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The thermodynamic relations in the Tsallis statistics were studied with physical quantities. An additive entropic variable related to the Tsallis entropy was introduced by assuming the form of the first law of the thermodynamics. The fluctuations in the Tsallis statistics were derived with physical quantities with the help of the introduced entropic variable. It was shown that the mean squares of the fluctuations of the physical quantities in the Tsallis statistics are the same as those in the conventional statistics. The mean square of the fluctuation of the Tsallis entropy and the mean square of the fluctuation of the Tsallis temperature were also derived. The mean square of the relative fluctuation of the Tsallis entropy and the mean square of the relative fluctuation of the Tsallis temperature are represented with heat capacities. It was shown that these fluctuations of the Tsallis quantities have the $q$-dependent terms in the Tsallis statistics of the entropic parameter $q$., Comment: 10 pages

A new method has been proposed to generalize Burr-XII distribution, also called Burr distribution, by adding an extra parameter to an existing Burr distribution for more flexibility. In this method, the exponent of the Burr distribution is modeled using a nonlinear function of the data and one additional parameter. The models of this newly introduced generalized Burr family can significantly increase the flexibility of the former Burr distribution with respect to the density and hazard rate shapes. Families expanded using the method proposed here is heavy-tailed and belongs to the maximum domain of attractions of the Frechet distribution. The method is further applied to yield three-parameter classical Pareto and generalized exponentiated distributions which shows the broader application of the proposed idea of generalization. A relevant model of the new generalized Burr family has been considered in detail, with particular emphasis on the hazard functions, stochastic orders, estimation procedures, and testing methods are derived. Finally, as empirical evidence, the new distribution is applied to the analysis of large-scale heavy-tailed network data and compared with other commonly used distributions available for fitting degree distributions of networks. Experimental results suggest that the proposed Burr distribution with nonlinear exponent better fits the large-scale heavy-tailed networks better than the popularly used Marhsall-Olkin generalization of Burr and exponentiated Burr distributions.

We improve both dimension compression and expansion bounds for homeomorphisms with $p$-exponentially integrable distortion. To the first direction we also introduce estimates for the compression multifractal spectra, which will be used to estimate compression of dimension, and for the rotational multifractal spectra. For establishing the expansion case we use the multifractal spectra of the inverse mapping and construct examples proving sharpness.

We construct infinitely many families of Lorenz knots that are satellites but not cables, giving counterexamples to a conjecture attributed to Morton. We amend the conjecture to state that Lorenz knots that are satellite have companion a Lorenz knot, and pattern equivalent to a Lorenz knot. We show this amended conjecture holds very broadly: it is true for all Lorenz knots obtained by high Dehn filling on a parent link, and other examples., 24 pages, 9 figures. V2: Details added to argument in section 5, formulae for patterns corrected in section 6, and half-twisting results removed from this paper. V3: Improved exposition throughout

We point out that the Neyman-Pearson lemma applies to Bayes factors if we consider expected type-1 and type-2 error rates. That is, the Bayes factor is the test statistic that maximises the expected power for a fixed expected type-1 error rate. For Bayes factors involving a simple null hypothesis, the expected type-1 error rate is just the completely frequentist type-1 error rate. Lastly we remark on connections between the Karlin-Rubin theorem and uniformly most powerful tests, and Bayes factors. This provides frequentist motivations for computing the Bayes factor and could help reconcile Bayesians and frequentists., Comment: 8 pages

This paper concerns an investigation on the multifractal features of liquidity in China's stock markets based on multifractal detrended fluctuation analysis, and an analysis on multifractal features of market liquidity and fractal degree by determining the generalized Hurst exponent. Also involved in this paper are the identification of the trended fluctuations by tendency entropy dimension and a study on the validity of the correct rate of identification by stochastic correct rate. The results show that the multifractal features of liquidity are obvious in China's stock markets and that the multifractal liquidity degree of the large-cap stock is lower than those of the medium and small-cap stocks. The trend entropy dimension can be used to identify the trend of liquidity effectively, serving as an effective method to identify the fluctuation trend of fractal market.

