Subexponential Behavior Research Articles

Decoder Ties Do Not Affect the Error Exponent of the Memoryless Binary Symmetric Channel

The generalized Poor-Verdú error lower bound established by Chang <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (2020) for multihypothesis testing is studied in the classical channel coding context. It is proved that for any sequence of block codes sent over the memoryless binary symmetric channel (BSC), the minimum probability of error (under maximum likelihood decoding) has a relative deviation from the generalized bound that grows at most linearly in blocklength. This result directly implies that for arbitrary codes used over the BSC, decoder ties can only affect the subexponential behavior of the minimum probability of error.

Power law behaviour in the saturation regime of fatality curves of the COVID-19 pandemic

We apply a versatile growth model, whose growth rate is given by a generalised beta distribution, to describe the complex behaviour of the fatality curves of the COVID-19 disease for several countries in Europe and North America. We show that the COVID-19 epidemic curves not only may present a subexponential early growth but can also exhibit a similar subexponential (power-law) behaviour in the saturation regime. We argue that the power-law exponent of the latter regime, which measures how quickly the curve approaches the plateau, is directly related to control measures, in the sense that the less strict the control, the smaller the exponent and hence the slower the diseases progresses to its end. The power-law saturation uncovered here is an important result, because it signals to policymakers and health authorities that it is important to keep control measures for as long as possible, so as to avoid a slow, power-law ending of the disease. The slower the approach to the plateau, the longer the virus lingers on in the population, and the greater not only the final death toll but also the risk of a resurgence of infections.

Sojourn time in a Processor Sharing queue with batch arrivals

For the Processor Sharing queue with batch job arrivals, the sojourn time W of a single job is investigated. This queuing model is motivated by the evaluation of Cloud Computing or Virtualized Network systems where the treatment of micro-services within requests determines the global system performance. We first show that the distribution of sojourn time W can be obtained from an infinite linear differential system; the structure of this system, however, makes the explicit derivation of this distribution generally difficult. When further assuming that the batch size has a geometric distribution with some given parameter , this differential system can be analyzed via a single generating function which is shown to verify a second-order partial differential equation involving a boundary term at point . Solving this partial differential equation for with required analyticity properties determines the one-sided Laplace transform for given u. Writing in terms of a multivariate hypergeometric function enables one to extend its analyticity domain to a cut-plane , for negative constants and . By means of a Laplace inversion of for a suitable value of u, the complementary distribution function is then given an explicit integral representation. This enables us to show that the tail of this distribution has an exponential decay with rate , together with a sub-exponential factor. A convergence in distribution is also asserted for W in heavy load condition, the limit distribution exhibiting a sub-exponential behavior itself. Using our exact results for the sojourn time of a single job, we finally discuss an approximation for the distribution of the sojourn time of an entire batch when assuming that the batch size is not too large.

Optical fluxes in coupled PT -symmetric photonic structures

In this work we first examine transverse and longitudinal fluxes in a $\cal PT$-symmetric photonic dimer using a coupled-mode theory. Several surprising understandings are obtained from this perspective: The longitudinal flux shows that the $\cal PT$ transition in a dimer can be regarded as a classical effect, despite its analogy to $\cal PT$-symmetric quantum mechanics. The longitudinal flux also indicates that the so-called giant amplification in the $\cal PT$-symmetric phase is a sub-exponential behavior and does not outperform a single gain waveguide. The transverse flux, on the other hand, reveals that the apparent power oscillations between the gain and loss waveguides in the $\cal PT$-symmetric phase can be deceiving in certain cases, where the transverse power transfer is in fact unidirectional. We also show that this power transfer cannot be arbitrarily fast even when the exceptional point is approached. Finally, we go beyond the coupled-mode theory by using the paraxial wave equation and also extend our discussions to a $\cal PT$ diamond and a one-dimensional periodic lattice.

Irreversibility and complex network behavior of stream flow fluctuations

Exploiting the duality between time series and networks, directed horizontal visibility graphs (DHVGs) are used to perform an unprecedented analysis of the dynamics of stream flow fluctuations with focus on time irreversibility and long range dependence. The analysis relies on a large quality-controlled data set consisting of 699 daily time series recorded in the continental United States (CONUS) that are not affected by human activity and primarily reflects meteorological conditions. DHVGs allow a clear visualization and quantification of time irreversibility of flow dynamics, which can be interpreted as a signature of nonlinearity, and long range dependence resulting from the interaction of atmospheric, surface and underground processes acting at multiple spatio-temporal scales. Irreversibility is explored by mapping the time series into ingoing, outgoing, and undirected graphs and comparing the corresponding degree distributions. Using surrogate data preserving up to the second order linear temporal dependence properties of the observed series, DHVGs highlight the additional complexity introduced by nonlinearity into flow fluctuation dynamics. We show that the degree distributions do not decay exponentially as expected, but tend to follow a subexponential behavior, even though sampling uncertainty does not allow a clear distinction between apparent or true power law decay. These results confirm that the complexity of stream flow dynamics goes beyond a linear representation involving for instance the combination of linear processes with short and long range dependence, and requires modeling strategies accounting for temporal asymmetry and nonlinearity.

