mathematical Society, 2006Co-Authors: Magda Peligrad, Sergey UtevAbstract:The paper aims to establish a new sharp Burkholder-type Maximal Inequality in Lp for a class of stationary sequences that includes martingale sequences, mixingales and other dependent structures. The case when the variables are bounded is also addressed, leading to an exponential Inequality for a maximum of partial sums. As an application we present an invariance principle for partial sums of certain maps of Bernoulli shifts processes.
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A new Maximal Inequality and invariance principle for stationary sequences
The Annals of Probability, 2005Co-Authors: Magda Peligrad, Sergey UtevAbstract:We derive a new Maximal Inequality for stationary sequences under a martingale-type condition introduced by Maxwell and Woodroofe [Ann. Probab. 28 (2000) 713-724]. Then, we apply it to establish the Donsker invariance principle for this class of stationary sequences. A Markov chain example is given in order to show the optimality of the conditions imposed.
Jiahui Zhu - One of the best experts on this subject based on the ideXlab platform.
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Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces
Annales de l'Institut Henri Poincaré Probabilités et Statistiques, 2017Co-Authors: Jiahui Zhu, Zdzislaw Brzeźniak, Erika HausenblasAbstract:Abstract. Assume that E is an M-type p Banach space with q-th, q ≥ p, power of the norm isof C 2 -class. We consider the stochastic convolutionu(t) =Z t0 Z Z S(t − s)ξ(s,z)N˜(ds,dz),where S is a C 0 -semigroup of contractions on E and N˜ is a compensated Poisson random measure.We derive a Maximal Inequality for a ca`dl`ag modification ˜u of uE sup 0≤s≤t |˜u(s)| q ′ E ≤ C EZ t0 Z Z |ξ(s,z)| pE N(ds,dz) q′p ,for every 0 0. Stochastic convolution and M-type p Banachspace and Poisson random measure 1. IntroductionThe Maximal Inequality for stochastic convolutions of a contraction C 0 -semigroup and right con-tinuous martingales in Hilbert spaces was studied by Ichikawa [8], see also Tubaro [13], for moredetails see [12]). A submartingale type Inequality for the stochastic convolutions of a contractionC 0 -semigroup and square integrable martingales, also in Hilbert spaces, were obtained by Kotelenez[10]. Kotelenez also proved the existence of a ca`dl`ag version of the stochastic convolution processesfor square integrable ca`dl`ag martingales. In the paper by Brze´zniak and Peszat [4], the authorsestablished a Maximal Inequality in a certain class of Banach spaces for stochastic convolution pro-cesses driven by a Wiener process. It is of interest to know whether the Maximal Inequality holdsalso for pure jump type processes. Here we extend the results from [4] to the case where the stochas-tic convolution is driven by a compensated Poisson random measure. We work in the framework ofstochastic integrals and convolutions driven by a compensated Poisson random measures recentlyintroduced by the first two named authours in [3].Let us now briefly present the content of the paper. In the first section, i.e. section 2 weset up notation and terminology and then summarize without proofs some of the standard factson stochastic integrals with values in martingale type p, p ∈ (1,2], Banach spaces, driven bycompensated Poisson random measures. In the following section 3, we proceed with the study ofstochastic convolution process (u(t))
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a Maximal Inequality for stochastic convolutions in 2 smooth banach spaces
Electronic Communications in Probability, 2011Co-Authors: Jan Van Neerven, Jiahui ZhuAbstract:Let $(e^{tA})_{t\geq0}$ be a $C_0$-contraction semigroup on a $2$-smooth Banach space $E$, let $(W_t)_{t\geq0}$ be a cylindrical Brownian motion in a Hilbert space $H$, and let $(g_t)_{t\geq0}$ be a progressively measurable process with values in the space $\gamma(H,E)$ of all $\gamma$-radonifying operators from $H$ to $E$. We prove that for all $0
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a Maximal Inequality for stochastic convolutions in 2 smooth banach spaces
arXiv: Probability, 2011Co-Authors: Jan Van Neerven, Jiahui ZhuAbstract:Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space \gamma(H,E) of all \gamma-radonifying operators from H to E. We prove that for all 0
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Maximal Inequality of stochastic convolution driven by compensated poisson random measures in banach spaces
arXiv: Probability, 2010Co-Authors: Jiahui Zhu, Zdzislaw Brzeźniak, Erika HausenblasAbstract:Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fr\'{e}chet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a c\`{a}dl\`{a}g modification a $\tilde{u}$ of the process $u$ which satisfies the following Maximal Inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E} \left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1
Erika Hausenblas - One of the best experts on this subject based on the ideXlab platform.
