arXiv research

New insights into tail behavior of heavy-tailed random vectors and processes.

Given $n$ samples from a population of individuals belonging to different types with unknown proportions, how do we estimate the probability of discovering a new type at the $(n+1)$-th draw? This is a classical problem in statistics, commonly referred to as the missing mass estimation problem. Recent results by Ohannes…

Investigates a new measure PELVE_n for risk assessment.

We provide a new extension of Breiman's Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a complete characterization of regular variation on cones in $[0,\infty)^d$ under random linear transformations. This allows us to compute probabilities of a…

The study shows how geometric Weyl bulk-density exponent rigidifies spectral encodings in O-regularly varying classes.

A new notion of stochastic ordering is introduced to compare multivariate stochastic risk models with respect to extreme portfolio losses. In the framework of multivariate regular variation comparison criteria are derived in terms of ordering conditions on the spectral measures, which allows for analytical or numerical…

Introduces Polar Depth for analyzing multivariate heavy-tailed data extremes.

Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to accou…

Characterizes term structure models driven by Lévy processes.

For a Riemann surface $X$ and the moduli of regularly stable $G$-bundles $M$, there is a naturally occuring "$adjoint$" vector bundle over $X \times M$. One can take the determinant of this vector bundle with respect to the projection map onto $M$. Our aim here is to study the curvature of the determinant bundle as the…

We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities…

SS-GEN simulates rare events in heavy and light-tailed data.

New DQ based on expectiles improves portfolio diversification.

New features from early battery cycles predict lifetime with high accuracy.

We examine random variables in the power law/regularly varying class with stochastic tail exponent, the exponent $α$ having its own distribution. We show the effect of stochasticity of $α$ on the expectation and higher moments of the random variable. For instance, the moments of a right-tailed or right-asymmetric varia…

A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent close to 3. We revisit this results and use standard tests as well as develop a battery of new non-parametric and parametric tests (in particular with stretche…

Study free energy in spherical spin glasses, proving universality dichotomy.

Every day, hundreds of millions of new Tweets containing over 40 languages of ever-shifting vernacular flow through Twitter. Models that attempt to extract insight from this firehose of information must face the torrential covariate shift that is endemic to the Twitter platform. While regularly-retrained algorithms can…

Proposes a method to adapt DNNs to drift in data distribution.

We give three formulas expressing the Smale invariant of an immersion f of a (4k-1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k-1)…

Volterra square-root process boundary behavior and martingale measures

New protocol evaluates synthetic data for temporal consistency.

We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied bet…

Sharp large deviations and Gibbs conditioning for portfolio credit risk models.

This paper analyzes M-estimators under infinite-variance noise in high dimensions.

Regular variation provides a convenient theoretical framework to study large events. In the multivariate setting, the dependence structure of the positive extremes is characterized by a measure - the spectral measure - defined on the positive orthant of the unit sphere. This measure gathers information on the localizat…

Paper develops sparse learning for heavy-tailed time series with locally stationary dynamics.

We consider the following problem in stochastic portfolio theory. Are there portfolios that are relative arbitrages with respect to the market portfolio over very short periods of time under realistic assumptions? We answer a slightly relaxed question affirmative in the following high dimensional sense, where dimension…

Stochastic gradient descent (SGD) is one of the most widely used optimization methods for parallel and distributed processing of large datasets. One of the key limitations of distributed SGD is the need to regularly communicate the gradients between different computation nodes. To reduce this communication bottleneck, …

In this paper, we consider a framework adapting the notion of cointegration when two asset prices are generated by a driftless Itô-semimartingale featuring jumps with infinite activity, observed regularly and synchronously at high frequency. We develop a regression based estimation of the cointegrated relations method …

Researchers prove Weyl laws for Schrödinger operators on noncompact manifolds.

Let F be a closed orientable surface. If i,i':F \to R^3 are two regularly homotopic generic immersions, then it has been shown in [N] that all generic regular homotopies between i and i' have the same number mod 2 of quadruple points. We denote this number by Q(i,i') \in Z/2. We show that for any generic immersion i:F\…

Employing data on the assessed value of land in 1974--2007 Japan, we exhibit a quasistatically varying log-normal distribution in the middle scale region. In the derivation, a Non-Gibrat's law under the detailed quasi-balance is adopted together with two approximations. The resultant distribution is power-law with the …

We introduce a simulation scheme for Brownian semistationary processes, which is based on discretizing the stochastic integral representation of the process in the time domain. We assume that the kernel function of the process is regularly varying at zero. The novel feature of the scheme is to approximate the kernel fu…

We study the asymptotic behavior of the difference $Δρ^{X, Y}_α:= ρ_α(X + Y) - ρ_α(X)$ as $α\rightarrow 1$, where $ρ_α$ is a risk measure equipped with a confidence level parameter $0 < α< 1$, and where $X$ and $Y$ are non-negative random variables whose tail probability functions are regularly varying. The case where …

We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding number.

Bayesian nonparametric approaches, in particular the Pitman-Yor process and the associated two-parameter Chinese Restaurant process, have been successfully used in applications where the data exhibit a power-law behavior. Examples include natural language processing, natural images or networks. There is also growing em…

New method learns complex, multimodal distributions in ADVI.

We show that a compact complex surface which fibers smoothly over a curve of genus >1 with fibers of genus >1 fibers holomorphically. We deduce an improvement of a result in [D Kotschick, Math. Research Letters, 5 (1998) 227-234], and a characterisation of fibered surfaces with zero signature.

We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n+2 crossings.

Develops statistical framework for analyzing functional data extremes.

A new method for calculating ES from VaR under Solvency II.

We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, …

Proposes a new policy gradient algorithm to improve reinforcement learning efficiency and stability.

We study the problem of predicting the future, though only in the probabilistic sense of estimating a future state of a time-varying probability distribution. This is not only an interesting academic problem, but solving this extrapolation problem also has many practical application, e.g. for training classifiers that …

Paper introduces MTCM to measure multivariate tail dependence.

Let F be a closed orientable surface. We give an explicit formula for the number mod 2 of quadruple points occurring in any generic regular homotopy between any two regularly homotopic embeddings e,e':F -> R^3. The formula is in terms of homological data extracted from the two embeddings.

This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being developed in conjunction with "KnotInfo" (www.indiana.edu/~knotinfo), a web-based tabl…