Search

We propose a novel flexible framework for the joint evolution of stock log-returns and their volatility based on time changed Levy processes. The novelty of the approach stems from the generality of the jump structure we endow our model with, and the ability of the model to generate leverage effects out of the pure jump component. We derive the characteristic function and the forward characteristic function of the log-returns, which allow for the efficient pricing of vanilla and forward-start-like option contracts by Fourier transform methods. The proposed framework achieves robust calibration performance properties especially in the case of pure jump specifications. The results offered in this paper could have potentially interesting implications in terms of design of models and hedging strategies, and their development.

Accurate and reliable machine performance degradation tracking and remaining useful life (RUL) prognosis establish the foundation for predictive maintenance scheduling toward improved safety and productivity of machine operations. In general, machine performance degradation exhibits a nonlinear and nonhomogeneous pattern that arises from time-varying degradation rate and abrupt performance changes. To address this challenge and improve the generalizability of degradation modeling, in this article, we present a stochastic modeling technique based on the Levy process, which generalizes system variations as the accumulations of successive and jump increments. The developed Levy process model consists of two terms: a linear Brownian motion term for capturing the gradual degradation with time-varying rates and a nonhomogenous compound Poisson process term for capturing transient performance changes. By calculating the moments of the characteristic function of the Levy model, explicit expressions for the probability distributions of predicted performance degradation and RUL are derived. To obtain the time-varying parameters in the Levy model, Markov chain Monte Carlo is investigated. The developed technique is evaluated through simulation and run-to-failure tests of roller and ball bearings, and the good performance of the developed Levy model is confirmed.

We examine the influence of using a restart mechanism on the stationary distributions of a particular class of Markov chains. Namely, we consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We show that the restart mechanism may change significantly the limiting distribution. We obtain a limit theorem with a novel type of limiting distribution, a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.

In this paper, we derive a higher-order asymptotic expansion of characteristic functions of an Ito semimartingale over asymptotically shrinking time intervals. The leading term in the expansion is determined by the value of the diffusive coefficient at the beginning of the interval. The higher-order terms are determined by the jump compensator as well as the coefficients appearing in the diffusion dynamics. The result is applied to develop a nearly rate-efficient estimator of the leverage coefficient of an asset price, i.e., the coefficient in its volatility dynamics that appears in front of the Brownian motion that drives also the asset price.

This study proposes a new acceptance sampling scheme, namely mean ranked sampling scheme, for efficient decision of statistical quality control. The proposed scheme is based on ranked set sampling under exponential distribution for acceptance or rejection of a lot. Two-point method on operating characteristic function is used for the derivation of plan parameters assuming approximation of exponential to normal distribution. The minimum sample sizes and acceptance numbers are obtained satisfying producer’s and consumer’s risk at certain levels of acceptable (AQL) and tolerable quality (RQL). The main advantages of proposed scheme are (i) data is selected by using free cost sampling method from a large lot and (ii) all units of selected lot are used in making decision. The probability & distribution functions for proposed mean ranked set sampling plan are derived. The OC values comparison with simple random sampling is presented. It is shown that the tables presented here can be used if mean life of units are chosen as a criterion.

Many problems in real life exist that are full of confusion, vagueness, and ambiguity. The quantification of such issues in a scientific way is the need of time. The negative binomial distribution is an important discrete probability distribution from the account of classical probability distribution theory. The distribution was used to study the chance of kth success in n trials before n − 1 failures for crisp data. The literature lacks in dealing with the situations for interval-valued data under negative binomial distribution. In this research, the neutrosophic negative binomial distribution is proposed to generalize the classical negative binomial distribution. The generalized proposed distribution considers the indeterminacy and crisp form from interval-valued. Several properties of the proposed distribution, such as moment generating function, characteristic function, and probability generating function, are also derived. Furthermore, the derivation of reliability analysis properties such as survival, hazard rate, reversed hazard rate, cumulative hazard rate, mills ratio, and odds ratio are also presented. In addition, order statistics for the proposed distribution, including w th , joint, median, minimum, and maximum order statistics are part of the paper. The proposed distribution is discussed from the real data applications perspective by considering the different case studies. This research opens the way to deal with the problems that follow conventional conveyances and include nonprecisely determined details simultaneously.

