Continuous Gaussian Research Articles

Optimal importance sampling for continuous Gaussian fields

Abstract We consider the problem of selecting a change of mean which minimizes the variance of Monte Carlo estimators for the expectation of a functional of a continuous Gaussian field, in particular continuous Gaussian processes. Functionals of Gaussian fields have taken up an important position in many fields including statistical physics of disordered systems and mathematical finance (see, for example, [A. Comtet, C. Monthus and M. Yor, Exponential functionals of Brownian motion and disordered systems, J. Appl. Probab. 35 1998, 2, 255–271], [D. Dufresne, The integral of geometric Brownian motion, Adv. in Appl. Probab. 33 2001, 1, 223–241], [N. Privault and W. I. Uy, Monte Carlo computation of the Laplace transform of exponential Brownian functionals, Methodol. Comput. Appl. Probab. 15 2013, 3, 511–524] and [V. R. Fatalov, On the Laplace method for Gaussian measures in a Banach space, Theory Probab. Appl. 58 2014, 2, 216–241]. Naturally, the problem of computing the expectation of such functionals, for example the Laplace transform, is an important issue in such fields. Some examples are considered, which, for particular Gaussian processes, can be related to option pricing.

Some Large Deviations Principles for Time-Changed Gaussian Processes

LetX= (X(t))(t >= 0)(X(0) = 0) be a continuous centered Gaussian process on a probability space (omega,F,P), and let (Y-t)(t is an element of)[0,1] (Y-0= 0) be a continuous process (on the same probability space) with nondecreasing paths, independent ofX. Define the time-changed Gaussian processZ(t)=X(Y-t),t is an element of [0,1]. In this paper, we investigate a problem of finite-dimensional large deviations and a problem of pathwise large deviations for time-changed continuous Gaussian processes. As applications, we considered subordinated Gaussian processes.

Experimental Assessment of Entropy Production in a Continuously Measured Mechanical Resonator.

The information on a quantum process acquired through measurements plays a crucial role in the determination of its nonequilibrium thermodynamic properties. We report on the experimental inference of the stochastic entropy production rate for a continuously monitored mesoscopic quantum system. We consider an optomechanical system subjected to continuous displacement Gaussian measurements and characterize the entropy production rate of the individual trajectories followed by the system in its stochastic dynamics, employing a phase-space description in terms of the Wigner entropy. Owing to the specific regime of our experiment, we are able to single out the informational contribution to the entropy production arising from conditioning the state on the measurement outcomes. Our experiment embodies a significant step towards the demonstration of full-scale control of fundamental thermodynamic processes at the mesoscopic quantum scale.

Multi-physics multi-scale simulation of the solidification process in the molten pool during laser welding of aluminum alloys

The solidification microstructures directly determine the performance of laser welds. However, the present understanding of the dynamic solidification behavior within the melt pool during laser welding is still limited. This paper developed a multi-physics multi-scale model to investigate the complex solidification process during laser welding of 5083 aluminum alloy. The model combines the macroscale model for heat and mass transfer, the microscale model for polycrystalline alloy solidification, and the continuous Gaussian nucleation distribution model for heterogeneous nucleation. The model is validated by comparing with experimental results from three aspects, namely, the fusion profile, the grain structure and the dendritic structure, and good agreements are achieved. The whole solidification process, from the planar growth to cellular growth, columnar dendritic growth and equiaxed dendritic growth, in the laser weld pool is studied through analyzing the evolution of the temperature, solute concentration and constitutional undercooling fields. In particular, the distribution of constitutional undercooling in the liquid ahead of the solidification front is predicted quantitatively. The relationship between the constitutional undercooling distribution and the nucleation/growth of equiaxed grains is demonstrated, which reveals the selective growth behavior of the columnar grains and equiaxed grains at different locations in the laser weld.

Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.

Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics

We study boundary non-crossing probabilities $$\begin{aligned} P_{f,u} := \mathrm {P}\big (\forall t\in {\mathbb {T}} X_t + f(t)\le u(t)\big ) \end{aligned}$$ for a continuous centered Gaussian process X indexed by some arbitrary compact separable metric space $${\mathbb {T}}$$ . We obtain both upper and lower bounds for $$P_{f,u}$$ . The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $$P_{{y}f,u}$$ , $${y}\rightarrow +\infty $$ , which are two-term approximations (up to $$o({y})$$ ). The asymptotics are formulated in terms of the solution $${\tilde{f}}$$ to the constrained optimization problem $$\begin{aligned} \left\Vert h\right\Vert _{{\mathbb {H}}_X}\rightarrow \min , \quad h\in {\mathbb {H}}_X, h\ge f \end{aligned}$$ in the reproducing kernel Hilbert space $${\mathbb {H}}_X$$ of X. Several applications of the results are further presented.

A spatial partial differential equation approach to addressing unit misalignments in Bayesian poisson space-time models

Spatial analyses using data from geographic areas that change shape and location over time, like US ZIP codes, produce biased results to the extent that unit misalignments are related to covariate effects. To address this issue, one method has incorporated a fixed effect measure of population shifts and a spatial structure as a block-diagonal neighborhood adjacency matrix within a Besag-York-Mollié (BYM) model. However, this approach assumes that spatial relationships among units change with time and precludes the assessment of temporal dynamic effects. Here, we assume that a continuous Gaussian random field underlies misaligned data and apply a stochastic partial differential equation (SPDE) approach to modeling area outcomes. We compare SPDE and BYM methods and show that both provide similar estimates of covariate effects. Importantly, we demonstrate that the SPDE approach can additionally identify autoregressive processes underlying the development of problematic health outcomes using data observed across Pennsylvania over 11 years.

Asymptotic error distributions of the Euler method for continuous-time nonlinear filtering

In this paper, we deduce the asymptotic error distribution of the Euler method for the nonlinear filtering problem with continuous-time observations. As studied in previous works by several authors, the error structure of the method is characterized by conditional expectations of some functionals of multiple stochastic integrals. Our main result is to prove the stable convergence of a sequence of such conditional expectations by using the techniques of martingale limit theorems in the spirit of Jacod (On continuous conditional Gaussian martingales and stable convergence in law, seminaire de probabilites, XXXI, lecture notes in mathematics, Springer, Berlin, 1997).

A functional limit theorem for general shot noise processes

AbstractBy a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes.

Log-Modulated Rough Stochastic Volatility Models

We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index H. The so-obtained log-modulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for H = 0. As a consequence, the resulting super-rough stochastic volatility models can be analysed over the whole range 0 0, H >= 0, so no flattening of the skew occurs as H -> 0.

Time-Inhomogeneous Gaussian Stochastic Volatility Models: Large Deviations and Super Roughness

We introduce time-inhomogeneous stochastic volatility models, in which the volatility is described by a nonnegative function of a Volterra type continuous Gaussian process that may have extremely rough sample paths. The drift function and the volatility function are assumed to be time-dependent and locally ω-continuous for some modulus of continuity ω. The main results obtained in the paper are sample path and small-noise large deviation principles for the log-price process in a Gaussian model under very mild restrictions. We use these results to study the asymptotic behavior of binary up-and-in barrier options and binary call options.

Decision making in numeracy tasks with spatially continuous scales

A diffusion model of decision making on continuous response scales is applied to three numeracy tasks. The goal is to explain the distributions of responses on the continuous response scale and the time taken to make decisions. In the model, information from a stimulus is spatially continuously distributed, the response is made by accumulating information to a criterion, which is a 1D line, and the noise in the accumulation process is continuous Gaussian process noise over spatial position. The model is fit to the data from three experiments. In one experiment, a one or two digit number is displayed and the task is to point to its location on a number line ranging from 1 to 100. This task is used extensively in research in education but there has been no model for it that accounts for both decision times and decision choices. In the second task, an array of dots is displayed and the task is to point to the position of the number of dots on an arc ranging from 11 to 90. In a third task, an array of dots is displayed and the task is to speak aloud the number of dots. The model we propose accounts for both accuracy and response time variables, including the full distributions of response times. It also provides estimates of the acuity of decisions (standard deviations in the evidence distributions) and it shows how representations of numeracy information are task-dependent. We discuss how our model relates to research on numeracy and the neuroscience of numeracy, and how it can produce more comprehensive measures of individual differences in numeracy skills in tasks with continuous response scales than have hitherto been available.

