Search Results (7)

This paper develops a rigorous inferential framework for a class of Gaussian stochastic processes driven by white noise with constant drift, whose temporal evolution is governed by a Caputo fractional derivative of order ${\displaystyle \alpha \in (1/2,1)}$. The model belongs to the family of fractional Volterra processes, where memory is generated by the dynamics themselves rather than by correlated noise. We derive explicit analytical expressions for the mean, variance, and covariance structure of the solution, thereby characterizing in a precise manner how the fractional order ${\displaystyle \alpha }$ governs both variance growth and the strength of temporal dependence. In particular, the process exhibits correlated increments and a power-law variance scaling of order ${\displaystyle t^{2\alpha -1}}$, highlighting the dual role of ${\displaystyle \alpha }$ as a regularity and memory parameter. Building on this structural analysis, we address the statistical problem of estimating the parameter vector ${\displaystyle (\mu ,\sigma ,\alpha )}$ from discrete-time observations. Two complementary procedures are proposed for the estimation of the fractional order: a variance-growth method based on log–log regression of empirical variances, and a wavelet-based estimator exploiting multi-scale scaling properties of the process. For the drift and diffusion parameters ${\displaystyle (\mu ,\sigma )}$, we construct explicit Gaussian pseudo-maximum likelihood estimators derived from the Volterra covariance structure of the increment process. We establish unbiasedness, ${\displaystyle L^2}$-convergence, strong consistency, and asymptotic normality for all estimators. Furthermore, we derive Berry–Esseen type bounds that quantify the rate of convergence toward the Gaussian law, providing sharp distributional approximations in a genuinely fractional and non-Markovian setting. A Monte Carlo study is carried out, using high-resolution Volterra discretizations, large-scale simulation budgets, covariance-structured linear algebra, and multi-scale diagnostic tools. The numerical experiments confirm the theoretical convergence rates, demonstrate the finite-sample reliability of the estimators, and illustrate the sensitivity of the process dynamics to the fractional order ${\displaystyle \alpha }$: smaller values of ${\displaystyle \alpha }$ produce stronger memory effects and higher variability, while values closer to one lead to smoother and more stable trajectories. The proposed methodology unifies statistical inference for long-memory Gaussian processes with fractional differential stochastic dynamics, offering a coherent analytical and computational framework applicable in areas such as quantitative finance, anomalous diffusion in physics, hydrology, and engineering systems with hereditary effects. Full article

This study establishes explicit Berry–Esseen bounds for residual kernel density estimators in AR(1) models with $\alpha$-mixing errors. Since the true innovations are unobservable, we introduce a residual-based estimator ${\widehat{f}}_n\left(x\right)$ and establish its normal approximation under stationarity. By imposing conditions on the bandwidth, mixing coefficients, and moments, we obtain Kolmogorov distance bounds between the standardized estimator and its Gaussian limit. These bounds explicitly depend on the bandwidth, block parameters, and mixing coefficients. A key corollary quantifies the convergence rate as $O\left(n^{(2c-2b+a)/4}\right)$. Our results generalize prior work, advancing theoretical foundations for nonparametric inference in high-dimensional time series. Full article

We consider a Student process based on independent copies of a random variable X. If X is in the domain of attraction of the normal law (DAN), a weighted version of the Student process is known to follow a functional Central Limit Theorem (FCLT). Accordingly, appropriate functionals of such a process converge in distribution to the same functionals of the similarly weighted standard Wiener process. We use such a convergence for an integral functional and derive asymptotic confidence intervals (CIs) for the mean of X. For right-skewed distributions of X in DAN, we show that the obtained CIs have higher finite-sample coverage probabilities than, and may be preferred over, a CI $I_1$ of the same asymptotic confidence level $1-\alpha$ that is based on the CLT for the Student t-statistic, since the finite-sample coverage probabilities of the latter CI may be lower than $1-\alpha$. Moreover, for such distributions, the finite-sample coverage probabilities of our best two CIs are also higher than those of their respective equal-expected-length $I_1$ counterparts. Full article

For each $t\in (-1,1)$, the exact value of the least upper bound $H\left(t\right)=sup\left\{\mathbb{E}\right|X{|}^3/\mathbb{E}\left|X-t{|}^3\right\}$ over all the non-degenerate distributions of the random variable X with a fixed normalized first-order moment $\mathbb{E}{X}_1/\sqrt{\mathbb{E}{X}_1^2}=t$, and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum $M:=sup{L}_1\left(X\right)/L_1(X-\mathbb{E}X)=\left(\sqrt{17+7\sqrt{7}}\right)/4$, where $L_1\left(X\right)=\mathbb{E}{\left|X\right|}^3/{\left(\mathbb{E}{X}^2\right)}^{3/2}$ is the non-central Lyapunov ratio, and hence proving S. Shorgin’s (2001) conjecture on the exact value of M. As a corollary, an analog of the Berry–Esseen inequality for the Poisson random sums of independent identically distributed random variables $X_1,X_2,\dots$ is proven in terms of the central Lyapunov ratio $L_1(X_1-\mathbb{E}{X}_1)$ with the constant $0.3031·H\left(t\right){(1-t^2)}^{3/2}\in [0.3031,0.4517)$, $t\in [0,1)$, which depends on the normalized first-moment $t:=\mathbb{E}{X}_1/\sqrt{\mathbb{E}{X}_1^2}$ of random summands and being arbitrarily close to $0.3031$ for small values of t, an almost $1.5$ size improvement from the previously known one. Full article

For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, this paper obtains the Berry–Esseen bounds on the rates of convergence to normality of the distributions of the quasi maximum likelihood estimators based on stochastic Taylor approximation, under some regularity conditions, when the diffusion is observed at equally spaced dense time points over a long time interval, the high-frequency regime. It shows that the higher-order stochastic Taylor approximation-based estimators perform better than the basic Euler approximation in the sense of having smaller asymptotic variance. Full article

This paper is concerned with the rate of convergence of the distribution of the sequence $\{F_n/G_n\}$, where $F_n$ and $G_n$ are each functionals of infinite-dimensional Gaussian fields. This form very frequently appears in the estimation problem of parameters occurring in Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs). We develop a new technique to compute the exact rate of convergence on the Kolmogorov distance for the normal approximation of $F_n/G_n$. As a tool for our work, an Edgeworth expansion for the distribution of $F_n/G_n$, with an explicitly expressed remainder, will be developed, and this remainder term will be controlled to obtain an optimal bound. As an application, we provide an optimal Berry–Esseen bound of the Maximum Likelihood Estimator (MLE) of an unknown parameter appearing in SDEs and SPDEs. Full article