Infinite Mean Research Articles

Explosion Rates for Continuous-State Branching Processes in a Lévy Environment

Here, we study the long-term behaviour of the non-explosion probability for continuous-state branching processes in a Lévy environment when the branching mechanism is given by the negative of the Laplace exponent of a subordinator. In order to do so, we study the law of this family of processes in the infinite mean case and provide necessary and sufficient conditions for the process to be conservative, i.e. that the process does not explode in finite time a.s. In addition, we establish precise rates for the non-explosion probabilities in the subcritical and critical regimes, first found by Palau et al. (ALEA Lat Am J Probab Math Stat 13(2):1235–1258, 2016) in the case when the branching mechanism is given by the negative of the Laplace exponent of a stable subordinator.

An Unexpected Stochastic Dominance: Pareto Distributions, Dependence, and Diversification

Diversification is generally regarded as an efficient tool to reduce portfolio risks. In “An unexpected stochastic dominance: Pareto distributions, dependence, and diversification,” Chen, Embrechts, and Wang showed that the weighted average of independent and identically distributed (i.i.d.) Pareto random variables with infinite mean is larger than one such random variable in the sense of first-order stochastic dominance, and thus diversification is, surprisingly, worse than no diversification. The relation implies superadditivity of value-at-risk, a regulatory risk measure used in the finance and insurance sectors. The obtained relation also holds under some form of negative dependence.

Prediction and estimation of random variables with infinite mean or variance

In this article, we propose an optimal predictor of a random variable that has either an infinite mean or an infinite variance. The method consists of transforming the random variable such that the transformed variable has a finite mean and finite variance. The proposed predictor is a generalized arithmetic mean which is similar to the notion of certainty price in utility theory. Typically, the transformation consists of a parametric family of bijections, in which case the parameter might be chosen to minimize the prediction error in the transformed coordinates. The statistical properties of the estimator of the proposed predictor are studied, and confidence intervals are provided. The performance of the procedure is illustrated using simulated and real data.

Tight-binding model subject to conditional resets at random times.

We investigate the dynamics of a quantum system subjected to a time-dependent and conditional resetting protocol. Namely, we ask what happens when the unitary evolution of the system is repeatedly interrupted at random time instants with an instantaneous reset to a specified set of reset configurations taking place with a probability that depends on the current configuration of the system at the instant of reset? Analyzing the protocol in the framework of the so-called tight-binding model describing the hopping of a quantum particle to nearest-neighbor sites in a one-dimensional open lattice, we obtain analytical results for the probability of finding the particle on the different sites of the lattice. We explore a variety of dynamical scenarios, including the one in which the resetting time intervals are sampled from an exponential as well as from a power-law distribution, and a setup that includes a Floquet-type Hamiltonian involving an external periodic forcing. Under exponential resetting, and in both the presence and absence of the external forcing, the system relaxes to a stationary state characterized by localization of the particle around the reset sites. The choice of the reset sites plays a defining role in dictating the relative probability of finding the particle at the reset sites as well as in determining the overall spatial profile of the site-occupation probability. Indeed, a simple choice can be engineered that makes the spatial profile highly asymmetric even when the bare dynamics does not involve the effect of any bias. Furthermore, analyzing the case of power-law resetting serves to demonstrate that the attainment of the stationary state in this quantum problem is not always evident and depends crucially on whether the distribution of reset time intervals has a finite or an infinite mean.

First-Passage Times for Random Partial Sums: Yadrenko’s Model for e and Beyond

M. I. Yadrenko discovered that the expectation of the minimum number N 1 of independent and identically distributed uniform random variables on (0, 1) that have to be added to exceed 1 is e. For any threshold a > 0, K. G. Russell found the distribution, mean, and variance of the minimum number Na of independent and identically distributed uniform random summands required to exceed a. Here we calculate the distribution and moments of Na when the summands obey the negative exponential and Lévy distributions. The Lévy distribution has infinite mean. We compare these results with the results of Yadrenko and Russell for uniform random summands to see how the expected first-passage time E ( N a ) , a > 0 , and other moments of Na depend on the distribution of the summand.

Quantum evolution with random phase scattering

We consider the quantum evolution of a fermion–hole pair in a d-dimensional gas of non-interacting fermions in the presence of random phase scattering. This system is mapped onto an effective Ising model, which enables us to show rigorously that the probability of recombining the fermion and the hole decays exponentially with the distance of their initial spatial separation. In contrast, without random phase scattering the recombination probability decays like a power law, which is reflected by an infinite mean square displacement. The effective Ising model is studied within a saddle point approximation and yields a finite mean square displacement that depends on the evolution time and on the spectral properties of the deterministic part of the evolution operator.