Let $$X_1,\ldots ,X_n$$ be an i.i.d. sample from symmetric stable distribution with stability parameter $$\alpha$$ and scale parameter $$\gamma$$ . Let $$\varphi _n$$ be the empirical characteristic function. We prove a uniform large deviation inequality: given preciseness $$\epsilon >0$$ and probability $$p\in (0,1)$$ , there exists universal (depending on $$\epsilon$$ and p but not depending on $$\alpha$$ and $$\gamma$$ ) constant $$\bar{r}>0$$ so that $$P\big (\sup _{u>0:r(u)\le \bar{r}}|r(u)-\hat{r}(u)|\ge \epsilon \big )\le p,$$ where $$r(u)=(u\gamma )^{\alpha }$$ and $$\hat{r}(u)=-\ln |\varphi _n(u)|$$ . As an applications of the result, we show how it can be used in estimation the unknown stability parameter $$\alpha$$ .

This paper studies the heavy-traffic limit of the moments of the stationary distribution in GI/G/1-type Markov chains. For these Markov chains, several researchers have derived heavy-traffic-limit formulas for the stationary distribution itself. However, for its moments, no such formulas have been reported in the literature. This paper presents a heavy-traffic-limit formula for the moments of the stationary distribution and a sufficient condition for the formula to hold, by using a characteristic function approach.

The aim of the present work is evaluation of absolute constants in the Arak inequalities for the concentration functions of convolutions of probability distributions. This result allows us to calculate the constant in the inequality for the uniform distance between n and (n + 1)-fold convolutions of one-dimensional symmetric probability distributions with a characteristic function separated from −1, as well as a number of other estimates, in particular, the accuracy of approximation of samples of rare events by the Poisson point process.

The reasonable distribution of revenue determines not only the coordination but also the development of supply chain. A hybrid channel supply chain model composed of manufacturer and traditional and online retailers was identified as the object of study, decision models under different cooperation modes were built, different revenues of supply chain members and overall were analyzed, satisfaction of characteristic function required by the Shapley value was proved, and the Shapley values of revenue distribution were calculated. In view of the shortcoming of the Shapley value method which takes contribution margin as the only consideration and ignores the impact of other factors such as cost on revenue distribution, the cost correction model was built by using production and sales cost to modify the traditional Shapley value; the reasonable distribution of individual benefits under the premise of maximizing the overall revenue of hybrid channel supply chain was realized.

This paper is devoted to the problems of studying the multiperiodic solution of some evolutionary equations. The article also discusses the existence and uniqueness of a multiperiodic solution with respect to vector functions for an evolutionary reduced equation. Studies have been conducted on the characteristic function of a certain system of the evolutionary equation. Some properties of the vector function are proved. They can be used in the further study of oscillatory bounded solutions of evolutionary equations. Based on the argumentation of the theorem on the existence and uniqueness of an almost multiperiodic solution of the specified system, considered using the method of shortening the characteristic function. All estimates of the characteristic function are based on the enhanced Lipschitz condition, first introduced by academician K. P. Persidskiy. The results will also be useful in the study of periodic solutions of evolutionary equations of mathematical physics

Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,\chi_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo $c$. For $\sigma>1/2$, we prove an asymptotic distribution function $F_{\sigma}$ for the values of the logarithm of \begin{equation*} L_c(s)= L(s,\chi_c)L(s,\overline{\chi}_{c}), \end{equation*} as $c$ varies. Moreover, the characteristic function of $F_{\sigma}$ is expressed explicitly as a product over the prime ideals of $\mathbb{Z}[i]$., Comment: 11 pages

In this paper, we investigate the growth and fixed points of solutions of higher-order linear differential equations in the unit disc. We extend the coefficient conditions to a type of one-constant-control coefficient comparison and obtain the same estimates of iterated order of solutions. We also obtain better estimates by providing a precise value of iterated order of solution instead of a range of that in the case of coefficient characteristic function comparison. Moreover, we utilize iteration to investigate and estimate the fixed points of solutions’ arbitrary-order derivatives with higher-order equations $f^{(k)}+A_{k-1}(z)f^{(k-1)}+{\cdots }+A_{1}(z)f'+A_{0}(z)f=0$ f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 and provide a concise method to judge if the items generated by the iteration do not vanish identically and ensure the iteration proceeds. Our results are an improvement over those by B. Belaïdi, T. B. Cao, G. W. Zhang and A. Chen.