Identifiability of models for time-resolved fluorescence with underlying distributions of rate constants

The deterministic identifiability analysis of photophysical models for the kinetics of excited-state processes, assuming errorless time-resolved fluorescence data, can verify whether the model parameters can be determined unambiguously. In this work, we have investigated the identifiability of several uncommon models for time-resolved fluorescence with underlying distributions of rate constants which lead to non-exponential decays. The mathematical functions used here for the description of non-exponential fluorescence decays are the stretched exponential or Kohlrausch function, the Becquerel function, the Förster type energy transfer function, decay functions associated with exponential, Gaussian and uniform distributions of rate constants, a decay function with extreme sub-exponential behavior, the Mittag-Leffler function and Heaviside's function. It is shown that all the models are uniquely identifiable, which means that for each specific model there exists a single parameter set that describes its associated fluorescence δ-response function.

Effective Laguerre Asymptotics

It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs $(a,z)$ one has the large-$n$ asymptotic behavior $L_n^{(-a)}(-z) \sim C(a,z) n^{-a/2-1/4} e^{2\sqrt{nz}}$. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an “exp-arc” method giving a natural bridge between converging series and effective asymptotics.

Local tail bounds for functions of independent random variables

It is shown that functions defined on (0, 1, . . r - 1} n satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger local sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand's [Ann. Probab. 22 (1994) 1576-1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on (0, 1, ..., r - 1) n for r ≥ 2.

Multivariate subexponential distributions and random sums of random vectors

Let F(x) denote a distribution function in Rd and let F*n(x) denote the nth convolution power of F(x). In this paper we discuss the asymptotic behaviour of 1 - F*n(x) as x tends to ∞ in a certain prescribed way. It turns out that in many cases 1 - F*n(x) ∼ n(1 - F(x)). To obtain results of this type, we introduce and use a form of subexponential behaviour, thereby extending the notion of multivariate regular variation. We also discuss subordination, in which situation the index n is replaced by a random index N.

Large-Time Behavior of Solutions for the Boltzmann Equation with Hard potentials

We study the quantitative behavior of the solutions of the one-dimensional Boltzmann equation for hard potential models with Grad’s angular cutoff. Our results generalize those of [5] for hard sphere models. The main difference between hard sphere and hard potential models is in the exponent of the collision frequency \(\nu(\xi)\approx (1+|\xi|)^\gamma\). This gives rise to new wave phenomena, particularly the sub-exponential behavior of waves. Unlike the hard sphere models, the spectrum of the Fourier operator \(-i\xi^1\eta+L\) is non-analytic in η for hard potential models. Thus the complex analytic methods for inverting the Fourier transform are not applicable and we need to use the real analytic method in the estimates of the fluidlike waves. We devise a new weighted energy function to account for the sub-exponential behavior of waves.

Dynamics of quark-gluon-plasma instabilities in discretized hard-loop approximation

Non-abelian plasma instabilities have been proposed as a possible explanation for fast isotropization of the quark-gluon plasma produced in relativistic heavy-ion colli- sions. We study the real-time evolution of these instabilities in non-abelian plasmas with a momentum-space anisotropy using a hard-loop effective theory that is discretized in the velocities of hard particles. We extend our previous results on the evolution of the most un- stable modes, which are constant in directions transverse to the direction of anisotropy, from gauge group SU(2) to SU(3). We also present first full 3+1-dimensional simulation results based on velocity-discretized hard loops. In contrast to the effectively 1+1-dimensional transversely constant modes we find subexponential behavior at late times.

Hard-Loop Dynamics of Non-Abelian Plasma Instabilities

Non-Abelian plasma instabilities may be responsible for the fast apparent quark-gluon thermalization in relativistic heavy-ion collisions if their exponential growth is not hindered by nonlinearities. We study numerically the real-time evolution of instabilities in an anisotropic non-Abelian plasma with an SU(2) gauge group in the hard-loop approximation. We find exponential growth of non-Abelian plasma instabilities both in the linear and in the strongly nonlinear regimes, with only a brief phase of subexponential behavior in between.

