Cumulant Research Articles

Effective distances for epidemics spreading on complex networks.

We show that the recently introduced logarithmic metrics used to predict disease arrival times on complex networks are approximations of more general network-based measures derived from random walks theory. Using the daily air-traffic transportation data we perform numerical experiments to compare the infection arrival time with this alternative metric that is obtained by accounting for multiple walks instead of only the most probable path. The comparison with direct simulations reveals a higher correlation compared to the shortest-path approach used previously. In addition our method allows to connect fundamental observables in epidemic spreading with the cumulant-generating function of the hitting time for a Markov chain. Our results provides a general and computationally efficient approach using only algebraic methods.

Pricing volatility swaps in the Heston’s stochastic volatility model with regime switching: A saddlepoint approximation method

Pricing a volatility swap is a highly nonlinear problem. Explicit solutions of the prices of volatility swaps are notoriously difficult to find. In this paper, we consider a saddlepoint approximation method for the valuation of a volatility swap under the Heston’s stochastic volatility model with regime switching. All the values of key parameters in our model are supposed to depend on the states of a continuous time observable Markov chain process. We present a closed-form exact cumulant generating functions (CGFs) of the continuous realized variance. Additionally, an approximated CGF is given. Then we approximate the volatility swaps by the saddlepoint approximation formula which derived from the Fourier inversion representation. The numerical results suggest that the alternative saddlepoint approximation method (ASAP) and the approximated ASAP method could both produce fairly accurate results for the given range of maturities.

Reliability analysis of structures using stochastic response surface method and saddlepoint approximation

Stochastic response surface method (SRSM) is a technique used for reliability analysis of complex structural systems having implicit or time consuming limit state functions. The main aspects of the SRSM are the collection of sample points, the approximation of response surface and the estimation of the probability of failure. In this paper, sample points are selected close to the most probable point of failure and the actual limit state surface (LSS). The response surface is fitted using the weighted regression technique, which allows the fitting points to be weighted based on their distance from the LSS. The cumulant generating function (CGF) of the response surface is derived analytically. The saddlepoint approximation (SPA) method is utilized to compute the probability of failure of the structural system. Finally, four numerical examples compare the proposed algorithm with the traditional quadratic polynomial SRSM, Kriging based SRSM and direct MCS.

Parameter Estimation in Stable Law

For general stable distribution, cumulant function based parameter estimators are proposed. Extensive simulation experiments are carried out to validate the effectiveness of the estimates over the entire parameter space. An application to non-life insurance losses distribution is made.

Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound Poisson process.

We study a stochastic model of gene expression, in which protein production has a form of random bursts whose size distribution is arbitrary, whereas protein decay is a first-order reaction. We find exact analytical expressions for the time evolution of the cumulant-generating function for the most general case when both the burst size probability distribution and the model parameters depend on time in an arbitrary (e.g., oscillatory) manner, and for arbitrary initial conditions. We show that in the case of periodic external activation and constant protein degradation rate, the response of the gene is analogous to the resistor-capacitor low-pass filter, where slow oscillations of the external driving have a greater effect on gene expression than the fast ones. We also demonstrate that the nth cumulant of the protein number distribution depends on the nth moment of the burst size distribution. We use these results to show that different measures of noise (coefficient of variation, Fano factor, fractional change of variance) may vary in time in a different manner. Therefore, any biological hypothesis of evolutionary optimization based on the nonmonotonic dependence of a chosen measure of noise on time must justify why it assumes that biological evolution quantifies noise in that particular way. Finally, we show that not only for exponentially distributed burst sizes but also for a wider class of burst size distributions (e.g., Dirac delta and gamma) the control of gene expression level by burst frequency modulation gives rise to proportional scaling of variance of the protein number distribution to its mean, whereas the control by amplitude modulation implies proportionality of protein number variance to the mean squared.