The goal of this work is to propose a new strategy for automatically quantifying the high-pass cut-off frequency of digital ground motion (GM) records. In previous studies, the high-pass filter cut-off frequency of GM records was often identified by visual inspection. For this, this article proposes a simple and efficient method to detect a correct cut-off frequency where the displacement waveform exactly does not change significantly. The displacement change is delineated as the change rate of the displacement at the ending of padded record with respect to the cut-off frequency. Based on the Short-Term-Average to the Long-Term-Average (STA/LTA) method, detecting the ratio of calculated average energy in two consecutive moving-time windows above a threshold, a critical frequency where the ending displacement starts deviating from zero level can be identified. Nevertheless, the cut-offs may be affected by GM intensities even though meticulous setting for window length or trigger threshold is adopted. In this paper, a new characteristic function (CF) on the basis of a modified cumulative envelope function is proposed. It has an appealing advantage to detect the critical cut-off frequency based on the CF shape and free of considering the intensity difference among GM records. Finally, the comparison of results with the cut-off frequencies of GMs from the 2008 Wenchuan mainshock calculated by the proposed method and those derived from two traditional methods is presented. Influence of the cut-off frequency on ground motion displacement, elastic, and inelastic spectrum is also studied.

The normality assumption is a significant part of the development of control charts. This underlying assumption of normality most likely does not hold true in real scenarios. One of such designs usually devised to observe the target parameter σ 2 of the Maxwell quality characteristics is the V -control chart. In general, quality practitioners preferably have to observe the scale parameter σ rather than σ 2 in examined processes. The contemporary V -control chart is relying on the V -statistic which does not hold the unbiasedness property with respective to parameter σ of the Maxwell probability model. In view of this, implementation of the V -chart is not an appropriate design in monitoring a real parameter of the underlying Maxwell data. To accommodate the monitoring of the parameter σ of the Maxwell model, a novel design of the V S Q -chart is mainly proposed in this work. To support a statistical understanding of the V S Q -chart, power function, characteristic function, and the average run length ARL have been essentially established. The parameters of the V S Q -chart are determined from the results of the sampling distribution of the derived statistic. Analytical findings are further applied to determine the performance of the study proposal with its existing counterpart. Substantially, the better performance of the proposed technique has been observed because of statistical power used as a performance measure. Eventually, the computational plan of the V S Q -chart is considered both for the simulated and real datasets with the aim of illustrating the theory of the proposed design.

A structure theorem is established which shows that an arbitrary multimode bosonic Gaussian observable can be represented as a combination of four basic cases whose physical prototypes are homodyne and heterodyne, noiseless or noisy, measurements in quantum optics. The proof establishes a connection between the descriptions of a Gaussian observable in terms of the characteristic function and in terms of the density of a probability operator-valued measure (POVM) and has remarkable parallels with the treatment of bosonic Gaussian channels in terms of their Choi–Jamiolkowski form. Along the way we give the “most economical” (in the sense of minimal dimensions of the quantum ancilla) construction of the Naimark extension of a general Gaussian observable. We also show that the Gaussian POVM has bounded operator-valued density with respect to the Lebesgue measure if and only if its noise covariance matrix is nondegenerate.

In this paper, we study the finite-time ruin probability in the risk model driven by a Levy subordinator, by incorporating the popular Fourier-cosine method. Our interest is to propose a general approximation for any specified precision provided that the characteristic function of the Levy Process is known. To achieve this, we derive an explicit integral expression for the finite-time ruin probability, which is expressed in terms of the density function and the survival function of L t . Moreover, we apply the rearrangement inequality to further improve our approximations. In addition, with only mild and practically relevant assumptions, we prove that the approximation error can be made arbitrarily small (actually an algebraic convergence rate up to 3, which is the fastest possible approximant known upon all in the literature), and has a linear computation complexity in a number of terms of the Fourier-cosine expansion. The effectiveness of our results is demonstrated in various numerical studies; through these examples, the supreme power of the Fourier-cosine method is once demonstrated.

