Infinite Mean Research Articles

On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean

In this work, we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions. More precisely, we verify convergence in probability as well as almost sure convergence to a strictly positive and finite constant without assuming any dependence structure or the existence of means. Results of this kind are known as exact weak and exact strong laws.

Number of edges in inhomogeneous random graphs

We study the number of edges in the inhomogeneous random graph when vertex weights have infinite mean and show that the number of edges is O(n log n). Central limit theorems for the number of edges are also established.

Singular PT-symmetry broken point with infinite transmittance and reflectance-a classical analytical demonstration.

To demonstrate the existence of singular parity-time symmetry (PT-symmetry) broken point in optics system, we designed a one-dimensional PT symmetric structure including N unit-cell with loss and gain materials in half. We performed an analytical deduction to obtain the transmittance and reflectance of the structure basing on Maxwell's equations. We found that with the exact structure unit-cell number and the imaginary part of refraction index, the transmittance and reflectance are both close to infinite. Such strict condition is called the singular point in this study. At the singular point position, both the transmission and reflection are direction-independent. Away from the singular point, the transmittance and reflectance become finite. In light of classical wave optics, the single unit and total structure both become the resonance units. The infinite transmittance and reflectance result from the resonance matching of single unit and total structure. In light of quantum theory, the singular point corresponds to the single eigenvalue of electromagnetic scattering matrix. The infinite transmittance and reflectance mean a huge energy transformation from pumping source to light waves. Numerical calculation and software simulation both demonstrate the result.

On the ratio of current age to total life for null recurrent renewal processes

A number of open problems associated with determining the limit distribution of the ratio of current age to total life for a null recurrent renewal process (i.e. where inter-arrival times have infinite mean) are solved. In particular, when the survival function for the inter-arrival times satisfies F̄(t)∼t−αL(t) as t→∞ with L slowly varying and 0≤α≤1, we prove that the limit distribution corresponds to that of U1∕α, where U is uniformly distributed on (0,1), with the limit distribution taken to be degenerate at 0 when α=0. By using direct methods instead of appealing to strong renewal theorems, we are able to prove this result without regard to whether the inter-arrival time distribution is latticed or not, and without extraneous constraints on the renewal function.

Regression to the Tail: Why the Olympics Blow Up

The Olympic Games are the largest, highest-profile, and most expensive megaevent hosted by cities and nations. Average sports-related costs of hosting are $12.0 billion. Non-sports-related costs are typically several times that. Every Olympics since 1960 has run over budget, at an average of 172 percent in real terms, the highest overrun on record for any type of megaproject. The paper tests theoretical statistical distributions against empirical data for the costs of the Games, in order to explain the cost risks faced by host cities and nations. It is documented, for the first time, that cost and cost overrun for the Games follow a power-law distribution. Olympic costs are subject to infinite mean and variance, with dire consequences for predictability and planning. We name this phenomenon regression to the tail: it is only a matter of time until a new extreme event occurs, with an overrun larger than the largest so far, and thus more disruptive and less plannable. The generative mechanism for the Olympic power law is identified as strong convexity prompted by six causal drivers: irreversibility, fixed deadlines, the Blank Check Syndrome, tight coupling, long planning horizons, and an Eternal Beginner Syndrome. The power law explains why the Games are so difficult to plan and manage successfully, and why cities and nations should think twice before bidding to host. Based on the power law, two heuristics are identified for better decision making on hosting. Finally, the paper develops measures for good practice in planning and managing the Games, including how to mitigate the extreme risks of the Olympic power law.

Heavy-tailed random walks, buffered queues and hidden large deviations

It is well-known that large deviations of random walks driven by independent and identically distributed heavy-tailed random variables are governed by the so-called principle of one large jump. We note that further subtleties hold for such random walks in the large deviations scale which we call hidden large deviations. Our results are illustrated using two examples. First, we apply this idea in the context of queueing processes with heavy-tailed service times and study approximations of probabilities of severe congestion times for (buffered) queues. We exhibit our techniques by using limit measures from different large deviation regimes to provide a unified estimate of rare event probabilities in a simulated queue. Furthermore, we use our result to provide probability estimates of rare events governed by more than one jump in case the innovations of a random walk have infinite mean.