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Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces
Annales de l'Institut Henri Poincaré Probabilités et Statistiques, 2017Co-Authors: Jiahui Zhu, Zdzislaw Brzeźniak, Erika HausenblasAbstract:Abstract. Assume that E is an M-type p Banach space with q-th, q ≥ p, power of the norm isof C 2 -class. We consider the stochastic convolutionu(t) =Z t0 Z Z S(t − s)ξ(s,z)N˜(ds,dz),where S is a C 0 -semigroup of contractions on E and N˜ is a compensated Poisson random measure.We derive a Maximal Inequality for a ca`dl`ag modification ˜u of uE sup 0≤s≤t |˜u(s)| q ′ E ≤ C EZ t0 Z Z |ξ(s,z)| pE N(ds,dz) q′p ,for every 0 0. Stochastic convolution and M-type p Banachspace and Poisson random measure 1. IntroductionThe Maximal Inequality for stochastic convolutions of a contraction C 0 -semigroup and right con-tinuous martingales in Hilbert spaces was studied by Ichikawa [8], see also Tubaro [13], for moredetails see [12]). A submartingale type Inequality for the stochastic convolutions of a contractionC 0 -semigroup and square integrable martingales, also in Hilbert spaces, were obtained by Kotelenez[10]. Kotelenez also proved the existence of a ca`dl`ag version of the stochastic convolution processesfor square integrable ca`dl`ag martingales. In the paper by Brze´zniak and Peszat [4], the authorsestablished a Maximal Inequality in a certain class of Banach spaces for stochastic convolution pro-cesses driven by a Wiener process. It is of interest to know whether the Maximal Inequality holdsalso for pure jump type processes. Here we extend the results from [4] to the case where the stochas-tic convolution is driven by a compensated Poisson random measure. We work in the framework ofstochastic integrals and convolutions driven by a compensated Poisson random measures recentlyintroduced by the first two named authours in [3].Let us now briefly present the content of the paper. In the first section, i.e. section 2 weset up notation and terminology and then summarize without proofs some of the standard factson stochastic integrals with values in martingale type p, p ∈ (1,2], Banach spaces, driven bycompensated Poisson random measures. In the following section 3, we proceed with the study ofstochastic convolution process (u(t))
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Maximal Inequality of stochastic convolution driven by compensated poisson random measures in banach spaces
arXiv: Probability, 2010Co-Authors: Jiahui Zhu, Zdzislaw Brzeźniak, Erika HausenblasAbstract:Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fr\'{e}chet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a c\`{a}dl\`{a}g modification a $\tilde{u}$ of the process $u$ which satisfies the following Maximal Inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E} \left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1
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stochastic convolutions driven by martingales Maximal inequalities and exponential integrability
Stochastic Analysis and Applications, 2007Co-Authors: Erika Hausenblas, Jan SeidlerAbstract:Abstract Stochastic convolutions driven by a local martingale M in a Hilbert space are studied in the case when S(t) is a strongly continuous semigroup of contractions. Very simple proofs of the Maximal Inequality and exponential tail estimates are given by using unitary dilations and Zygmund's extrapolation theorem. Applications to stochastic convolutions driven by Poisson random measures are provided. A part of the results is then generalized to stochastic convolutions in L q -spaces.
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A note on Maximal Inequality for stochastic convolutions
Czechoslovak Mathematical Journal, 2001Co-Authors: Erika Hausenblas, Jan SeidlerAbstract:Using unitary dilations we give a very simple proof of the Maximal Inequality for a stochastic convolution $$\int_0^t {S(t - s)} \psi (s){\text{d}}W(s)$$ driven by a Wiener process W in a Hilbert space in the case when the semigroup S(t) is of contraction type.