A new generalization of the skew-t distribution was proposed. The two-parameter lifetime model called the odd exponentiated skew-t distribution has the ability of fitting skewed, long and heavy tailed datasets. It is considered to be more flexible than the skew-t distribution as it contains it as a special case. Some basic properties of the distribution such as the order statistics, entropy, asymptotic behaviour, moment, incomplete moment, characteristic function and quantile function were derived. The odd exponentiated skew-t distribution parameter estimates were derived using the maximum likelihood estimation method and simulation studies performed to evaluate the finite sample performance of these parameter estimates showed that the parameter estimates were consistent and approached the arbitrary selected parameter values as the sample size is increased. The application using a real-life dataset indicated that the new distribution outperformed the other competing distributions. The hazard rate shape of the odd exponentiated skew-t distribution was found to be increasing and J-shaped which was also reflected in the application result.

Consider a discrete-time supercritical branching random walk, which is a branching process combined with random spatial motion of particles, the number of the descendants tending to infinity with positive probability. Let $$ Z_n(\cdot ) $$ be the counting measure which counts the number of particles of generation n in a given set. Revesz (1994) studied the convergence rates in the central and local limit theorems for $$ Z_n(\cdot ) $$ in some special cases, and then, the topic was further developed in various cases. In this paper, we give the exact convergence rates of the central and local limit theorems for $$ Z_n(\cdot ) $$ in the case the underlying motion is governed by a general non-lattice random walk on $${\mathbb {R}}^d$$ with the characteristic function of the motion law satisfying the weak Cramer condition.

The capacitance is a characteristic function of an electrical energy storage device that relates the applied voltage on the device to the accumulated electric charge. It is inconsistently taken in some studies as a multiplicative function in the time domain [i.e., ${q}_{t}{(}{t}{)}={c}_{t}{(}{t}{)} \times {v}_{t}{(}{t}{)}$ ], and in others as a multiplicative function in the frequency domain [i.e., ${Q}_{f}{(}{s}{)}={C}_{f}{(}{s}{)} \times \, {V}_{f}{(}{s}{)}$ derived from the definition of admittance ${I}_{f}{(}{s}{)}/{V}_{f}{(}{s}{)} = {s} {C}_{f}{(}{s}{)}$ ], despite the fact that the capacitance is time- and frequency-dependent. However, the convolution theorem states that multiplication of functions in the time domain is equivalent to a convolution operation in the frequency domain, and vice versa. In this work, we revisit and compare the two outlined definitions of capacitance for an ideal capacitor and for a lossy fractional-order capacitor. Although ${c}_{t}{(}{t}{)} = {C}_{f}{(}{s}{)} = {C}$ for an ideal constant capacitor, we show that this is not the case for fractional-order capacitors which exhibit frequency-dispersed impedance, memory effects, and nonexponential relaxation functions. This fact is crucial in the accurate modeling and characterization of supercapacitors and batteries. For these devices, and for being consistent with measurements using conventional impedance analyzers, it is recommended to apply the integral convolution definition in the time domain which reverts to the multiplicative definition in the frequency domain.

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our approach consists of replacing the characteristic function $\chi_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < \alpha < n/2$., Comment: 31 pages. Minor revision

We propose generalization of the spatial Simultaneous Autoregressive (SAR) model on a lattice towards modelling for asymmetry. Under the assumption of skew-normal error structure, expression for density and characteristic function for the induced distribution of response are obtained. Full-likelihood based implementation of the proposed model to a real data set is performed using Differential Evolution (DE). The relevant results are reported and compared with the results from existing models.

To model data in angular form, circular distributions are the most appropriate mathematical tools. In this paper, we introduce a new one-parameter circular distribution based on the wrapping method...

