arXiv subjects
Publications and source records attributed to Leonid V. Bogachev.
We propose a citation index $ν$ (``nu'') and show that it lies between the classical $h$-index and $g$-index. This idea is then generalized to a monotone parametric family $(ν_α)$ ($α\ge 0$), whereby $h=ν_0$ and $ν=ν_1$, while the limiting value $ν_\infty$ is expressed in terms of the maximum citation.
The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set $\check{\varLambda}^{q}$ of strict integer partitions (i.e., with unequal parts) into perfect $q$-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition weight (the sum of parts) and length (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters $\langle N\rangle$ and $\langle M\rangle$ controlling the expected weight and length, respectively. We study ``short'' partitions, where the parameter $\langle M\rangle$ is either fixed or grows slower than for typical plain (unconstrained) partitions. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed $\langle M\rangle$ and a limit shape result in the case of slow growth of $\langle M\rangle$. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyse their performance.
The generalized inverse Gaussian-Poisson (GIGP) distribution proposed by Sichel in the 1970s has proved to be a flexible fitting tool for diverse frequency data, collectively described using the item production model. In this paper, we identify the limit shape (specified as an incomplete gamma function) of the properly scaled diagrammatic representations of random samples from the GIGP distribution (known as Young diagrams). We also show that fluctuations are asymptotically normal and, moreover, the corresponding empirical random process is approximated via a rescaled Brownian motion in inverted time, with the inhomogeneous time scale determined by the limit shape. Here, the limit is taken as the number of production sources is growing to infinity, coupled with an intrinsic parameter regime ensuring that the mean number of items per source is large. More precisely, for convergence to the limit shape to be valid, this combined growth should be fast enough. In the opposite regime referred to as "chaotic", the empirical random process is approximated by means of an inhomogeneous Poisson process in inverted time. These results are illustrated using both computer simulations and some classic data sets in informetrics.
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 pro-actively 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.
The class of minimal difference partitions MDP($q$) (with gap $q$) is defined by the condition that successive parts in an integer partition differ from one another by at least $q\ge 0$. 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 classic Bose-Einstein ($q=0$) and Fermi-Dirac ($q=1$) cases. This was done by formally allowing values $q \in (0,1)$ 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 $(q_i)$, whereby the (limiting) gap $q$ is naturally interpreted as the Cesàro mean of $(q_i)$. In this model, we find the family of limit shapes parameterized by $q \in [0,\infty)$ confirming the earlier answer, and also obtain the asymptotics of the number of parts.
The paper concerns the $q$-state Potts model (i.e., with spin values in $\{1,\dots,q\}$) on a Cayley tree $\mathbb{T}^k$ of degree $k\geq 2$ (i.e., with $k+1$ edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invariant SGMs and prove a novel criterion of translation invariance. Assuming a ferromagnetic nearest-neighbour spin-spin interaction, we obtain various sufficient conditions for uniqueness. For a model with constant external field, we provide in-depth analysis of uniqueness vs.\ non-uniqueness in the subclass of completely homogeneous SGMs by identifying the phase diagrams on the "temperature--field" plane for different values of the parameters $q$ and $k$. In a few particular cases (e.g., $q=2$ or $k=2$), the maximal number of completely homogeneous SGMs in this model is shown to be $2^q-1$, and we make a conjecture (supported by computer calculations) that this bound is valid for all $q\ge 2$ and $k\ge2$.
Predicting the occurrence, level and duration of high air pollution concentrations exceeding a given critical level enables researchers to study the health impact of road traffic on local air quality and to inform public policy action. Precise estimates of the probabilities of occurrence and level of extreme concentrations are formidable due to the combination of complex physical and chemical processes involved. This underpins the need for developing sophisticated extreme value models, in particular allowing for non-stationarity of environmental time series. In this paper, extremes of nitrogen oxide (NO), nitrogen dioxide (NO$_2$) and ozone (O$_3$) concentrations are investigated using two models. Model I is based on an extended peaks-over-threshold (POT) approach developed by A. C. Davison and R. L. Smith, whereby the parameters of the underlying generalized Pareto distribution (GPD) are treated as functions of covariates (i.e., traffic and meteorological factors). The new Model II resolves the lack of threshold stability in the Davison--Smith model by constructing a special functional form for the GPD parameters. For each of the models, the effects of trafic and meteorological factors on the frequency and size of extreme values are estimated using Markov chain Monte Carlo methods. Finally, appropriate goodness-of-fit tests and model selection criteria confirm that Model II significantly outperforms Model I in estimation and forecasting of extremes.