On the Diversification Effect in Solvency II for Extremely Dependent Risks

In this article, we investigate the validity of diversification effect under extreme-value copulas, when the marginal risks of the portfolio are identically distributed, which can be any one having a finite endpoint or belonging to one of the three maximum domains of attraction. We show that Value-at-Risk (V@R) under extreme-value copulas is asymptotically subadditive for marginal risks with finite mean, while it is asymptotically superadditive for risks with infinite mean. Our major findings enrich and supplement the context of the second fundamental theorem of quantitative risk management in existing literature, which states that V@R of a portfolio is typically non-subadditive for non-elliptically distributed risk vectors. In particular, we now pin down when the V@R is super or subadditive depending on the heaviness of the marginal tail risk. According to our results, one can take advantages from the diversification effect for marginal risks with finite mean. This justifies the standard formula for calculating the capital requirement under Solvency II in which imperfect correlations are used for various risk exposures.

The asymptotic distribution of a truncated sample mean for the extremely heavy-tailed distributions

This article deals with the asymptotic distribution for the extremely heavy-tailed distributions with infinite mean or variance by using a truncated sample mean. We obtain three necessary and sufficient conditions under which the asymptotic distribution of the truncated test statistics converges to normal, neither normal nor stable or converges to − ∞ or the combination of stable distributions, respectively. The numerical simulation illustrates an application of the theoretical results above in the hypothesis testing.

From Semi-Markov Random Evolutions to Scattering Transport and Superdiffusion

We here study random evolutions on Banach spaces, driven by a class of semi-Markov processes. The expectation (in the sense of Bochner) of such evolutions is shown to solve some abstract Cauchy problems. Further, the abstract telegraph (damped wave) equation is generalized to the case of semi-Markov perturbations. A special attention is devoted to semi-Markov models of scattering transport processes which can be represented through these evolutions. In particular, we consider random flights with infinite mean flight times which turn out to be governed by a semi-Markov generalization of a linear Boltzmann equation; their scaling limit is proved to converge to superdiffusive transport processes.

Branching random walk with infinite progeny mean: A tale of two tails

We study the extremes of branching random walks under the assumption that the underlying Galton–Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the nth generation when the tail of the displacement behaves like exp(−K(x)), where either K is a regularly varying function of index r>0, or K has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when r>1.

Catching the Killer: Dynamic Disorder Design Rules for Small‐Molecule Organic Semiconductors

AbstractSmall‐molecule organic semiconductors are promising materials for applications ranging from solar cells to medical sensors. Their nearly infinite design space means that, in theory, it is possible to tailor the material to the exact specifications of a particular application. In reality; however, design rules to improve mobility (μ) remain elusive because of the complex calculations required to understand its limiters. Transient localization theory posits that charge carriers are slowed down by the collective phonon motions, called dynamic disorder, which localize charge carriers temporarily. Traditionally, a mode‐by‐mode analysis is performed to try and identify “killer” modes. Herein, the flaws are demonstrated with this analysis and present a streamlined simulation workflow that simplifies simulation of dynamic disorder and enables engineering of new molecular structures based on phonon dynamics. This workflow is combined with a novel visualization technique that enables per‐atom insights into how μ is limited. This workflow is applied and analyzed to a series of ‐acenes and a series of BTBT‐based molecules. Then design rules are identified in each series that identify possible ways to improve the μ beyond current experimental limits using phonon engineering.

Limit theorems for negatively superadditive-dependent random variables with infinite or finite means

<abstract><p>The author studies the laws of large numbers for weighted sums of negatively superadditive-dependent random variables. The obtained results in this paper extend and improve the corresponding theorems of Yang et al. [Commun. Stat. Theor. M., 48 (2019), 3044-3054]. Moreover, the author obtains a new theorem of mean convergence for weighted sums of negatively superadditive-dependent random variables, which was not considered in Yang et al. (2019).</p></abstract>

Indeterminacy of Meaning in William Blake’s “The Sick Rose”

This study attempts to make deconstructive reading of William Blake’s poem “The Sick Rose”. It also shows how the literary text can be interpreted from multiple perspectives deriving infinite meanings from the same text. The major motive of this research is to demonstrate potentiality of every text to be creatively misread generating various possible meanings. The paper also projects the fact that deconstruction is not destruction, rather it is reconstruction. It displays how the use of language in the text creates contradiction, un-decidability and multiplicity opening up possibility for new meanings. To justify the argument, deconstructive ideas of the philosophers Jacques Derrida and Roland Barthes have been used. The four steps of deconstructive reading- a. Reading the Text b. Finding the Binaries c. Hierarchizing the Binaries and d. Creative Misreading-have been followed for the study of the poem “The Sick Rose”. This research can be a contributing source for promotion of deconstructive reading of anything prevailing in the society and thereby decentering the center and re-centering the margin opening up possibilities for new perspectives, meanings and truths.