Hydration thermodynamics beyond the linear response approximation

The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, and , with the solvent environment. Throughout the A B transformation of the solute, the solvation system is described by a Hamiltonian that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density that the dimensionless perturbational solute–solvent interaction energy has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density . The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute–solvent electrostatic interactions, or (2) the shorter-range attractive interactions are switched-on in the A B process.

On the strong Brillinger-mixing property of α-determinantal point processes and some applications

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function C(x, y) defining an α-determinantal point process (DPP). Assuming absolute integrability of the function C 0(x) = C(o, x), we show that a stationary α-DPP with kernel function C 0(x) is “strongly” Brillinger-mixing, implying, among others, that its tail-σ-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications to statistical second-order analysis of α-DPPs.

A Weighted Saddlepoint Approximation Method for System Reliability Analysis

The relationship between the reliability probabilities of the component and the system is hard to get. If this relationship can be obtained easily, the reliability of the system can be calculated by using the reliability structures of the components. The common method to express this relationship is using the linear correlation index, which only shows the linear correlation between the components failure rather than the relationship between high nonlinear functions. In order to describe this relationship accurately and calculate the system reliability using the component reliability structures, a Uniform Design (UD)-Saddlepoint Approximation (SA)-based system reliability analysis method is proposed. The system reliability analysis method is decomposed to three simple steps: (1) calculating the weight coefficient which represents the contribution rate of each component to system reliability, (2) approximating Cumulant Generation Function (CGF) of each component, (3) calculating CGF of the system and approximating the system reliability with SA method. The weight coefficient of each component is derived from UD method, and a variable interval selection method is developed to decrease the required number of samples and increase the accuracy of the weight coefficients. First-Order Saddlepoint Approximation (FOSA) method or Mean-Value First-Order Saddlepoint Approximation (MVFOSA) method is used to analyze the CGF of a component performance function. Then the CGF of the system can be obtained by the weighted addition law by combining the CGFs of components performance functions with the weight coefficients. Finally, the system reliability can be approximated by SA method. Four examples are employed to demonstrate that the new method outperforms other methods for system reliability analysis in terms of efficiency and accuracy.

The large-scale correlations of multicell densities and profiles: implications for cosmic variance estimates

In order to quantify the error budget in the measured probability distribution functions of cell densities, the two-point statistics of cosmic densities in concentric spheres is investigated. Bias functions are introduced as the ratio of their two-point correlation function to the two-point correlation of the underlying dark matter distribution. They describe how cell densities are spatially correlated. They are computed here via the so-called large deviation principle in the quasi-linear regime. Their large-separation limit is presented and successfully compared to simulations for density and density slopes: this regime is shown to be rapidly reached allowing to get sub-percent precision for a wide range of densities and variances. The corresponding asymptotic limit provides an estimate of the cosmic variance of standard concentric cell statistics applied to finite surveys. More generally, no assumption on the separation is required for some specific moments of the two-point statistics, for instance when predicting the generating function of cumulants containing any powers of concentric densities in one location and one power of density at some arbitrary distance from the rest. This exact "one external leg" cumulant generating function is used in particular to probe the rate of convergence of the large-separation approximation.

Reprint of : Full counting statistics of Majorana interferometers

We study the full counting statistics of interferometers for chiral Majorana fermions with two incoming and two outgoing Dirac fermion channels. In the absence of interactions, the FCS can be obtained from the 4×4 scattering matrix S that relates the outgoing Dirac fermions to the incoming Dirac fermions. After presenting explicit expressions for the higher-order current correlations for a modified Hanbury Brown–Twiss interferometer, we note that the cumulant-generating function can be interpreted such that unit-charge transfer processes correspond to two independent half-charge transfer processes, or alternatively, to two independent electron-hole conversion processes. By a combination of analytical and numerical approaches, we verify that this factorization property holds for a general SO(4) scattering matrix, i.e. for a general interferometer geometry.

