Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion. Our main result is to show that for each fixed $t$ $${\int (L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx-24h^{2}t\over h^2} \stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int (L^x_t)^3dx)^{1/2}η$$ as $h\to 0$, where $η$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments
A quenched central limit theorem is derived for the super-Brownian motion with super-Brownian immigration, in dimension $d\geq 4$. At the critical dimension $d=4$, the quenched and annealed fluctuations are of the same order but are not equal.
Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion and \[ α_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $η=N(0,1)$ be independent of $α_{t}$. For each fixed $t$ \[ {\int_{-\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{α_{t}} η, \] as $h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}}{\to}({64 \over 3} )^{1/2}\sqrt{α_{1}} η, \] as $t\rar\infty$.
Let $B$ be a bifractional Brownian motion with parameters $H\in (0, 1)$ and $K\in(0,1]$. For any $n\geq1$, set $Z_n =\sum_{i=0}^{n-1}\big[n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\E((B_{i+1}-B_{i})^2)\big]$. We use the Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati \cite{NP09} to derive, in the case when $0<HK\leq3/4$, Berry-Esséen-type bounds for the Kolmogorov distance between the law of the correct renormalization $V_n$ of $Z_n$ and the standard normal law. Finally, we study almost sure central limit theorems for the sequence $V_n$.
Consider a random walk among random conductances on $\mathbb{Z}^d$ with $d\geq 2$. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate positive. We show that the conditional limit law is the product of a Brownian meander and a $(d-1)$-dimensional Brownian motion.
We construct an iterated stochastic integral with fractional Brownian motion with H > 1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric stochastic integral is equal to the Malliavin divergence integral. By a version of the Fourth Moment theorem of Nualart and Peccati, we show that a family of such integrals converges in distribution to a scaled Brownian motion. An application is an approximation to the windings for a planar fBm, previously studied by Baudoin and Nualart.
Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric Lévy process with local time $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the Lévy exponent $ψ(\la)$ is regularly varying at zero with index $1<β\leq 2$, and satisfies some additional regularity conditions, \begin{eqnarray*} && {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- E(\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx)\over t\sqrt{ψ^{-1}(1/t)}}\label{r5.0tweaksabs} && \stackrel{\mathcal{L}}{\Longrightarrow}(8c_{ψ,1 })^{1/2}(\int_{-\infty}^{\finfty} (L_{β,1}^{x})^{2} dx)^{1/2} η\end{eqnarray*} as $t\rar\infty$, where …
In this paper, we consider the explicit bound for the second-order approximation of the quadratic variation of a general fractional Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by $(ts)^{H-1}$ up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or have stationary increments. %Some examples include the su…
In a paper of Jason Swanson, a CLT for the sample median of independent Brownian motions with value 0 at 0 was proved. Here we extend this result in two ways. We prove such a result for a collection of self-similar processes which include the fractional Brownian motions, all stationary, independent increment symmetric stable processes tied down at 0 as well as iterated and integrated Brownian motions. Second, our results hold uniformly over all quantiles in a compact sub-interval of (0,1). We also examine sample function properties connected with these CLTs.
We give a stochastic calculus proof of the Central Limit Theorem \[ {\int (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c(\int (L^{x}_{t})^{2} dx)^{1/2} η\] as $h\to 0$ for Brownian local time $L^{x}_{t}$. Here $η$ is an independent normal random variable with mean zero and variance one.
We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension $d\ge 3$, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a Brownian motion in $d\ge4$. In $d=3$, the limit process is a fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussian process when starting from a Poisson field. For (dependent) branching random walks with state dependent branching rate we obtain convergence in f.d.d. to the same limit process, and for $d=3$ al…
In this short article, we will focus on the different links between some stochastic processes resulting from Brownian motion and two notions of probability theory (proportional increments and last hitting times).
We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet. This Gaussian field is an extension of fractional Brownian motion. We prove some properties of covariance function for self-similar fields with rectangular increments. Using Lamperti transformation we obtain properties of covariance function for the corresponding stationary fields. We present an example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet.
In this paper, we show an approximation in law of the complex Brownian motion by processes constructed from a stochastic process with independent increments. We give sufficient conditions for the characteristic function of the process with independent increments that ensure the existence of the approximation. We apply these results to Lévy processes. Finally we extend this results to the $m$-dimensional complex Brownian motion.
The Lévy-Ciesielski Construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process $(W_{t})_{t\in \left[ 0,T\right] }$ normalized by the global modulus function, for all positive $\varepsilon $ and $δ$. Additionally, uniform results over $δ$ are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in…
Let $W$ be a one-dimensional Brownian motion starting from 0. Define $Y(t)= \int_0^t{\d s \over W(s)} := \lim_{ε\to0} \int_0^t 1_{(|W(s)|> ε)} {\d s \over W(s)} $ as Cauchy's principal value related to local time. We prove limsup and liminf results for the increments of $Y$.