This paper proposes an innovative method to improve the attribute weighting approaches for naive Bayes text classifiers using the improved distance correlation coefficient. The resulted model is called improved distance correlation coefficient attribute weighted multinomial naive Bayes, denoted by IDCWMNB. Unlike the traditional correlation statistical measurements that consider the cumulative distribution function of random vectors, the improved distance correlation coefficient tests the joint correlation of random vectors by describing the distance between the joint characteristic function and the product of the marginal characteristic functions. Specifically, a measurement of inverse document frequency that considers the distribution information of document concentrating and scattering has been proposed. Then, the measurement and the distance correlation coefficient between attributes and categories have been combined to measure the importance of attributes to categories, to allocate different weights to different terms. Meanwhile, the learned attribute weights are incorporated into the posterior probability estimates of the multinomial naive Bayes model, which is known as deep attribute weighting. This measurement is more effective than the traditional statistical measurements in the presence of nonlinear relationship between two random vectors. Experimental results taking benchmark and real-world data indicate that the new attribute weighting method can achieve an effective balance between classification accuracy and execution time.

Matrix variate generalizations of Pareto distributions are proposed. Several properties of these distributions including cumulative distribution functions, characteristic functions and relationship to matrix variate beta type I and matrix variate type II distributions are studied.

We propose an extension of the $$\Gamma $$ Γ -OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the $$\Gamma $$ Γ -OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the $$\Gamma $$ Γ -OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the $$\Gamma $$ Γ -OU Barndorff-Nielsen and Shephard model.

The concept of extended slash distribution is introduced. For the extended slash distribution of the sum of two independent logistic random variables, the characteristic function, moments, absolute...

Using Bochner's theorem, we derive a necessary and sufficient hierarchy of measurable conditions in terms of the characteristic function of quasi-moments of a weakly measured observable. A quantum ...

In this paper, we study the value distribution of the differential polynomial $\varphi f^2f^{(k)}-1$, where $f(z)$ is a transcendental meromorphic function, $\varphi (z)\;(\not\equiv 0)$ is a small function of $f(z)$ and $k\;(\geq 2)$ is a positive integer. We obtain an inequality concerning the Nevanlinna Characteristic function $T(r,f)$ estimated by reduced counting function only. Our result extends the result due to J.F. Xu and H.X. Yi [J. Math. Inequal., 10 (2016), 971-976].

Markov Chain Monte Carlo methods become increasingly popular in applied mathematics as a tool for numerical integration with respect to complex and high-dimensional distributions. However, application of MCMC methods to heavy-tailed distributions and distributions with analytically intractable densities turns out to be rather problematic. In this paper, we propose a novel approach towards the use of MCMC algorithms for distributions with analytically known Fourier transforms and, in particular, heavy-tailed distributions. The main idea of the proposed approach is to use MCMC methods in Fourier domain to sample from a density proportional to the absolute value of the underlying characteristic function. A subsequent application of the Parseval’s formula leads to an efficient algorithm for the computation of integrals with respect to the underlying density. We show that the resulting Markov chain in Fourier domain may be geometrically ergodic even in the case of heavy-tailed original distributions. We illustrate our approach by several numerical examples including multivariate elliptically contoured stable distributions.

Statistical characteristics of narrow-band signal in the presence of modulating noise are considered, including case of determining the univariate density of signal distribution. The relationship between the characteristic function of instantaneous signal values and the probability distribution density (PDD) envelope with a uniform phase distribution law is determined. Expressions describing a PDD of instantaneous values of signal and additive noise under action of modulating noise are obtained. Cases where there is no and is functional relationship between amplitude and phase fluctuations have been analyzed. It is shown that with the law of phase distribution other than uniform at the interval $$[0,2\pi]$$ , the signal is a nonstationary random function tending to stationary at increasing the depth of phase distortions. With uniform phase distribution, the signal is a stationary random function. It is shown that in case of functionally connected amplitude-phase distortions, the PDD signal is described through statistical characteristics of its envelope.