Second-Order Subexponential Behavior of Subordinated Sequences

Suppose that $$\left\{ {a\left( n \right),n{\text{ }} \in {\text{ }}N_0 } \right\}$$ , $$\left\{ {b\left( n \right),n{\text{ }} \in {\text{ }}N_0 } \right\}$$ , and $$\left\{ {p\left( n \right),n{\text{ }} \in {\text{ }}N_0 } \right\}$$ are three discrete probability distributions related by the equation (E): $$b\left( n \right) = \sum\nolimits_{k = 0}^\infty {p\left( k \right)} a^{*k} \left( n \right)$$ , where $$\left\{ {a^{*k} \left( n \right),n \in N_0 } \right\}$$ denotes the k-fold convolution of $$\left\{ {a\left( n \right),n \in {\text{ }}N_{\text{0}} } \right\}$$ In this paper, we investigate the relation between the asymptotic behaviors of $$a$$ and $$b$$ . It turns out that, for wide classes of sequences $$a$$ and $$p$$ , relation (E) implies that $$b\left( n \right)/a\left( n \right) \to {\text{E}}N$$ , where $${\text{E}}N$$ is the mean of $$p$$ . The main object of this paper is to discuss the rate of convergence in this result. In our main results, we obtain O-estimates and exact asymptotic estimates for the difference $$b\left( n \right) - {\text{E}}N{\text{ }}a\left( n \right)$$ .

Value-at-Risk-efficient portfolios for a class of super- and sub-exponentially decaying assets return distributions

Using a family of modified Weibull distributions encompassing both sub-exponentials and super-exponentials to parametrize the marginal distributions of asset returns and their multivariate generalizations with Gaussian copulas, we offer exact formulae for the tails of the distribution P(S) of returns S of a portfolio of arbitrary composition of these assets. We find that the tail of P(S) is also asymptotically a modified Weibull distribution with a characteristic scale χ function of the asset weights with different functional forms depending on the super- or sub-exponential behaviour of the marginals and on the strength of the dependence between the assets. We then treat in detail the problem of risk minimization using the Value-at-Risk and expected shortfall which are shown to be (asymptotically) equivalent in this framework.

Two-state model for sub-exponential fluorescence

We outline a theory of fluorescence in systems where the decay of the fluorescence takes place more slowly than exponential, a behavior we refer to as `sub-exponential'. We develop a two-state model for the fluorescence where the two states are the trapped state and the fluorescing level. The sub-exponential behavior is associated with a distribution of detrapping rates. Asymptotic inverse power-law behavior of the fluorescence, i.e., t − δ , arises from a distribution that varies near the origin as the δ−2 power of the detrapping rate. Stretched exponential behavior, where the asymptotic time dependence of the fluorescence is dominated by the factor exp[−(t/τ) β], 0<β<1 , is associated with a steep but continuous decrease in the density of transition rates. A one-parameter thermal activation model is introduced in which the distribution of detrapping rates arises from a distribution of activation energies. Power-law behavior of the fluorescence is associated with an exponential decrease in the activation energy distribution at high energies, while stretched exponential dynamics reflects an even faster decrease. The thermal activation model is used to make predictions about the temperature dependence of the exponents characterizing the fluorescence decay. Possible experimental tests of the theory are discussed.

Network multiplexer with many truncated heavy-tailsources

An exactly solvable discrete-time model with multiplexed truncated heavy-tail sources is used to investigate queueing behaviour in a network multiplexer as a function of both source number N and the subexponential behaviour of individual sources. In particular, the model illustrates the large N asymptotic description for the waiting time probabilities in such multiplexers.

Periodic orbits in a simple ray-splitting system.

We study ray dynamics in a square billiard that allows mode conversion and is parametrized by \ensuremath{\kappa}, the ratio of the velocities of the two modes. At \ensuremath{\kappa}\ensuremath{\rightarrow}${1}^{+}$, conversion occurs at every reflection and periodic orbits proliferate exponentially. As \ensuremath{\kappa} increases beyond \ensuremath{\surd}2, the collection of daughter rays explore only three momentum directions and mode conversion is progressively inhibited. We provide an algorithm for determining periodic orbits when \ensuremath{\kappa}\ensuremath{\gtrsim}\ensuremath{\surd}2 and show numerically that exponential proliferation persists around \ensuremath{\kappa}\ensuremath{\simeq}\ensuremath{\surd}2 but as \ensuremath{\kappa} increases, a crossover to subexponential behavior occurs for short periods. We discuss these results in the light of conservation laws. \textcopyright{} 1996 The American Physical Society.

A semi-empirical model for cell kill kinetics during hyperthermia

A three-parameter mathematical model is developed which describes the available data on cell-kill kinetics during hyperthermia. The sub-exponential behaviour of the kinetics suggests that the cell-kill is not a one-step process.