We derive a nonlinear integral equation to calculate Root's solution of the Skorokhod embedding problem for atom-free target measures. We then use this to efficiently generate bounded time-space increments of Brownian motion and give a parabolic version of Muller's classic "Random walk over spheres" algorithm.
We present a random walk approximation to fractional Brownian motion where the increments of the fractional random walk are defined as a weighted sum of the past increments of a Bernoulli random walk.
Our purpose is to investigate properties for processes with stationary and independent increments under $G$-expectation. As applications, we prove the martingale characterization to $G$-Brownian motion and present a decomposition for generalized $G$-Brownian motion.
Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$
Université du Littoral Côte d’Opale（法国北部海岸大学）；Université de Lyon, Université Lyon 1, Institut de Science Financière et d’Assurances（里昂大学，里昂第一大学，金融与保险科学学院）；Laboratory in Finance and Insurance - LFA CREST - Center for Research in Economics and Statistics, ENSAE（金融与保险实验室-LFA CREST-经济与统计研究中心，巴黎国立高等经济学院）
We proof a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (Séminaire de Probabilités XLIII, Springer, 2011) are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exc…
Braverman, Mallows and Shepp (1995), showed that if the absolute moments of partial sums of i.i.d. symmetric variables are equal to those of normal variables, then the marginals have normal distribution. This fact suggested the conjecture that probably the absolute moments alone characterize the homogeneous process with independent increments. In this paper we prove a more general result that gives a positive answer to this conjecture, and then apply it in order to obtain the CLT for a class of dependent random variables under a normalization involving the absolute moments of partial sums.
Let G=\{G(x),x\in R^1\} be a mean zero Gaussian processes with stationary increments and set \si ^2(|x-y|)= E(G(x)-G(y))^2. Let f be a symmetric function with Ef(η)<\ff, where η=N(0,1). When \si^2(s) is concave or when \si^2(s)=s^r$, $1<r\leq 3/2 we have \lim_{h\downarrow 0}{\int_a^b f(\frac{G(x+h)-G(x)}{\si (h)}) dx - (b-a)Ef(η)\over \sqrt{Φ(h,\si(h),f,a,b)}}= N(0,1) in law where Φ(h,\si(h),f,a,b) is the variance of the numerator. This result continues to hold when \si^2(s)=s^r, 3/2<r<2, for certain functions f, depending on the nature of the coefficients in their Hermite polynomial expansion…
Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics
Université du Littoral Côte d’Opale（滨海大学）；Université de Lyon, Université Lyon 1, Institut de Science Financière et d’Assurances（里昂大学，里昂第一大学，金融与保险科学学院）；Laboratory in Finance and Insurance - LFA CREST - Center for Research in Economics and Statistics, ENSAE（金融与保险实验室-LFA CREST-经济与统计研究中心，国家经济统计学院）
The study of random polynomials has a long and rich history. This paper studies random algebraic polynomials $P_n(x) = a_0 + a_1 x + \ldots + a_{n-1} x^{n-1}$ where the coefficients $(a_k)$ are correlated random variables taken as the increments $X(k+1) - X(k)$, $k\in \mathbb{N}$, of a fractional Brownian motion $X$ of Hurst index $0< H < 1$. This reduces to the classical setting of independent coefficients for $H = 1/2$. We obtain that the average number of the real zeros of $P_n(x)$ is~$\sim K_H \log n$, for large $n$, where $K_H = (1 + 2 \sqrt{H(1-H)})/π$ (a generalisation of a classical re…
We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standard geodesic distance. The radial process is closely related to a Wright-Fisher diffusion, increments of which can be simulated exactly using the recent work of Jenkins & Spanò (2017) [JS17]. The rapid spinning phenomenon of the skew-product decomposition then yields the algorithm for the increments of the process on the sphere.
We show that the global fluctuations of spectra of GOE and GUE matrices and their principal submatrices executing Dyson's Brownian motion are Gaussian in the limit of large matrix dimensions. For nested submatrices one obtains a limiting three-dimensional generalized Gaussian process; its restrictions to two-dimensional sections that are monotone in matrix sizes and time moments coincide with the two-dimensional Gaussian Free Field with zero boundary conditions. The proof is by moment convergence, and it extends to more general Wigner matrices and their stochastic evolution.
For stochastic processes $\{X_t:t\in E\}$, we establish sufficient conditions for the empirical process based on $\{I_{X_t\le y}-\operatorname{Pr}(X_t\le y):t\in E,y\in\mathbb{R}\}$ to satisfy the CLT uniformly in $t\in E,y\in\mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E=[0,1]$.
The problems considered in the present paper have their roots in two different cultures. The 'true' (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit, Parisi and Peliti. This is a nearest neighbor non-Markovian random walk in Z^d which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (independently of the physics community), is the continuous space-time counterpart: a diffusion in R^d pushed by the negative gradient …
We investigate here the Central Limit Theorem of the Increment Ratio Statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer-Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.