Saddle Point Approximation for Outage Probability Using Cumulant Generating Functions

This letter proposes the use of saddle point approximation (SPA) to evaluate the outage probability of wireless cellular networks. Unlike traditional numerical integration-based approaches, the SPA approach relies on cumulant generating functions (CGFs) and eliminates the need for explicit numerical integration. The approach is generic and can be applied to a wide variety of distributions, given that their CGFs exist. We illustrate the usefulness of SPA on channel fading distributions such as Nakagami-$m$, Nakagami-$q$ (Hoyt), and Rician distributions. Numerical results validate the accuracy of the proposed SPA approach.

Ward identities and the analogous Goldberger-Treiman relation in a three-flavor Spectral Quark Model

This work presents the first results of an extension of the spectral quark model which includes different flavors. The spectral quark model is an approach based on a generalization of the Lehmann representation for the quark propagator. Gauge and chiral invariance are ensured with the help of gauge technique which provides particular solutions to the Ward-Takahashi identities. General conditions on the quark spectral function follow from natural physical requirements. In particular, the function is normalized, its positive momenta must vanish, while the physical observables depend on negative moments and the so-called log moments. As a consequence, the model is made finite. To allow the description of mesons constituted by different flavors of quarks we introduce different spectral functions and obtain vertex functions constructed from Ward-Takahashi identities that includes two different spectral (constituent) quark masses, allowing the physical description of strange mesons, for example. We obtain some observables based on the current approach and, in particular, the spectral version of the Kaon analogous Goldberger-Treiman relation.

Large-deviation statistics of vorticity stretching in isotropic turbulence.

A key feature of three-dimensional fluid turbulence is the stretching and realignment of vorticity by the action of the strain rate. It is shown in this paper, using the cumulant-generating function, that the cumulative vorticity stretching along a Lagrangian path in isotropic turbulence obeys a large deviation principle. As a result, the relevant statistics can be described by the vorticity stretching Cramér function. This function is computed from a direct numerical simulation data set at a Taylor-scale Reynolds number of Re(λ)=433 and compared to those of the finite-time Lyapunov exponents (FTLE) for material deformation. As expected, the mean cumulative vorticity stretching is slightly less than that of the most-stretched material line (largest FTLE), due to the vorticity's preferential alignment with the second-largest eigenvalue of strain rate and the material line's preferential alignment with the largest eigenvalue. However, the vorticity stretching tends to be significantly larger than the second-largest FTLE, and the Cramér functions reveal that the statistics of vorticity stretching fluctuations are more similar to those of the largest FTLE. In an attempt to relate the vorticity stretching statistics to the vorticity magnitude probability density function in statistically stationary conditions, a model Kramers-Moyal equation is constructed using the statistics encoded in the Cramér function. The model predicts a stretched-exponential tail for the vorticity magnitude probability density function, with good agreement for the exponent but significant difference (35%) in the prefactor.

The Poisson boundary of Out(FN)

Let $\mu$ be a probability measure on $\text{Out}(F_N)$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $\text{Out}(F_N)$. We show that almost every sample path of the random walk on $(\text{Out}(F_N),\mu)$, when realized in Culler and Vogtmann's outer space, converges to the simplex of a free, arational tree. We then prove that the space $\mathcal{FI}$ of simplices of free and arational trees, equipped with the hitting measure, is the Poisson boundary of $(\text{Out}(F_N),\mu)$. Using Bestvina-Reynolds' and Hamenst\"adt's description of the Gromov boundary of the complex $\mathcal{FF}_N$ of free factors of $F_N$, this gives a new proof of the fact, due to Calegari and Maher, that the realization in $\mathcal{FF_N}$ of almost every sample path of the random walk converges to a boundary point. We get in addition that $\partial\mathcal{FF}_N$, equipped with the hitting measure, is the Poisson boundary of $(\text{Out}(F_N),\mu)$.

