We consider tensor product random fields $Y_d$, $d\in\mathbb{N}$, whose covariance funtions are Gaussian kernels. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate $Y_d$, with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$.
Namely, we find criteria of boundedness for $n^{Y_d}(\varepsilon)$ on $d$ and of tending $n^{Y_d}(\varepsilon)\to\infty$, $d\to\infty$, for any fixed $\varepsilon\in(0,1)$. In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics
\begin{eqnarray*}
\ln n^{Y_d}(\varepsilon)= a_d+q(\varepsilon)b_d+o(b_d),\quad d\to\infty,
\end{eqnarray*}
with any $\varepsilon\in(0,1)$. Here $q\colon (0,1)\to\mathbb{R}$ is a non-decreasing function, $(a_d)_{d\in\mathbb{N}}$ is a sequence and $(b_d)_{d\in\mathbb{N}}$ is a positive sequence such that $b_d\to\infty$, $d\to\infty$.
We show that only special quantiles of self-decomposable distribution functions appear as functions $q$ in a given asymptotics. These general results apply to $n^{Y_d}(\varepsilon)$ under particular assumptions on the length scale parameters.
In this paper we present a general framework for Stein's method for multivariate continuous distributions. The approach gives a collection of Stein characterisations, among which we highlight score-Stein operators and kernel Stein operators. Applications include copulas and distance between posterior distributions. We give a general construction for Stein kernels for elliptical distributions and discuss Stein kernels in generality, highlighting connections with Fisher information and mass transport. Finally, a goodness-of-fit test based on Stein discrepancies is given.
In this paper, we consider the nonparametric estimation of the multivariate probability density function and its partial derivative with a support on $[0,\infty)$. To this end we use the class of kernel estimators with asymmetric gamma kernel functions. The gamma kernels are nonnegative. They change their shape depending on the position on the semi-axis and are robust to the boundary bias problem. We investigate the mean integrated squared error (MISE) assuming dependent data with strong mixing and find the optimal bandwidth of the kernel as a minimum of the MISE. We derive the bias, the varia…
In this paper, we address the problem of fitting multivariate Hawkes processes to potentially large-scale data in a setting where series of events are not only mutually-exciting but can also exhibit inhibitive patterns. We focus on nonparametric learning and propose a novel algorithm called MEMIP (Markovian Estimation of Mutually Interacting Processes) that makes use of polynomial approximation theory and self-concordant analysis in order to learn both triggering kernels and base intensities of events. Moreover, considering that N historical observations are available, the algorithm performs l…
This article studies asymptotic approximations of ruin probabilities of multivariate random walks with heavy-tailed increments. Under our assumptions, the distributions of the increments are closely connected to multivariate subexponentiality and admit dependence between components. Keywords: subexponential distribution, ruin probability, multivariate random walk
A multivariate version of Spearman's rho for testing independence is considered. Its asymptotic efficiency is calculated under a general distribution model specified by the dependence function. The efficiency comparison study that involves other multivariate Spearman-type test statistics is made. Conditions for Pitman optimality of the test are established. Examples that illustrate the quality of the multivariate Spearman's test are included.
For a multivariate Lévy process satisfying the Cramér moment condition and having a drift vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by the multivariate ruin problem introduced in F. Avram et al. (2008) in the two-dimensional case. Our solution relies on the analysis from Y. Pan and K. Borovkov (2017) for multivariate random walks and an appropriate time discretization.
This paper develops asymptotics and approximations for ruin probabilities in a multivariate risk setting. We consider a model in which the individual reserve processes are driven by a common Markovian environmental process. We subsequently consider a regime in which the claim arrival intensity and transition rates of the environmental process are jointly sped up, and one in which there is (with overwhelming probability) maximally one transition of the environmental process in the time interval considered. The approximations are extensively tested in a series of numerical experiments.
Given a multivariate generating function F, we determine asymptotics for the coefficients. Our approach is to use Cauchy's integral formula near singular points of F, resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers will treat singular points of F where the local geometry is more complicated, and for which other methods of analysis are not known.
We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of $X$, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in $A$ is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence inter…
This paper considers population processes in which general, not necessarily Markovian, multivariate Hawkes processes dictate the stochastic arrivals. We establish results to determine the corresponding time-dependent joint probability distribution, allowing for general intensity decay functions, general intensity jumps, and general sojourn times. We obtain an exact, full characterization of the time-dependent joint transform of the multivariate population process and its underlying intensity process in terms of a fixed-point representation and corresponding convergence results. We also derive …
In this paper we introduce the \textit{multivariate} Brownian semistationary (BSS) processes and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general stationary multivariate Gaussian processes, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.
We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R}^d$ having finite first moment. Building on previous univariate findings, we solve an integro-partial differential Stein equation by a mixture of semigroup and Fourier analytic methods. Then, under a second moment assumption, we introduce a notion of Stein kernel and an associated Stein discrepancy specifically designed for infinitely divisible distributions. Combining these new tools, we obtain quantitative bounds on smooth-Wasserstein distances between a probability measure in $\…
In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kernel estimators introduced by Aitchison & Lauder (1985) and studied theoretically in Ouimet (2020). Another potential application related to the asymptotic equivalence between the Gaussian variance regression problem and the Gaussian white noise problem…
A multivariate fractional Poisson process was recently defined in Beghin and Macci (2016) by considering a common independent random time change for a finite dimensional vector of independent (non-fractional) Poisson processes; moreover it was proved that, for each fixed $t\geq 0$, it has a suitable multinomial conditional distribution of the components given their sum. In this paper we consider another multivariate process $\{\underline{M}^ν(t)=(M_1^ν(t),\ldots,M_m^ν(t)):t\geq 0\}$ with the same conditional distributions of the components given their sums, and different marginal distributions…
This paper studies the properties of the probability density function $p_{α,ν, n}(\mathbf{x})$ of the $n$-variate generalized Linnik distribution whose characteristic function $φ_{α,ν,n}(\boldsymbol{t})$ is given by φ_{α,ν,n}(\boldsymbol{t})=\frac{1} {(1+\Vert\boldsymbol{t}\Vert^α)^ν}, α\in (0,2], ν>0, \boldsymbol{t}\in \mathbb{R}^n, where $\Vert\boldsymbol{t}\Vert$ is the Euclidean norm of $\boldsymbol{t}\in\mathbb{R}^n$. Integral representations of $p_{α,ν, n}(\mathbf{x})$ are obtained and used to derive the asymptotic expansions of $p_{α,ν, n}(\mathbf{x})$ when $\Vert\mathbf{x}\Vert\to 0$ a…
It is known that the normalized maxima of a sequence of independent and identically distributed bivariate normal random vectors with correlation coefficient $ρ\in (-1,1)$ is asymptotically independent, which may seriously underestimate extreme probabilities in practice. By letting $ρ$ depend on the sample size and go to one with certain rate, Hüsler and Reiss (1989) showed that the normalized maxima can become asymptotically dependent. In this paper, we extend such a study to a triangular array of multivariate Gaussian sequence, which further generalizes the results in Hsing, Hüsler and Reiss …
We present an optimal investment theorem for a currency exchange model with random and possibly discontinuous proportional transaction costs. The investor's preferences are represented by a multivariate utility function, allowing for simultaneous consumption of any prescribed selection of the currencies at a given terminal date. We prove the existence of an optimal portfolio process under the assumption of asymptotic satiability of the value function. Sufficient conditions for asymptotic satiability of the value function include reasonable asymptotic elasticity of the utility function, or a gr…