The basic assumption that is often used in bond valuations is the assumption on the Black-Scholes model. The practical assumption of the Black-Scholes model is the return of assets with normal distribution, but in reality there are many conditions where the return of assets of a company is not normally distributed and causing improperly developed bond valuation modeling. The Fast-Fourier Transform model (FFT) was developed as a solution to this problem. The Fast-Fourier Transformation Model is a Fourier transformation technique with high accuracy and is more effective because it uses characteristic functions. In this research, a modeling will be carried out to calculate bond valuations designed to take advantage of the computational power of the FFT. The characteristic function used is the Variance Gamma, which has the advantage of being able to capture data return behavior that is not normally distributed. The data used in this study are Sustainable Bonds I of Bank Danamon Phase I Year 2019 Series B, Sustainable Bonds II of Bank CIMB Niaga II Phase IV Year 2018 Series C, Sustainable Subordinated Bonds II of Bank UOB Indonesia Phase II 2019. The results obtained are FFT model using the Variance Gamma characteristic function gives more precise results for the return of assets with not normal distribution. Keywords: Bonds, Bond Valuation, Black-Scholes, Fast-Fourier Transform, Variance Gamma

Here we consider a generalized log-transformed version of the Weibull distribution and investigate some of its important properties like expressions for the cumulative distribution function hazard rate function, quantile function, characteristic function, raw moments, incomplete moments, etc. The distribution and moments of order statistics are obtained along with some results on certain structural properties of the distribution. The maximum likelihood estimation of the parameters of the distribution is attempted for both complete and censored data sets and the usefulness of the distribution is illustrated with the help of real-life data sets from biomedical fields.

Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an $n$-splitter, which is fed with $n$ copies of a centred state $\rho$ with finite second moments, converges to the Gaussian state with the same first and second moments as $\rho$. Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate $\mathcal{O}\left(n^{-1/2}\right)$ in the Hilbert--Schmidt norm whenever the third moments of $\rho$ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given $m$-mode state enters a cascade of $n$ beam splitters of equal transmissivities $\lambda^{1/n}$ fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate $\mathcal{O}\Big(n^{-\frac{1}{2(m+1)}}\Big)$. This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function $\chi_\rho$ is uniformly bounded by some $\eta_\rho, Comment: 52 pages, 4 figures. The presentation in v2 has been improved extensively; the proofs in Sections VI and VIII have been re-written, although their mathematical content is almost unchanged; the statement of Corollary 13 has been slightly modified; we added a Remark on p.41

During the last several decades, the results related to geometric random sums of independent identically distributed random variables have become interesting results in probability theory and in insurance, risk theory, queuing theory and stochastic finance, etc. The negative–binomial random sums are extensions of geometric random sums. Up to the present, the negative–binomial random sums of independent identically distributed random variables have attracted much attention since they appear in many fields. However, for the strictly stationary m-dependent summands, such negative–binomial random sums are rarely studied. It seems that the complexity of the dependent structure has limited the use of classical tools based on independent random variables such as characteristic function. Up to now, a number of methods have been used to overcome the dependency such as the truncate method, Stein method, and Heinrich’s method. The paper deals with the order of approximation in weak limit theorems for normalized negative–binomial sums of strictly stationary m-dependent random variables generated by a sequence of independent, identically distributed random variables, using the moving average techniques. The orders of approximation in desired theorems are established in terms of the so-called Zolotarev ζ-metric. The obtained results are extensions and generalizations of several known results related to negative–binomial random sums and geometric random sums of independent, identically distributed random variables.

A matrix variate generalization of the two-sided power distribution is proposed. Several properties of this distribution including probability density function, cumulative distribution function and characteristic function are derived.

In this paper two basic random variables constructed from Quasi Lindley distribution have been introduced. One of these variables is defined as the sum of two independent random variables following the Quasi-Lindley distribution with the same parameter (2SQLindley). The second one is defined as the difference of two independent random variables following the Quasi-Lindley distribution with also the same parameter (2DQLindley). For both cases, we provided some statistical properties such as moments, incomplete moments and characteristic function. The parameters are estimated by maximum likelihood method. From simulation studies, the performance of the maximum likelihood estimators has been assessed. The usefulness of the corresponding models is proved using goodness-of-fit tests based on different real datasets. The new models provide consistently better fit than some classical models used in this research.