In this article we propose a new explicit Euler-type approximation method for stochastic differential equations (SDEs). In this method, Brownian increments in the recursion of the Euler method are replaced by suitable bounded functions of the Brownian increments. We prove strong convergence rate one-half for a large class of SDEs with polynomial coefficient functions whose local monotonicity constant grows at most like the logarithm of a Lyapunov-type function.
We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments. We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments.…
Let $W_t$ be a standard Brownian motion. It is well-known that the Langevin equation $d U_t = -θU_td t + d W_t$ defines a stationary process called Ornstein-Uhlenbeck process. Furthermore, Langevin equation can be used to construct other stationary processes by replacing Brownian motion $W_t$ with some other process $G$ with stationary increments. In this article we prove that the converse also holds and all continuous stationary processes arise from a Langevin equation with certain noise $G=G_θ$. Discrete analogies of our results are given and applications are discussed.
Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson's integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of m…
We propose some class of statistics suitable for estimation of the Hurst index of the fractional Brownian motion based on the second order increments of an observed discrete trajectory.
Sub-fractional Brownian motion is a process analogous to fractional Brownian motion but without stationary increments. In \cite{GGL1} we proved a strong uniform approximation with a rate of convergence for fractional Brownian motion by means of transport processes. In this paper we prove a similar type of approximation for sub-fractional Brownian motion.
We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be applied in a recursive manner. We establish an upper bound on the mean squared maximal distance between the paths and determine a suitable mesh size in various asymptotic regimes. The analysis proceeds by reduction to the comonotonic coupling of inc…
We construct the least-square estimator for the unknown drift parameter in the multifractional Ornstein-Uhlenbeck model and establish its strong consistency in the non-ergodic case. The proofs are based on the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of multifractional Brownian motion (mBm) and of some other functionals of mBm, including increments and fractional derivatives. As the auxiliary results having independent interest, we produce the asymptotic bounds with probability 1 for the rate of the growth of the trajectories of the general Gaussian p…
Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.
The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dime…
We prove the following result: For $(Z_t)_{t \in \mathbf{R}}$ a fractional Brownian motion with arbitrary Hurst parameter, there does not exist any stopping time $τ$ adapted to the natural filtration of the increments of $Z$ such that, with positive probability, $τ$ a local minimum at right of the trajectory of $Z$.
We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predominance of the Brownian component, the presence of exchangeable structures, and fast convergence of normal empirical quantile functions. The second procedure amounts to filtering the increments and compensating with the final value. It requires a care…
Gonzalez Cazares and Ivanovs (2021) suggested a new method for "recovering" the Brownian motion component from the trajectory of a Levy process that required sampling from an independent Brownian motion process. We show that such a procedure works equally well without any additional source of randomness if one uses normal quantiles instead of the ordered increments of the auxiliary Brownian motion process.
We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the Brownian meander.
We introduce a natural family of random walks on the set of integers that scale to fractional Brownian motion. The increments X_n have the property that given {X_k: k < n}, the conditional law of X_n is that of X_{n-k_n}, where k_n is sampled independently from a fixed law μon the positive integers. When μhas a roughly power law decay (precisely, when it lies in the domain of attraction of an αstable subordinator, for 0 < α< 1/2) the walk scales to fractional Brownian motion with Hurst parameter α+ 1/2. The walks are easy to simulate and their increments satisfy an FKG inequality. In a sense w…
We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter fractional Bro…
Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter $H \in (0, 1)$ called the Hurst index. The use of time-changed processes in modeling often requires the knowledge of their second order properties such as covariance function. This paper provides the explicit expression for the correlation structure for time-changed fractional Brownian motion. Several examples useful in applications are discussed.
In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Hölder regularity conditions, for some Hölder exponent greater than 1/2. This result will be applied to the infinite dimensional fractional Brownian motion.
Error Distribution of the Local Linearization Method for Stochastic Differential Equations with Additive Brownian Noise
Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科）；Department of Statistics, London School of Economics and Political Science（伦敦政治经济学院统计学系）
The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and non-stationary noises with a power-law variance function. In this paper we study sample path properties of the generalized fractional Brownian motion, including Holder continuity, path differentiability/non-differentiability, and functional and local Law of the Iterated Logarithms.
We prove a Chung-type law of the iterated logarithm for a multiparameter extension of the fractional Brownian motion which is not increment stationary. This multiparameter fractional Brownian motion behaves very differently at the origin and away from the axes, which also appears in the Hausdorff dimension of its range and in the measure of its pointwise Hölder exponents. A functional version of this Chung-type law is also provided.
We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Levy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no "really nice" set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, …
We consider fractional Brownian motion with the Hurst parameters from (1/2,1). We found that the increment of a fractional Brownian motion can be represented as the sum of a two independent Gaussian processes one of which is smooth in the sense that it is differentiable in mean square. We consider fractional Brownian motion and stochastic integrals generated by the Riemann sums. As an example of applications, this results is used to find an optimal pre-programmed strategy in the mean-variance setting for a Bachelier type market model driven by a fractional Brownian motion.