An autoregressive–moving-average (ARMA) point process model is introduced, which combines self-exciting and shot-noise cluster mechanisms, both useful in a variety of applications. The process is analogous to the ARMA for integer-valued time series, sharing methodological and mathematical similarities. A maximum likelihood estimation procedure, based on MCEM (Monte Carlo Expectation Maximization), is derived and studied. This approach conveniently allows for: (i) trends in immigration/background intensity, (ii) multiple parametric specifications of memory functions and mark distributions, as well as (iii) cases where marks and immigrants are not observed. As such, the ARMA point process provides a flexible framework to disentangle cluster mechanisms in continuously observed count data., Econometrics and Statistics, 24, ISSN:2452-3062

Polar format algorithm (PFA) is an efficient image formation algorithm for spotlight-mode synthetic aperture radar (SAR). However, the derivation of PFA relies upon the approximation of planar wave...

Micro-holes drilling of titanium alloy using ultrasonic-assisted high-speed drilling (UAHD) has been proven to be effective in improving the height of the exit burr. The height of the exit burr is considered as one of the key output variables to evaluate the machining quality of micro-holes. Investigating the height of the exit burr and its modeling development provides great help to understand the effects of processing parameters on the burr height. This investigation will present an analytical model for the height of micro-holes exit burr. The acoustic softening effect is considered in this model. The approaches of the modeling development start from the work done by each part, including the axial force work, material deformation work, and total work. Then the burr height is calculated according to the conservation of energy. To verify the model, a single factor experiment of titanium alloy micro-holes was designed. The trends of analytical burr heights were consistent well with those of experimental results. Both the analytical model and the experimental results showed that the height of the exit burr was negatively correlated with the ultrasonic amplitude and spindle speed, and positively correlated with the feed speed. Also, the micro-holes exit morphology was observed under different processing parameters. The results showed that larger amplitude, spindle speed, and smaller feed speed were beneficial to the micro-holes exit with higher shape accuracy.

This article situates Descartes’ physical thinking within the nexus of machine science, which rests upon different foundational piers than regular classical mechanics of a Newtonian stripe. In part...

A statistical approach, based on artificial neural networks, is pro- posed for the post-calibration of weather radar rainfall estimation. Tested artificial neural networks include multilayer feedforward networks and radial basis functions. The multilayer feedforward training algorithms consisted of four variants of the gradient descent method, four variants of the conju- gate gradient method, Quasi-Newton, One Step Secant, Resilient backprop- agation, Levenberg-Marquardt method and Levenberg-Marquardt method using Bayesian regularization. The radial basis networks were the radial basis functions and the generalized regression networks. In general, results showed that the Levenberg-Marquardt algorithm using Bayesian regulariza- tion can be introduced as a robust and reliable algorithm for post-calibration of weather radar rainfall estimation. This method benefits from the conver- gence speed of the Levenberg-Marquardt algorithm and from the over fitting control of Bayes' theorem. All the other multilayer feedforward training al- gorithms result in failure since they often lead to over fitting or converged to a local minimum, which prevents them from generalizing the data. Radial basis networks are also problematic since they are very sensitive when used with sparse data.

In 2011, Jin Qinian has proposed a Levenberg–Marquardt method, by making use of duality mapping and the Bregman distance, to get an approximate solution of a nonlinear ill-posed operator equation i...

The present paper reports a rigorous study of the Indian Summer Monsoon Rainfall (ISMR) through Multifractal Detrended Fluctuation Analysis (MF-DFA). It has been observed that the ISMR is characterized by multifractality and Hurst Exponent above 0.5. It has been interpreted from the Hurst Exponent value that the ISMR time series is characterized by long term positive auto-correlation. Studying the strong correlation between the qthorder fluctuation and the length scale the multifractality has been confirmed within the ISMR time series. Finally, the entropy associated with ISMR has been computed using the principle of entropy maximization and the potential of entropy maximizing principle has been established over conventional fitting of normal distribution to ISMR.