Decompositions of log-correlated fields with applications

In this article, we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a Holder continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields whose covariance kernel has a logarithmic singularity on the diagonal—or log-correlated Gaussian fields. The theorem states that any log-correlated Gaussian field $X$ can be decomposed locally into a sum of a Holder continuous function and an independent almost $\star $-scale invariant field (a special class of stationary log-correlated fields with ’cone-like’ white noise representations). This decomposition holds whenever the term $g$ in the covariance kernel $C_{X}(x,y)=\log (1/|x-y|)+g(x,y)$ has locally $H^{d+\varepsilon }$ Sobolev smoothness. We use these decompositions to extend several results that have been known basically only for $\star $-scale invariant fields to general log-correlated fields. These include the existence of critical multiplicative chaos, analytic continuation of the subcritical chaos in the so-called inverse temperature parameter $\beta $, as well as generalised Onsager-type covariance inequalities which play a role in the study of imaginary multiplicative chaos.

Asymmetric Einstein–Podolsky–Rosen steering manipulating among multipartite entangled states

Asymmetric Einstein–Podolsky–Rosen (EPR) steering means the ability of one party to nonlocally “steer” the state in the other side through local measurements. The asymmetric property of EPR steering provides adequate security in one-sided device-independent secure quantum communication protocols. In this paper, we theoretically investigate the asymmetric EPR steering schemes between two subsystems of multipartite entangled states by an asymmetric modulation method. Two schemes, including tripartite entangled state to bipartite steering and quadripartite entangled state to tripartite steering, are investigated. By reconstructing the covariance matrix of continuous variable multipartite Gaussian states, we quantify the steerability of manipulating two different subsystems and demonstrate the EPR steering parameter versus various types of states’ parameters. Our work provides a reference for asymmetric quantum information processing and enriches resources for secure quantum networks.

Turbo Message Passing-Based Receiver Design for Time-Varying OFDM Systems

In this paper, we study time-varying orthogonal frequency division multiplexing (OFDM) systems, and propose a joint channel-and-signal estimation receiver based on turbo message passing (TMP) to efficiently suppress inter-carrier interference (ICI). We establish a factor graph representation of the problem and divide the whole factor graph into two parts, one for channel estimation and the other for signal detection. For the first part, we use Gaussian message passing (GMP) for channel estimation; for the second part, a discrete state space (DSS) model is employed to describe the transition of signal states, and a forward-backward algorithm is adopted for message passing over the transition trellis in signal detection. The resulting algorithm is referred to as DSS-GMP. The complexity of DSS-GMP quickly becomes the bottleneck as the increase of the signal constellation size and the ICI width. To address this issue, we further develop a continuous-state-space (CSS) model based turbo message passing algorithm, where the messages of modulated signals are approximated as continuous Gaussian messages. Numerical results demonstrate that the TMP based scheme significantly outperforms the state-of-the-art schemes.

Entanglement transition from variable-strength weak measurements

We show that weak measurements can induce a quantum phase transition of interacting many-body systems from an ergodic thermal phase with a large entropy to a nonergodic localized phase with a small entropy, but only if the measurement strength exceeds a critical value. We demonstrate this effect for a one-dimensional quantum circuit evolving under random unitary transformations and generic positive operator-valued measurements of variable strength. As opposed to projective measurements describing a restricted class of open systems, the measuring device is modeled as a continuous Gaussian probe, capturing a large class of environments. By employing data collapse and studying the enhanced fluctuations at the transition, we obtain a consistent phase boundary in the space of the measurement strength and the measurement probability, clearly demonstrating a critical value of the measurement strength below which the system is always ergodic, irrespective of the measurement probability. These findings provide guidance for quantum engineering of many-body systems by controlling their environment.

