arXiv subjects

Elina Moldavskaya

Publications and source records attributed to Elina Moldavskaya.

7 recordsLinked to original sources

arXiv2026

Constants in the Weighted Law of the Iterated Logarithm under Long-Range Dependence: Hermite Rank Two

At Hermite rank two, the constant in the weighted law of the iterated logarithm under long-range dependence is represented as the largest eigenvalue of a suitably normalized positive integral operator on the unit interval. The same eigenvalue determines the exponential-moment threshold and the logarithmic right-tail rate of the weighted second-chaos limit, which in the unweighted case is the Rosenblatt law. The weighted third spectral moment is evaluated in closed form. Convergent two-sided spectral enclosures are obtained, and in the weight-concentration limit the leading eigenvalue is characterized by a scalar equation with a uniform geometric remainder. At the unweighted memory boundary, the constant decays like the square root of the distance to criticality. Its leading coefficient is determined to $25$ decimal places with certified full-operator residual bounds. After division by the square-root boundary factor, the weight-concentration and memory limits commute. Their common coefficient differs from the unweighted boundary coefficient and lies in the certified interval $(1.370323114331,\,1.370323114332)$. The joint asymptotic formula is established with a uniform two-parameter remainder bound.

math.PR↗

arXiv2026

Malliavin Smoothness of Hermite Processes of Arbitrary Order and Their Wiener Integrals

We establish Malliavin nondegeneracy for every weighted integral against a singular Hermite kernel, with all negative moments of the Malliavin derivative norm finite, uniformly over admissible compact families of bounded deterministic weights. In particular, all finite-dimensional distributions of Hermite processes of every fixed finite order have Schwartz densities, as do vectors of non-overlapping increments and, more generally, finite families of Wiener integrals with respect to a Hermite process. For such vectors, linear independence of the weights is both necessary and sufficient for absolute continuity; under this condition, all inverse Malliavin determinant moments exist and the joint density belongs to the Schwartz space, with bounds that are again uniform over compact families. The proof rests on a nonvanishing condition on compact weight families that is preserved under directional differentiation. An analytic-tail property of fractional transforms of the directions verifies this condition at every chaos level. Uniform Malliavin estimates then convert lower-order gradient bounds into bounded joint densities of arbitrarily many first directional derivatives, and Bessel's inequality closes the induction. The argument works directly with fixed non-Gaussian laws and singular kernels, without a Gaussian-limit assumption and without a separate local-nondeterminism transfer.

math.PR↗

arXiv2026

Weighted Empirical Risk Minimization for Machine Learning under Long-Range Dependence: Exact Pathwise Rates and Learning-Error Geometry

We develop an exact almost-sure learning theory for smooth parametric models trained by regularly weighted empirical risk minimization on long-range dependent data. The training observations are generated from a fixed finite window of a stationary Gaussian sequence, and the sample weights are regularly varying. If the loss gradient at the population minimizer has Wiener-chaos rank $m$ and a nonzero low-frequency coefficient, then, in the long-memory interior regime, the finite-lag score reduces on the iterated-logarithm scale to a single weighted Hermite chaos. This yields an almost-sure Bahadur representation, an exact limsup law for the learned parameter, and, for $m\ge2$, the functional cluster set of the complete learning trajectory. The polynomial learning exponent is determined by the memory parameter and the chaos rank and is invariant under the admissible power weighting, whereas the sharp pathwise constant and cluster geometry depend on the weights. In the rank-one case, global optimization over the admissible power exponents shows that every optimizer is positive. Time-series prediction and classification examples illustrate the results.

stat.ML↗

arXiv2026

Malliavin smoothness and density estimates for the third-order Hermite process

We prove Malliavin nondegeneracy for the third-order Hermite process. The key step is to show that, for every nonzero test direction $h\in C_c^\infty(0,1)$, the directional Malliavin derivative of a third-order Hermite random variable is an infinite-rank Gaussian quadratic form. Using arbitrarily large orthonormal families of such directions, we derive a self-contained Fourier bound for their joint characteristic function and obtain polynomial small-ball estimates of arbitrary order for the Malliavin norm. This yields negative moments of every order. Combining the one-time estimate with determinant factorization and Malliavin strong local nondeterminism, we obtain negative moments of all orders for finite-dimensional Malliavin determinants. Consequently, all finite-dimensional distributions, as well as arbitrary vectors of non-overlapping increments, admit Schwartz densities. We further establish grid-uniform Sobolev bounds for inverse Malliavin determinants of normalized increment vectors and derive stretched-exponential estimates for all partial derivatives of their densities. The decay exponent is $2/3$, reflecting the third Wiener chaos. This settles the next non-Gaussian Hermite order after the Rosenblatt case and provides a quantitative counterpart to the order-two smoothness theory.

math.PR↗

arXiv2026

Law of the Iterated Logarithm for Weighted Sums of Functionals of Long-Memory Gaussian Sequences

We establish laws of the iterated logarithm for weighted sums of linear and nonlinear functionals of long-memory stationary Gaussian sequences. The leading Wiener chaos determines the scale of the almost-sure fluctuations; all higher chaoses are negligible on it and the exponent of the iterated-logarithm factor is half the Hermite rank. In the linear case, we obtain the exact LIL constant, and it is the classical one. In the nonlinear case, we identify a functional cluster set and give a variational formula for the LIL constant. We also present explicit upper and lower bounds for this constant. The lower bound is expressed through beta functions and depends on the memory parameter, the weight exponent, and the Hermite rank; numerically it matches the constant to within a fraction of one percent, and can therefore be used in its place. Thus, the weights affect the normalization, the geometry of the limiting cluster set, and the nonlinear LIL constant itself.

math.PR↗

arXiv2018

Searching for, and quantifying, non-convexity of functions

Convexity plays a prominent role in a number of problems, but practical considerations frequently give rise to non-convex functions. We suggest a method for determining convex regions, and also for assessing the lack of convexity in the other regions. The method relies on a specially constructed decomposition of symmetric matrices, such as the Hessian. We illustrate theoretical results using several examples, one of which analyses a problem arising in risk measurement and management in insurance and finance.

math.FA↗

arXiv2014

On the existence of paths between points in high level excursion sets of Gaussian random fields

The structure of Gaussian random fields over high levels is a well researched and well understood area, particularly if the field is smooth. However, the question as to whether or not two or more points which lie in an excursion set belong to the same connected component has constantly eluded analysis. We study this problem from the point of view of large deviations, finding the asymptotic probabilities that two such points are connected by a path laying within the excursion set, and so belong to the same component. In addition, we obtain a characterization and descriptions of the most likely paths, given that one exists.

math.PR↗