The dynamics of Pareto distributed wealth in a small open economy

We study a small open economy with labor, capital accumulation, random death, taxation and a government budget balanced in the long run. We offer methods that provide ordinary differential equations for means and analytical expressions for densities. The latter is achieved by solving stochastic differential equations analytically and deriving the density from this solution. Starting from any distribution, the aggregate distribution converges, both on a transition path towards a steady state and on a transition path towards balanced growth, to a Pareto-distribution. We provide an intuitive economic interpretation for a stationary long-run density with an infinite mean in an economy on a balanced growth path. We also show how government tax policy can lead to non-monotonic links between the equilibrium growth rate of the economy and risk aversion of households.

Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the ‘aging effect’ as a hallmark of non-Markovianity where the discrete-time version of the ‘aging renewal process’ comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a t2-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (‘narrow’ with finite mean and variance) the walk exhibits normal diffusion and for ‘broad’ waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

Relevance between faith to God and health dimensions in the Holy Quran: A content analysis

Introduction: Islam has the largest number of followers in the world after Christianity. Islamic religions and sects differ in their beliefs, ceremonies, rites, and leaders, but all have in common the faith to God and the Quran. God places great emphasis on the health of his servants in the holy Quran. This study aimed to analysis the relevance between faith to God and the dimensions of health in the holy Quran. Methods: This qualitative study was conducted by using content analysis of the original text of the holy Quran in eight steps. Meanings of the verses of the holy Quran were used in five Persian and five English translations. Content analysis were done through conceptual coding, categorization of the codes, and achievement of the main themes. Results: 800 verses of the holy Quran showed appurtenances of the faith including faith to God, God's verses, God's Books, God's angels, the unseen word, and the Last day are related to five dimensions of the heath including physical, mental, social, spiritual, and moral. Conclusion: This study offers new messages about the infinite meanings and concepts of the holy Quran verses by a new outlook at the relevance between faith to God and health.

Limit theorems for dependent random variables with infinite means

We investigate necessary and sufficient conditions for the convergence in probability of weighted averages of random variables with infinite means. The obtained results are valid for a wide range of dependence structures, they extend and improve the corresponding theorems of Adler (2012) and Nakata (2016), established in the independent case.

Hawkes processes with infinite mean intensity.

The stability condition for Hawkes processes and their nonlinear extensions usually relies on the condition that the mean intensity is a finite constant. It follows that the total endogeneity ratio needs to be strictly smaller than unity. In the present Letter we argue that it is possible to have a total endogeneity ratio greater than unity without rendering the process unstable. In particular, we show that, provided the endogeneity ratio of the linear Hawkes component is smaller than unity, quadratic Hawkes processes are always stationary, although with infinite mean intensity when the total endogenity ratio exceeds 1. This results from a subtle compensation between the inhibiting realizations (mean reversion) and their exciting counterparts (trends).

The Discrete Inverse Burr Distribution with Characterizations, Properties, Applications, Bayesian and Non-Bayesian Estimations

A new one-parameter heavy tailed discrete distribution with infinite mean is defined and studied. The probability mass function of the new distribution can be "unimodal and right skewed" and its failure rate can be monotonically decreasing. Some of its relevant properties are discussed. Some characterizations based on: (i) the conditional expectation of a certain function of the random variable and (ii) in terms of the reversed hazard function are presented. Different Bayesian and non-Bayesian estimation methods are described and compared using simulations and two real data applications are given. The new model is used to model carious teeth data and counts of cysts in kidneys datasets, and it outperforms many well-known competitive discrete models.

On asymptotic structure of critical Galton-Watson branching processes allowing immigration with infinite variance

We consider the Galton-Watson branching process allowing immigration. We are dealing with the critical case, in which the immigration law has infinite mean and the offspring law have an infinite variance. An explicit-integral form of the generating function of a stationary measure for the process without immigration is found. We study the asymptotic properties of transition probabilities and their convergence to stationary measures in the case of processes with immigration, when the process is ergodic. And also we define a rate of speed of this convergence.