Power-law distributions of corporate innovative output: evidence from U.S. patent data

This study aims to examine the existence and the characteristics of power laws in the distribution of corporate innovative output. Using a dataset containing information on 1,102,839 U.S. patent applications by 94,103 U.S. private firms during the period of 1976–2000, we find that corporate innovative output, as measured by either simple or quality-adjusted patent counts, follows power-law distributions that theoretically have infinite variances and, in some cases, infinite means. In addition, we find that corporate innovative output is power-law distributed in all technological fields. We further find that the power-law distribution of corporate innovative output tends to be stronger (i.e., the concentration of corporate innovative output in a few firms is more pronounced) in technological fields with more abundant technological opportunities and for firms with higher technological competence.

Random walks in a strongly sparse random environment

The integer points (sites) of the real line are marked by the positions of a standard random walk with positive integer jumps. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity and assuming additionally that the distribution tail of the jumps is regularly varying at infinity we consider a nearest neighbor random walk on the set of integers having jumps ±1 with probability 1∕2 at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2019) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.

Simulating an infinite mean waiting time

We consider a hybrid method to simulate the return time to the initial state in a critical-case birth--death process. The expected value of this return time is infinite, but its distribution asymptotically follows a power-law. Hence, the simulation approach is to directly simulate the process, unless the simulated time exceeds some threshold and if it does, draw the return time from the tail of the power law.

The laws of large numbers for Pareto-type random variables with infinite means

In this paper, we consider the laws of large numbers for NSD random variables satisfying Pareto-type distributions with infinite means. Based on the Pareto-Zipf distributions, some weak laws of large numbers for weighted sums of NSD random variables are obtained. Meanwhile, we show that a weak law for Pareto-Zipf distributions cannot be extended to a strong law. Furthermore, based on the two tailed Pareto distribution, a strong law of large numbers for weighed NSD random variables is presented. Our results extend the corresponding earlier ones.

Exact sampling of the infinite horizon maximum of a random walk over a nonlinear boundary

AbstractWe present the first algorithm that samples maxn≥0{Sn − nα}, where Sn is a mean zero random walk, and nα with $\alpha \in ({1 \over 2},1)$ defines a nonlinear boundary. We show that our algorithm has finite expected running time. We also apply this algorithm to construct the first exact simulation method for the steady-state departure process of a GI/GI/∞ queue where the service time distribution has infinite mean.

Transport properties and first-arrival statistics of random motion with stochastic reset times.

Stochastic resets have lately emerged as a mechanism able to generate finite equilibrium mean-square displacement (MSD) when they are applied to diffusive motion. Furthermore, walkers with an infinite mean first-arrival time (MFAT) to a given position x may reach it in a finite time when they reset their position. In this work we study these emerging phenomena from a unified perspective. On one hand, we study the existence of a finite equilibrium MSD when resets are applied to random motion with 〈x^{2}(t)〉_{m}∼t^{p} for 0<p≤2. For exponentially distributed reset times, a compact formula is derived for the equilibrium MSD of the overall process in terms of the mean reset time and the motion MSD. On the other hand, we also test the robustness of the finiteness of the MFAT for different motion dynamics which are subject to stochastic resets. Finally, we study a biased Brownian oscillator with resets with the general formulas derived in this work, finding its equilibrium first moment and MSD and its MFAT to the minimum of the harmonic potential.

Laws of large numbers with infinite mean

Laws of large numbers with infinite mean

Strong laws of large numbers for intermediately trimmed Birkhoff sums of observables with infinite mean

We consider dynamical systems on a finite measure space fulfilling a spectral gap property and Birkhoff sums of a non-negative, non-integrable observable. For such systems we generalize strong laws of large numbers for intermediately trimmed sums only known for independent random variables. The results split up in trimming statements for general distribution functions and for regularly varying tail distributions. In both cases the trimming rate can be chosen in the same or almost the same way as in the i.i.d. case. As an example we show that piecewise expanding interval maps fulfill the necessary conditions for our limit laws. As a side result we obtain strong laws of large numbers for truncated Birkhoff sums.