Random walks on weakly hyperbolic groups

Abstract Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

Synthesis and structural study of the Linde Type-A zeolite prepared from kaolinite

The goal of this work is to synthesize and measure the crystal structuras of sodalite-related zeolites using X-ray power diffraction technique. We made a successful attempt to recover the Linde Type A (LTA) zeolite framework using X-ray diffraction data sets. Cation exchange was done in this structure with Cs+ ions, in order to demonstrate the usability of LTA in the nuclear technique. Linde Type A zeolite crystals were synthetized using (NaOH) method from metakaolinite as the starting material. The resulted LTA crystals exhibit cubic and spherical shapes as showned by scanning electron microscopy. In the LTA framework the interchangeable cations are Na+ ions, which were replaced by (H3O)+ and Cs+ ions respectively. We measured the resulted structural changes, using X-ray powder diffraction data in the SIR (Semi Invariant Reconstruction) program package. The measured hkl and intensity data sets are the inputs for structure determination. Due to overlapping reflections, some samples resulted in high R-factor values. However, the resulted LTA-type framework proved to be evident. We refined the atom position in the SHELX program package.

On the Moment-Generating Functions of Extrema and Their Complements for Almost Semicontinuous Integer-Valued Poisson Processes on Markov Chains

For an integer-valued compound Poisson process with geometrically distributed jumps of a certain sign [these processes are called almost upper (lower) semicontinuous] defined on a finite regular Markov chain, we establish relations (without projections) for the moment-generating functions of extrema and their complements. Unlike the relations obtained earlier in terms of projections, the proposed new relations for the moment-generating functions are determined by the inversion of the perturbed matrix cumulant function. These matrix relations are expressed via the moment-generating functions for the distributions of the corresponding jumps.

Saddlepoint approximations to option price in a regime-switching model

In this paper we consider the saddlepoint approximation for the valuation of a European-style call option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. The standard option pricing procedure under this model becomes problematic as the occupation time of chains for a given state cannot be evaluated easily. In the case of two-state Markov chains, we present an explicit analytic formula of the cumulant generating function (CGF). When the process has more than two states, an approximate formula of the CGF is provided. We adopt a splitting method to reduce the complexity of computing the matrix exponential function. Then we use these CGFs to develop the use of the saddlepoint approximations. The numerical results show that the saddlepoint approximation is an efficient and reliable approach for option pricing under a multi-state regime-switching model.

Fluctuating currents in stochastic thermodynamics. I. Gauge invariance of asymptotic statistics.

Stochastic thermodynamics uses Markovian jump processes to model random transitions between observable mesoscopic states. Physical currents are obtained from antisymmetric jump observables defined on the edges of the graph representing the network of states. The asymptotic statistics of such currents are characterized by scaled cumulants. In the present work, we use the algebraic and topological structure of Markovian models to prove a gauge invariance of the scaled cumulant-generating function. Exploiting this invariance yields an efficient algorithm for practical calculations of asymptotic averages and correlation integrals. We discuss how our approach generalizes the Schnakenberg decomposition of the average entropy-production rate, and how it unifies previous work. The application of our results to concrete models is presented in an accompanying publication.

Full counting statistics in the not-so-long-time limit

The full counting statistics of charge transport is the probability distribution that n electrons have flown through the system in measuring time tm. The cumulant generating function (CGF) of this distribution has been well studied in the long time limit , however there are relatively few results on the finite measuring time corrections to this. In this work, we study the leading finite time corrections to the CGF of interacting Fermi systems with a single transmission channel at zero temperature but driven out of equilibrium by a bias voltage. We conjecture that the leading finite time corrections are logarithmic in tm with a coefficient universally related to the long time limit. We provide detailed numerical evidence for this with reference to the self-dual interacting resonant level model. This model further contains a phase transition associated with the fractionalization of charge at a critical bias voltage. This transition manifests itself technically as branch points in the CGF. We provide numerical results of the dependence of the CGF on measuring time for model parameters in the vicinity of this transition, and thus identify features in the time evolution associated with the phase transition itself.