Maximum entropy (MAXENT) is a widespread method for recovering unknown probabilities of a random variable Z . The method uses first few (empiric) moments of Z and leads to non-parametric estimators. Here we study MAXENT in a Bayesian set-up assuming that there exists a well-defined Dirichlet density for unknown probabilities. This allows to employ the average Kullback-Leibler (KL) distance in evaluating various MAXENT constraints, and comparing MAXENT with parametric estimators: the regularized maximum likelihood (ML) and the Bayesian estimator. KL distance is singled out among other distances from demanding a weak consistency of MAXENT with respect to a shrinkage. We show that MAXENT applies to sparse data, provided that there are certain types of prior information, e.g. the probabilities of Z are most probably deterministic, or there exist prior conditional rank correlations between Z and its probabilities. In the latter case MAXENT can outperform the optimally regularized ML and produce results that are close to the globally optimal Bayes estimator. Predictions of MAXENT are meaningless (i.e. worse than a random guess) if the needed prior information is absent.

A systematic review protocol of the neurophysiological properties underlying HEPs in both diagnostic conditions and healthy controls.

We consider the computability and complexity of decision questions for Probabilistic Finite Automata (PFA) with sub-exponential ambiguity. We show that the emptiness problem for strict and non-strict cut-points of polynomially ambiguous commutative PFA remains undecidable, implying that the problem is undecidable when inputs are from a letter monotonic language. We show that the problem remains undecidable over a binary input alphabet when the input word is over a bounded language, in the noncommutative case. In doing so, we introduce a new technique based upon the Turakainen construction of a PFA from a Weighted Finite Automata which can be used to generate PFA of lower dimensions and of subexponential ambiguity. We also study freeness/injectivity problems for polynomially ambiguous PFA and study the border of decidability and tractability for various cases., Comment: Amended an issue regarding commutativity of matrices in Theorem 1 to show why the matrices commute

Public engagement in HEP

The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg–Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link. Hyperspectral anomaly detection is a key technique of unsupervised target detection. In the hyperspectral anomaly detection based on spectral dimensional transformation, the feature projection makes it easy to distinguish the ground objects which are not distinguishable in the original feature space. Although the means of spectral dimensional transformation can improve the distinguishable between diverse categories, it cannot highlight the anomalous targets. To be able to highlight anomalous targets while improving the diversity between different ground objects, an unsupervised network model of Redundant Difference-Value Network (RDVN) is proposed and applied to hyperspectral anomaly detection. RDVN is composed of multiple single-layer neural networks with the same structure and hyper-parameters. A group of training samples is used as the input of the networks, and the difference between the activation values of any networks and benchmark network is used as the error for Back-propagation. After the training is completed, the difference-value between the activation values of the two networks is used as a distinguishing feature (DF). Finally, DF is used as the input of the anomaly detector to obtain the detection results. Experimental results demonstrate that the proposed algorithm can achieve higher detection accuracy. DF not only highlights the anomalous target to increase the true positive rate but also increases the discriminability between different categories, thereby reducing the false positive rate.

The multifractal behaviors in financial markets results from temporal correlations as well as broad distribution. To evaluate the intrinsic multifractality caused by temporal correlations, surrogate approach is employed under the rank order remapping technique and sign randomization. In contrast to raw multifractality, it is found that intrinsic multifractality is more stable across many years. In this work, we utilize ANFIS model for estimating the intrinsic multifractality in financial returns. Furthermore, the intrinsic multifractality of serial major instruments are highly correlated, which can be served as an index of global market.

We construct complex a-priori bounds for certain infinitely renormalizable Lorenz maps. As a corollary, we show that renormalization is a real-analytic operator on the corresponding space of Lorenz maps.

Robot deburring is an effective method for improving the surface quality of the high‐voltage copper contact. The first step of robot deburring is to acquire the burr images. We propose a new burr mathematical model and build a real burr image dataset for burr image denoising. In order to improve burr image denoising effects of the high‐voltage copper contact, this study proposes an online burr image denoising algorithm, that is, block cosparsity overcomplete learning transform algorithm (BCOLTA). The penalty term and the condition number are affected by the burr parameter. The clustering and transform alternate minimisation algorithms are adopted to achieve lower computational cost and better denoising effect. In addition, BCOLTA also has a good adaptibility to inherent noise images, especially in Gaussian noise. Compared with other traditional and deep learning algorithms by no reference and full reference image quality assessment methods, BCOLTA has state‐of‐the‐art denoising effects and computational complexity on dealing with burr images. This research will play an important role in the intelligent manufacturing field.

We propose an example for which the multifractal Hausdorff and packing measures are mutually singular.