Several researchers have addressed the problem of testing the homogeneity of the scale parameters of several independent inverse Gaussian distributions based on the likelihood ratio test. However, only approximations of the distribution function of the test statistic are available in the literature. In this note, we present the exact distribution of the likelihood ratio test statistic for testing the equality of the scale parameters of several independent inverse Gaussian populations in a closed form. To this end, we apply the Mellin inverse transform and the Jacobi polynomial expansion to the moments of the likelihood ratio test statistic. We also propose an approximate method based on the Jacobi polynomial expansion. Finally, we apply an accurate numerical method, which is based on the inverse of characteristic function, to obtain a near-exact approximation of the likelihood ratio test statistic distribution. The proposed methods are illustrated via numerical and real data examples.

This research work is based on the concept of the one-factor copula model together with the discrete Fourier transform, which is applied to reduce the dimensionality problems associated with the basket default swap pricing. We employ the Gaussian, the student-t and the Clayton one-factor copula to estimate the conditional probability of default. Incorporating the Fourier transform together with the distribution function of a counting process, we derive the quasi-analytical expression for the computation of the swap payment legs. We compute the conditional characteristic function for the corresponding portfolio loss distribution using the fast Fourier transform. Then, employ numerical integration with the aid of the inverse fast Fourier transform to retrieve the distribution function or the unconditional characteristic function. Our results show that in the absence of the trending simulation method, a semi-analytic method which involves the applications of the discrete Fourier transform can be utilized to price the basket credit default swaps.

In this paper, we investigate the fast growing solutions of higher-order linear differential equations where $ A_0 $, the coefficient of $ f $, dominates other coefficients near a point on the boundary of the unit disc. We improve the previous results of solutions of the equations where the modulus of $ A_{0} $ is dominant near a point on the boundary of the unit disc, and obtain extensive version of iterated order of solutions of the equations where the characteristic function of $ A_{0} $ is dominant near the point. We also obtain a general result of the iterated exponent of convergence of the fixed points of the solutions of higher-order linear differential equations in the unit disc. This work is an extension and an improvement of recent results of Hamouda and Cao.

This article develops a beta-exponentiated Ishita distribution that extends the exponentiated Ishita distribution. Expansions for the cumulative distribution and probability density functions are given. Various properties of the new distribution such as hazard function, moments, cumulants, skewness, kurtosis, mean deviations, Bonferroni and Lorenz curves, Renyi and Tsallis entropies, and stress-strength reliability are discussed. Moment generating function and characteristic function of the new model were derived. Distribution and the moment of order statistic have been derived. The method of maximum likelihood was used for estimation of parameters. The new model is quite flexible in analysing positively skewed data. Two real datasets are used to demonstrate the flexibility of the new distribution.

The arbitrary linear quantization is considered for definition of Wigner function. The evolution equation of this function is derived with the use of inversion of linear quantization kernel. In general case the Moyal equation depends on the characteristic function of quantization. It is shown that only for Weyl quantization this equation does not consist the source of quasi-probability. The stationary solutions of Moyal equation are constructed for an arbitrary linear quantization of a harmonic oscillator. The case of anharmonic oscillator is presented as a practical example of quantum system model in the so-called post-newtonian approximation. This model exhibits the intersection effect between quantization dependences of Hamiltonian and Wigner function.

A characteristic function is a special operator-valued analytic function defined on the open unit ball of $\mathbb{C}^n$ associated with an $n$-tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of $n$-tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason's problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of the noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions., 26 pages