Simulation of Gaussian channels via teleportation and error correction of Gaussian states

Gaussian channels are the typical way to model the decoherence introduced by the environment in continuous-variable quantum states. It is known that those channels can be simulated by a teleportation protocol using as a resource state either a maximally entangled state passing through the same channel, i.e., the Choi-state, or a state that is entangled at least as much as the Choi-state. Since the construction of the Choi-state requires infinite mean energy and entanglement, i.e. it is unphysical, we derive instead every physical state able to simulate a given channel through teleportation with finite resources, and we further find the optimal ones, i.e., the resource states that require the minimum energy and entanglement. We show that the optimal resource states are pure and equally entangled to the Choi-state as measured by the entanglement of formation. We also show that the same amount of entanglement is enough to simulate an equally decohering channel, while even more entanglement can simulate less decohering channels. We, finally, use that fact to generalize a previously known error correction protocol by making it able to correct noise coming not only from pure loss but from thermal loss channels as well.

Infinite measure renewal theorem and related results

We present abstract conditions under which a special flow over a probability preserving map with a non-integrable roof function is Krickeberg mixing. Our main condition is some version of the local central limit theorem for the underlying map. We check our assumptions for iid random variables (renewal theorem with infinite mean) and for suspensions over Pomeau-Manneville maps.

On Solving Mean Payoff Games Using Pivoting Algorithms

Two classical pivoting algorithms, due to Lemke and Cottle–Dantzig, are studied on linear complementarity problems (LCPs) and their generalizations that arise from infinite duration two-person mean payoff games (MPGs) under zero-mean partition problem. Lemke’s algorithm was studied in solving MPGs via reduction to discounted payoff games or to simple stochastic games. We provide an alternative and efficient approach for studying the LCPs arising from the MPGs without any reduction. A binary MPG can easily be formulated as an LCP which has always terminated in a complementary solution in numerical experiments, but has not yet been proven either the processability of MPG’s by Lemke’s algorithm or a counter example that it will not terminate with a solution. Till now, the processability of MPG’s by Lemke’s algorithm remains open. A general MPG (with arbitrary outgoing arcs) naturally reduces to a generalized linear complementarity problem (GLCP) involving a rectangular matrix where the vertices are represented by the columns and the outgoing arcs from each vertex are represented by rows in a particular block. The noteworthy result in this paper is that the GLCP obtained from an MPG is processable by Cottle–Dantzig principal pivoting algorithm which terminates with a solution. Several properties of the matrix which arise in this context are also discussed.

Speeding up non-Markovian first-passage percolation with a few extra edges

Abstract One model of real-life spreading processes is that of first-passage percolation (also called the SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with independent and identically distributed heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow due to bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power-law distribution ℙ(ξ&gt;t)∼t-α with infinite mean. For any finite connected graph G with a root s, we find the largest number of vertices κ(G,s) that are infected in finite expected time, and prove that for every k≤κ(G,s), the expected time to infect k vertices is at most O(k1/α). Then we show that adding a single edge from s to a random vertex in a random tree 𝒯 typically increases κ(𝒯,s) from a bounded variable to a fraction of the size of 𝒯, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton--Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erdős‒Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.

Система H2|GI|&lt;» с бесконечным значением среднего времени обслуживания

Система H2|GI|&lt;» с бесконечным значением среднего времени обслуживания

Weak and one-sided strong laws for random variables with infinite mean

Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,…) and Sn(a)=∑i=1naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bn→∞ such that W¯n(a)=bn−1Sn(a)→p1, i.e., a weak law of large numbers holds. If the weights comprise a regularly varying sequence, then under some additional technical conditions, this outcome can be strengthened to a strong law if and only if the index of regular variation is −1. This paper addresses a case where the Xj are not identically distributed, but rather the tail probability P(ajXj>x) is asymptotically proportional to aj(1−F(x)), where F is a relatively stable distribution function. Here the weak law holds but the strong law does not: under typical conditions almost surely lim infn→∞W¯n(a)=1 and lim supn→∞W¯n(a)=∞.