Measurements and methods of mitigation for small-angle neutron scattering instrument background caused by extraneous scattering from surfaces are presented., Measurements, calculations and design ideas to mitigate background caused by extraneous scattering in small-angle neutron scattering (SANS) instruments are presented. Scattering includes processes such as incoherent scattering, inelastic scattering and Bragg diffraction. Three primary sources of this type of background are investigated: the beam stop located in front of the detector, the inside lining of the detector vessel and the environment surrounding the sample. SANS measurements were made where materials with different albedos were placed in all three locations. Additional measurements of the angle-dependent scattering over the angular range of 0.7π–0.95π rad were completed on 16 different shielding materials at five wavelengths. The data were extrapolated to cover scattering angles from π/2 to π rad in order to estimate the materials’ albedos. Modifications to existing SANS instruments and sample environments to mitigate extraneous scattering from surfaces are discussed.

The Levenberg-Marquardt method is one of the most important methods for solving systems of nonlinear equations and nonlinear least-squares problems. It enjoys a quadratic convergence rate under the local error bound condition. Recently, to solve nonzero-residue nonlinear least-squares problem, Behling et al. propose a modified Levenberg-Marquardt method with at least superlinearly convergence under a new error bound condtion [3]. To extend their results for systems of nonlinear equations, by choosing the LM parameters adaptively, we propose an efficient variant of the Levenberg-Marquardt method and prove its quadratic convergence under the new error bound condition. We also investigate its global convergence by using the Wolfe line search. The effectiveness of the new method is validated by some numerical experiments.

In this article, we review fractal calculus ( $ F^{\alpha } $ -calculus) and define generalized Tsallis entropy on the fractal sets which is called fractal Tsallis entropy. We define q-fractal calculus to obtain q-Gaussian or generalized stable and Lévy distribution on fractal sets. The conditions for the nonlinear coupling of the statistical states are given. The relationship of fractal dimension which is defined in $ F^{\alpha } $ -calculus with q-parameter of the Tsallis entropy of the Hadron system is proposed.

In this paper, the circulatory integral and Routh?s equations of Lagrange systems are established with Riemann-Liouville fractional derivatives, and the circulatory integral of Lagrange systems is obtained by making use of the relationship between Riemann-Liouville fractional integrals and fractional derivatives. Thereafter, the Routh?s equations of Lagrange systems are given based on the fractional circulatory integral. Two examples are presented to illustrate the application of the results.

An efficient method for Electrocardiogram (ECG) signal denoising based on synchronous detection and Hilbert transform techniques is presented. The goal of the method is to decompose a noisy ECG signal into two components classified according to their energy: (1) component with high energy representing the dominant component which is the clean ECG signal, and (2) component with low energy representing the sub-dominant component which is the contaminant noise. The investigated approach is validated through out some experimentations on MIT-BIH ECG database. Experimental results show that random noises can be effectively suppressed from ECG signals.

Cumulative Tsallis entropy and its residual have developed by many others, and they proposed alternative (second) forms of them. In this article, we illustrate the differences and the relation between the proposed model and its alternative form. Furthermore, we study some properties and features for the concept of cumulative Tsallis entropy and its residual. In addition, we propose a dynamic past form of cumulative Tsallis entropy and obtain some of its properties.

Slides for talk 'PDFs@LHC' to be given at SM@LHC 2022

A multiscaling equation in the Fourier domain accommodates a trigonometric matrix polynomial. This trigonometric matrix polynomial is known as the symbol function. The existence and properties of a multiscaling function, which is the solution of a multiscaling equation, depend on the symbol function. It is possible to construct symbol functions corresponding to compactly supported and symmetric multiscaling functions from standard pairs. A standard pair carries the spectral information about the symbol function. In this paper, we briefly explain the construction of compactly supported and symmetric multiscaling functions and the corresponding mulitwavelets using standard pairs. We derive the necessary as well as sufficient condition, on the eigenspace of the square matrix in the standard pair, for the existence of a symbol function corresponding to a multiscaling equation with a compactly supported solution. We create a pseudo bi-orthogonal pair of symmetric and compactly supported multiscaling functions and the corresponding multiwavelets using standard pairs.