Research, development and extensive use of generalized form of distributions in order to analyze and modeling of applied sciences research data has been growing tremendously. Weibull and Frechet distribution are widely discussed for reliability and survival analysis using experimental data from physical, chemical, environmental and engineering sciences. Both the distributions are applicable to extreme value theory as well as small and large data sets. Recently researchers develop several probability distributions to model experimental data as these parent models are not adequate to fit in some experiments. Modified forms of the Weibull distribution and Frechet distribution are more flexible distributions for modeling experimental data. This article aims to introduce a generalize form of Weibull distribution known as Frechet-Weibull Distribution (FWD) by using the T-X family which extends a more flexible distribution for modeling experimental data. Here the pdf and cdf with survival function [S(t)], hazard rate function [h(t)] and asymptotic behaviour of pdf and survival function and the possible shapes of pdf, cdf, S(t) and h(t) of FWD have been studied and the parameters are estimated using maximum livelihood method (MLM). Some statistical properties of FWD such as mode, moments, skewness, kurtosis, variation, quantile function, moment generating function, characteristic function and entropies are investigated. Finally the FWD has been applied to two sets of observations from mechanical engineering and shows the superiority of FWD over other related distributions. This study will provide a useful tool to analyze and modeling of datasets in Mechanical Engineering sciences and other related field.

Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv-\infty$ on the complex plane. An integral of the maximum on circles centered at zero of $U^+:=\sup\{0,U\} $ or $|u|$ over $E$ with a function-multiplier $g\in L^p(E)$ in the integrand is estimated, respectively, in terms of the characteristic function $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, and also in terms of the linear Lebesgue measure of $E$ and the $ L^p$-norm of $g$. Our main theorem develops the proof of one of the classical theorems of Rolf Nevanlinna in the case $E=[0,R]$, given in the classical monograph by Anatolii A. Gol'dberg and Iosif V. Ostrovskii, and also generalizes analogs of the Edrei-Fuchs Lemma on small arcs for small intervals from the works of A.F. Grishin, M.L. Sodin, T.I. Malyutina. Our estimates are uniform in the sense that the constants in these estimates do not depend on $U$ or $u$, provided that $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively., Comment: 10 pages

In the year 1978, Ismail and May studied operators of exponential type and proposed some new operators which are connected with a certain characteristic function $$p\left( x\right) $$ p x . Several of these operators were not separately studied by researchers due to its unusual behavior. The topic of the present paper is the local rate of approximation of a sequence of exponential type operators $$R_{n}$$ R n belonging to $$p\left( x\right) =x\left( 1+x\right) ^{2}$$ p x = x 1 + x 2 . As the main result we derive a complete asymptotic expansion for the sequence $$\left( R_{n}f\right) \left( x\right) $$ R n f x as n tends to infinity.

This paper is devoted to a nonparametric estimation of the probability θ : = ℙ ( X < Y ) , where X, Y are continuous univariate random variables of interest and observed with additional random erro...

In the present paper, we study in the harmonic analysis associated to the Weinstein operator, the boundedness on L p of the uncentered maximal function. First, we establish estimates for the Weinstein translation of characteristic function of a closed ball with radius ɛ centered at 0 on the upper half space ℝd–1× ]0, +∞ [. Second, we prove weak-type L 1-estimates for the uncentered maximal function associated with the Weinstein operator and we obtain the L p -boundedness of this operator for 1 < p ≤ + ∞. As application, we define a large class of operators such that each operator of this class satisfies these L p -inequalities. In particular, the maximal operator associated respectively with the Weinstein heat semigroup and the Weinstein-Poisson semigroup belong to this class.

The so-called l0 pseudonorm on Rd counts the number of nonzero components of a vector. It is used in sparse optimization, either as criterion or in the constraints, to obtain solutions with few nonzero entries. For such problems, the Fenchel conjugacy fails to provide relevant analysis: indeed, the Fenchel conjugate of the characteristic function of the level sets of the l0 pseudonorm is minus infinity, and the Fenchel biconjugate of the l0 pseudonorm is zero. In this paper, we display a class of conjugacies that are suitable for the l0 pseudonorm. For this purpose, we suppose given a (source) norm on Rd. With this norm, we define, on the one hand, a sequence of so-called coordinate-k norms and, on the other hand, a coupling between Rd and Rd , called Capra (constant along primal rays). Then, we provide formulas for the Capra-conjugate and biconjugate, and for the Capra subdifferentials, of functions of the l0 pseudonorm (hence, in particular, of the l0 pseudonorm itself and of the characteristic functions of its level sets), in terms of the coordinate-k norms. As an application, we provide a new family of lower bounds for the l0 pseudonorm, as a fraction between two norms, the denominator being any norm.