Iterative Gaussian-Approximated Message Passing Receiver for MIMO-SCMA System

Sparse code multiple access (SCMA) and multiple-input multiple-output (MIMO) are viewed as two key techniques to address the high spectral efficiency requirements for the 5G wireless system. However, the complexity of standard message passing algorithm (MPA) for the MIMO-SCMA detection yet increases exponentially with the degree of resource nodes $d_f$ and the number of transmit antennas $N_t$ . In this paper, to tackle this problem, we design two types of low-complexity MPA over the MIMO-SCMA extended factor graph to perform joint multiuser and MIMO detection. The main idea of the proposed algorithms is approximating the discrete interferences at function nodes (FNs) as the continuous Gaussian messages so that the traversal enumeration at the FN update can be avoided. The symbol level Gaussian approximated MPA (GAMPA-S) updates the symbol likelihood during the detector's inner-loop iterations which is similar to the standard MPA, and this algorithm is compatible to any SCMA codebooks. The bit level Gaussian approximated MPA (GAMPA-B) updates the bit likelihood ratio which shows lower complexity compared to the GAMPA-S. However, the GAMPA-B requires that the SCMA codebooks can satisfy the bit significance expansion property. In the receiver, the proposed detectors are combined with users’ decoders to form the outer-loop structure, in which a critical damping factor (DF) is designed to compress the amplitude of extrinsic information exchanged between the detector and decoders. The extrinsic information transform chart is introduced to optimize the DF. Consequently, the complexity of the proposed receiver just increases linearly with $d_f$ and $N_t$ while its performance is comparable or even better than the conventional receiver. This would effectively boost the practical use of MIMO-SCMA techniques.

Positive definite matrix functions on spheres defined by hypergeometric functions

ABSTRACTPositive definite matrix functions on spheres arise naturally in multivariate approximation and spatial statistics. The construction of strictly positive definite models has become one of the most important goals for the analysis and study of vector valued random fields on spheres. This paper derives strictly positive definite models based on hypergeometric functions. In statistical nomenclature, it describes strictly positive definite covariance functions of stationary, isotropic and mean square continuous Gaussian vector random fields on spheres in which the isotropic part of the covariance functions are set through hypergeometric functions.

Firm Decisions under Jump-Diffusive Dynamics

We present a model of firm investment under uncertainty and partial irreversibility in which uncertainty is represented by a jump diffusion. This allows to represent both the continuous Gaussian volatility and the discontinuous uncertainty related to information arrival, sudden changes and large shocks. The model shows how both sources of uncertainty negatively impact the optimal investment and disinvestment policies, and how the presence of large negative jumps can drastically affect the firm’s ability to recover. Our results show that the standard Gaussian framework consistently underestimates the negative effect of uncertainty on firm investment decisions. We test these predictions on a panel dataset of UK firms: we first structurally estimate the uncertainty parameters using multinomial maximum likelihood and differential evolution techniques and subsequently study their impact on firm investment rates, validating our model predictions.

Comparison of photothermal responses under spatial and temporal modulations of CW Gaussian beam excitation: A numerical study

Obtaining stronger photothermal response of optical materials is of great interest in the measurement of optical losses using the photothermal tools, in particular the thermal mirror. In our previous work, we experimentally demonstrated a more effective method of thermal mirror detection employing the spatial modulation (SM) of a continuous wave Gaussian beam excitation instead of the conventional temporal modulation (TM). A complete analytical model is presented to describe the laser-induced three-dimensional dynamic thermal fields in a semi-infinite sample and the time-dependent thermal mirror due to the thermoelastic response of the material as well as the evolution of the intensity distribution of a probe beam after interacting with the thermal mirror, in both the SM and TM schemes under the low absorption and exponentially-decaying absorption cases. With the model, we obtained the variations of temperature, surface displacement, and diffraction intensity giving rise to the photothermal response of the material. The results of the comparison between the two schemes reveal that the SM scheme is more sensitive than the TM scheme originating from a large variation range of the temperature in the SM scheme. We also explained the complex relation between the surface displacement and the diffraction intensity of the probe beam at the center of the detection plane on the basis of the phase shift caused by the thermal mirror. The presented model and the results of comparison are of great importance to gain a deeper insight into the photothermal characterization in both the SM and TM schemes.