Based on the well-known Detrended Fluctuation Analysis (DFA) for time series, in this work we describe a DFA for continuous real variable functions. Under certain conditions, DFA accurately predicts the long-term auto-correlation of the time series, depending on the value of certain scaling parameter. We show that for continuous functions, the proposed continuous DFA also exhibits fractal properties and approximates a power law with scaling exponent one., 9 pages

Proposal to develop magnetrons as RF sources for HEP, contribution to Snowmass 2021

In this paper for the first time, a numerical code based on Levenberg–Marquardt method is presented for solving inverse heat transfer problem of axisymmetric stagnation flow impinging on a cylinder...

Electrocardiogram (ECG) is the graphical representation of electrical activity of the heart and is used to detect certain structural and functional heart conditions. Segmenting ECG waveforms and annotating constituent components is required for analysis of ECG and to arrive at a diagnosis. This paper proposes a Convolutional Long Short-Term Memory (ConvLSTM) neural network to segment the ECG waves. It consists of a convolutional layer followed by a Bidirectional LSTM architecture. The segmentation is achieved by adding additional features such as derivative of the ECG wave as well as the smoothened ECG wave and the model outperforms traditional Markov models.

The BFGS method is one of the most efficient quasi-Newton methods for solving small- and medium-size unconstrained optimization problems. For the sake of exploring its more interesting properties, a modified two-parameter scaled BFGS method is stated in this paper. The intention of the modified scaled BFGS method is to improve the eigenvalues structure of the BFGS update. In this method, the first two terms and the last term of the standard BFGS update formula are scaled with two different positive parameters, and the new value of yk is given. Meanwhile, Yuan-Wei-Lu line search is also proposed. Under the mentioned line search, the modified two-parameter scaled BFGS method is globally convergent for nonconvex functions. The extensive numerical experiments show that this form of the scaled BFGS method outperforms the standard BFGS method or some similar scaled methods.

In recorded Electrocardiogram (ECG) signal, clinical information is masked by several noises and distortion resulting in low signal-to-noise-ratio (SNR). In this situation, an efficient pre-process...

In this paper, a new class for generating lifetime distributions is introduced, called generalized Burr XII (GBXII) family of distributions. Numerous properties of the GBXII distribution are discus...

The Hurst exponent H is used to determine the measure of predictability of a time series. The value between 0 and 1 with 0.5 representative of a random or uncorrelated series, H > 0.5 and H 0.5 reflect a data set which is persistent and anti-persistent respectively. The fractal dimension can be given from the Hurst exponent. The fractal dimension is a factor of the complexity of which the system is being repeated at various scales. If the fractal dimension does not change with scale it is deemed monofractal if not, multifractal. The Hurst exponents were determined in this study using the Rescale Range Analysis (R/S Analysis) and Multifractal Detrended Fluctuation Analysis (MF-DFA) for monofractal and multifractal investigations respectively. These methods were applied to daily 10 min wind speed time series data for the year 2009 from three locations within Missouri: Columbia, Neosho and Blanchard for three tall tower stations. The results obtained from the monofractal analysis showed minor variations in the Hurst exponents for the three stations and heights for all the months in 2009. These values ranged from 0.7 to 0.9 and its corresponding fractal dimension was ranged between 1.3 and 1.1. The results for the MF-DFA showed that the wind speed time series were multifractal in nature as the Hurst exponents were functions of the scaling parameters. Also, the plots of the Renyi Exponent were non-linear for the stations and the various channels; this is representative of multifractal signals. The fractal dimensions of the time series using multifractal analysis were determined to be greater than these values determined using monofractal analysis. However, there were no indications of consistent increases in the complexity of the systems’ multifractality with increasing heights for the various stations’ tall towers.

A functional version of the Kato one-parametric regularisation for the construction of a dynamical semigroup generator of a relative bound one perturbation is introduced. It does not require that the minus generator of the unperturbed semigroup is a positivity preserving operator. The regularisation is illustrated by an example of a boson-number cut-off regularisation.

We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe $\mathrm{C}^*$-algebras which depend only on their Banach algebra structure.

Hyperspectral image sensors capture surface reflectance over a range of wavelengths. The fine spectral information is recorded in terms of hundreds of bands. Hyperspectral image classification has ...

