On bounded continuous solutions of the archetypal equation with rescaling

The `archetypal' equation with rescaling is given by $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, with random $\alpha,\beta$ and $\mathbb{E}$ denoting expectation. Examples include: (i) functional equation $y(x)=\sum_{i} p_{i} y(a_i(x-b_i))$; (ii) functional-differential (`pantograph') equation $y'(x)+y(x)=\sum_{i} p_{i} y(a_i(x-c_i))$ ($p_{i}>0$, $\sum_{i} p_{i}=1$). Interpreting solutions $y(x)$ as harmonic functions of the associated Markov chain $(X_n)$, we obtain Liouville-type results asserting that any bounded continuous solution is constant. In particular, in the `critical' case $\mathbb{E}\{\ln|\alpha|\}=0$ such a theorem holds subject to uniform continuity of $y(x)$; the latter is guaranteed under mild regularity assumptions on $\beta$, satisfied e.g.\ for the pantograph equation (ii). For equation (i) with $a_i=q^{m_i}$ ($m_i\in\mathbb{Z}$, $\sum_i p_i m_i=0$), the result can be proved without the uniform continuity assumption. The proofs utilize the iterated equation $y(x)=\mathbb{E}\{y(X_\tau)\,|\,X_0=x\}$ (with a suitable stopping time $\tau$) due to Doob's optional stopping theorem applied to the martingale $y(X_n)$.Comment: Substantially revised. The title is modifie

Optimal Stopping and Utility in a Simple Model of Unemployment Insurance

Managing unemployment is one of the key issues in social policies. Unemployment insurance schemes are designed to cushion the financial and morale blow of loss of job but also to encourage the unemployed to seek new jobs more proactively due to the continuous reduction of benefit payments. In the present paper, a simple model of unemployment insurance is proposed with a focus on optimality of the individual’s entry to the scheme. The corresponding optimal stopping problem is solved, and its similarity and differences with the perpetual American call option are discussed. Beyond a purely financial point of view, we argue that in the actuarial context the optimal decisions should take into account other possible preferences through a suitable utility function. Some examples in this direction are worked out

Limit Shape of Minimal Difference Partitions and Fractional Statistics

The class of minimal difference partitionsMDP(q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least q≥0. In a recent series of papers by A. Comtet and collaborators, the MDP(q) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classical Bose–Einstein (q=0) and Fermi–Dirac (q=1) cases. This was done by formally allowing values q∈(0,1) using an analytic continuation of the limit shape of the corresponding Young diagrams calculated for integer q. To justify this “replica-trick”, we introduce a more general model based on a variable MDP-type condition encoded by an integer sequence q=(qi), whereby the (limiting) gap q is naturally interpreted as the Cesàro mean of q. In this model, we find the family of limit shapes parameterized by q∈[0,∞) confirming the earlier answer, and also obtain the asymptotics of the number of parts

A Generalized Power Law Model of Citations

The classical power law model is widely used in informetrics to describe citations of scientific papers, although it is not addressing variability across individual authors. We report our preliminary results for a novel model based on a certain parametric form of the expected individual citation profile which generalizes the power law frequency formula. The new model interpolates between large citation numbers, where the power law tail is reproduced, and low citation numbers, which are usually truncated when fitting the power law model to the data. In addition, we derive a deterministic limit shape of the citation profile, which can be used to make predictions about various citation function such as the h-index

Limit laws for sums of random exponentials

We study the limiting distribution of the sum S-N(t) = Sigma(i=1)(N) e(tXi) as t -> infinity, N -> infinity, where (X-i) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in random media theory. Examples include the quenched mean population size of a colony of branching processes with random branching rates and the partition function of Derrida's Random Energy Model. In this paper, the problem is considered under the assumption that the log-tail distribution function h(x) = -log P{X-I > x} is regularly varying at infinity with index 1 < rho < infinity. An appropriate scale for the growth of N relative to t is of the form e(lambda H0(t)), where the rate function Ho(t) is a certain asymptotic version of the cumulant. generating function H(t) = log E[e(tXi)] provided by Kasahara's exponential Tauberian theorem. We have found two critical points, 0 < lambda(1) < lambda(2) < infinity, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below lambda(2), we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent alpha = alpha(rho, lambda) ranging from 0 to 2 and with skewness parameter beta = 1. A limit theorem for the maximal value of the sample {e(tXi), i = 1,...,N} is also proved

Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence

We discuss a relation between the Kantorovich-Wasserstein (KW) metric and the Kullback-Leibler (KL) divergence. The former is defined using the optimal transport problem (OTP) in the Kantorovich formulation. The latter is used to define entropy and mutual information, which appear in variational problems to find optimal channel (OCP) from the rate distortion and the value of information theories. We show that OTP is equivalent to OCP with one additional constraint fixing the output measure, and therefore OCP with constraints on the KL-divergence gives a lower bound on the KW-metric. The dual formulation of OTP allows us to explore the relation between the KL-divergence and the KW-metric using decomposition of the former based on the law of cosines. This way we show the link between two divergences using the variational and geometric principles

Limit shape of random convex polygonal lines: even more universality

The paper concerns the limit shape (under some probability measure) of convex polygonal lines with vertices on Z+2, starting at the origin and with the right endpoint n=(n1,n2)→∞. In the case of the uniform measure, an explicit limit shape γ⁎:={(x1,x2)∈R+2:1−x1+x2=1} was found independently by Vershik (1994) [19], Bárány (1995) [3], and Sinaĭ (1994) [16]. Recently, Bogachev and Zarbaliev (1999) [5] proved that the limit shape γ⁎ is universal for a certain parametric family of multiplicative probability measures generalizing the uniform distribution. In the present work, the universality result is extended to a much wider class of multiplicative measures, including (but not limited to) analogs of the three meta-types of decomposable combinatorial structures — multisets, selections, and assemblies. This result is in sharp contrast with the one-dimensional case where the limit shape of Young diagrams associated with integer partitions heavily depends on the distributional type

Extreme value theory for random exponentials

We study the limit distribution of upper extreme values of i.i.d. exponential samples {e^(tX_i), i=1,...,N} as t->infty, N->infty. Two cases are considered: (A) ess supX = 0 and (B) ess supX = 1. We assume that the function h(x) = &#8722;log P{X > x} (case B) or h(x) = &#8722;log P{X > &#8722;1/x} (case A) is (normalized) regularly varying at 1 with index 1 < rho < infty (case B) or 0 < rho < 1 (case A). The growth scale of N is chosen in the form N = exp(lambda H_0^rho(t)) (0 < lambda < infty), where H_0(t) is a certain asymptotic version of the function H(t) := log E[e^(tX)] (case B) or H(t) = &#8722;log E[e^(tX)] (case A). As shown earlier by Ben Arous et al.(2005), there are critical points lambda_1 < lambda_2, below which the LLN and CLT, respectively, break down, whereas for 0 < lambda < 2 the limit laws for the sum S_N(t) = e^(tX_1) + ··· + e^(tX_N) prove to be stable, with characteristic exponent alpha = alpha(rho,lambda) in (0,2). In this paper, we obtain the (joint) limit distribution of the upper order statistics of the exponential sample. In particular, M_{1,N} = max{e^(tX_i), i=1,...,N} has asymptotically the Frechet distribution with parameter alpha. We also show that the empirical extremal measure converges (in fdd) to a Poisson random measure with intensity d(x^{&#8722;alpha}). These results are complemented by explicit representations of the joint limit distribution of S_N(t) and M_{1,N}(t) (and in particular of their ratio) in terms of i.i.d. random variables with standard exponential distribution

Cities as evolutionary systems in random media

The purpose of the paper is to discuss some potential applications of random media theory to urban modelling, with the emphasis on the intermittency phenomenon. The moment test of intermittency is explained using the model of continuous-time branching random walk on the integer lattice Z^d with random branching rates. Statistical moments of the population density are studied using a Cauchy problem for the Anderson operator with random potential. The Feynman–Kac representation of the solution is discussed, and Lyapunov exponents responsible for the super-exponential growth of the moments are evaluated. The higher-order Lyapunov exponents are also obtained. The results suggest that the higher-order intermittency is reduced, in a sense, to that of the mean population density

Limit laws for sums of random exponentials

We study the limiting distribution of the sum S-N(t) = Sigma(i=1)(N) e(tXi) as t -> infinity, N -> infinity, where (X-i) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in random media theory. Examples include the quenched mean population size of a colony of branching processes with random branching rates and the partition function of Derrida's Random Energy Model. In this paper, the problem is considered under the assumption that the log-tail distribution function h(x) = -log P{X-I > x} is regularly varying at infinity with index 1 < rho < infinity. An appropriate scale for the growth of N relative to t is of the form e(lambda H0(t)), where the rate function Ho(t) is a certain asymptotic version of the cumulant. generating function H(t) = log E[e(tXi)] provided by Kasahara's exponential Tauberian theorem. We have found two critical points, 0 < lambda(1) < lambda(2) < infinity, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below lambda(2), we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent alpha = alpha(rho, lambda) ranging from 0 to 2 and with skewness parameter beta = 1. A limit theorem for the maximal value of the sample {e(tXi), i = 1,...,N} is also proved