In the present work, we are concerned with some multifractal dimensions estimations of vector-valued measures in the framework of the so-called mixed multifractal analysis. We precisely consider some Borel probability measures that are no longer Gibbs and introduce some mixed multifractal generalizations of Hausdorff and packing dimensions of measures in a framework of relative mixed multifractal analysis. As an application, we are interested in the '-unidimensionality of those measures and to the calculus of its mixed multifractal Hausdorff and packing dimensions. In particular, we give a necessary and sufficient condition for the existence of the '-mixed multifractal Hausdorff and packing dimensions of a Borel probability measure. Finally, concrete examples satisfying the above property are developed.

A symmetric Lorenz map is obtained by ‘flipping’ one of the two branches of a symmetric unimodal map. We use this to derive a Sharkovsky-like theorem for symmetric Lorenz maps, and also to find cas...

In this study, a wavelet denoising method is first used to eliminate the influence of noise. Then, an overlapping smooth window technique is introduced into the asymmetric multifractal detrended cross-correlation analysis method, which was combined with the multiscale multifractal analysis method, resulting in the proposed asymmetric multiscale multifractal detrended cross-correlation analysis method. This method not only remedies the pseudo-fluctuation defect of the traditional method, but also explores the asymmetric multifractal cross-correlation between China’s rebar futures and spot markets at different scales. The results show the existence of an asymmetric multifractal cross-correlation between rebar futures and spot markets with upward and downward trends at different scales. This cross-correlation is highly complex at the small-scale, and more pronounced when the futures market is in an uptrend.

Previous research on analytical models used to predict burr size in machining operations has usually been limited to simple geometries and mechanics models. This paper investigates the material side-flow and Poisson burr formation mechanism. An improved analytical model is developed to describe the surface (subsurface) plastic deformation and Poisson burr morphology. First, the theoretical model is established based on mechanical, geometrical, and material property considerations. Then, machining experiments are performed on high-purity copper, and the proposed model is validated by comparing the predicted burr height and thickness to the experimental measurements. The results show that the developed model provides two new methods to predict burr dimensions with considerable accuracy and that this model is suitable for ductile materials. Additionally, this article offers a basis for minimizing the energy required during deburring, suppressing the formation of burrs, and optimizing the machining parameters.

Recent research in the area of univariate mixture modeling indicated that the finite mixture models based on Burr and Inverse Burr component distributions perform well in the modeling of heavy-tail...

Cloud-to-Ground (CG) lightning intensity time series recorded in northern Benin, during days of monsoon season (summer 2006), has been deeply explored using multifractal framework. The results suggest the existence of strong multifractal characteristics in lightning intensity. However, detrending the data reduces the degree of multifractality. The multifractality arises from both a fat-tailed probability density function and long-range temporal correlations. But, the most dominant multifractality in lightning intensity series depends strongly on the kind of detrending that is retained from the profile during the multifractal detrended fluctuation analysis (MFDFA). These findings have allowed us to understand and characterize the complexity of lightning intensity structure in the network.

Bus bunching resulting from initially small headway irregularities is a widely-known and studied problem. A variety of headway-prediction approaches, as well as corrective strategies, have been developed to identify and correct headway irregularity in real time. Instead of predicting an exact value for future headways, this study explores a probabilistic predictive methodology to forecast whether or not a bus will be bunched during its dwelling at a downstream stop, using a logistic regression model based on GPS records of buses at least k stops upstream to allow for sufficient time to possibly implement control strategies. A case study is conducted on a circular bus route in Kyoto City. Compared to two headway-based prediction approaches using linear regression and support vector machine, the superior performance of the proposed tool in detecting bunching is illustrated by Receiver Operator Characteristic (ROC) analysis. The high reliability in long-term prediction gives adequate time for operators to employ countermeasures. Besides, the proposed method provides operators with tradeoff options. We find that a bunching-averse operator can obtain 95% “sensitivity”, that is the ratio of correctly identified bunching events, at the cost of decreasing “specificity”, which is the ratio of correct non-bunching predictions over all events. This is true even if the prediction horizon is more than 10 stops.

In this paper we prove converse theorems to obtain usual convergence of improper integrals from Cesàro summability.