The `archetypal' equation with rescaling is given by $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,μ(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $μ$ is a probability measure; equivalently, $y(x)=\mathbb{E}\{y(α(x-β))\}$, with random $α,β$ 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 $\mathbb{E}\{\ln|α|\}=0$ such a theorem holds subject to uniform continuity of $y(x)$; the latter is guaranteed under mild regularity assumptions on $β$, 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 $y(x)=\mathbb{E}\{y(X_τ)\,|\,X_0=x\}$ (with a suitable stopping time $τ$) due to Doob's optional stopping theorem applied to the martingale $y(X_n)$.
We study the archetypal functional equation of the form $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,μ(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $μ$ is a probability measure on $\mathbb{R}^2$; equivalently, $y(x)=\mathbb{E}\{y(α(x-β))\}$, where $\mathbb{E}$ is expectation with respect to the distribution $μ$ of random coefficients $(α,β)$. Existence of non-trivial (i.e., non-constant) bounded continuous solutions is governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,μ(\mathrm{d}a,\mathrm{d}b)=\mathbb{E}\{\ln|α|\}$; namely, under mild technical conditions no such solutions exist whenever $K<0$, whereas if $K>0$ (and $α>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(α,β)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $\mathbb{P}(α<0)>0$ is drastically different from that with $α>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrowα(x-β)$.
The paper concerns the limit shape (under some probability measure) of convex polygonal lines with vertices on $\mathbb{Z}_+^2$, starting at the origin and with the right endpoint $n=(n_1,n_2)\to\infty$. In the case of the uniform measure, an explicit limit shape $γ^*:=\{(x_1,x_2)\in\mathbb{R}_+^2\colon \sqrt{1-x_1}+\sqrt{x_2}=1\}$ was found independently by Vershik (1994), Bárány (1995), and Sinai (1994). Recently, Bogachev and Zarbaliev (2011) 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.
We propose an extension of the Ewens measure on permutations by choosing the cycle weights to be asymptotically proportional to the degree of the symmetric group. This model is primarily motivated by a natural approximation to the so-called spatial random permutations recently studied by V. Betz and D. Ueltschi (hence the name "surrogate-spatial"), but it is of substantial interest in its own right. We show that under the suitable (thermodynamic) limit both measures have the similar critical behaviour of the cycle statistics characterized by the emergence of infinitely long cycles. Moreover, using a greater analytic tractability of the surrogate-spatial model, we obtain a number of new results about the asymptotic distribution of the cycle lengths (both small and large) in the full range of subcritical, critical and supercritical domains. In particular, in the supercritical regime there is a parametric "phase transition" from the Poisson-Dirichlet limiting distribution of ordered cycles to the occurrence of a single giant cycle. Our techniques are based on the asymptotic analysis of the corresponding generating functions using Polya's Enumeration Theorem and complex variable methods.
We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form $\mathcal{F}(z)=\prod_{\ell=1}^\infty \mathcal{F}_0(z^\ell)$ (which entails equal weighting among possible parts $\ell\in\mathbb{N}$). Under mild technical assumptions on the function $H_0(u)=\ln(\mathcal{F}_0(u))$, we show that the limit shape $ω^*(x)$ exists and is given by the equation $y=γ^{-1}H_0(\mathrm{e}^{-γx})$, where $γ^2=\int_0^1 u^{-1}H_0(u)\,\mathrm{d}u$. The wide class of partition measures covered by this result includes (but is not limited to) representatives of the three meta-types of decomposable combinatorial structures --- assemblies, multisets and selections. Our method is based on the usual randomization and conditioning; to this end, a suitable local limit theorem is proved. The proofs are greatly facilitated by working with the cumulants of sums of the part counts rather than with their moments.
Let ${\varPi}_n$ be the set of convex polygonal lines $\varGamma$ with vertices on $\mathbb {Z}_+^2$ and fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit shape, as $n\to\infty$, of "typical" $\varGamma\in {\varPi}_n$ with respect to a parametric family of probability measures $\{P_n^r,0<r<\infty\}$ on ${\varPi}_n$, including the uniform distribution ($r=1$) for which the limit shape was found in the early 1990s independently by A. M. Vershik, I. Bárány and Ya. G. Sinai. We show that, in fact, the limit shape is universal in the class $\{P^r_n\}$, even though $P^r_n$ ($r\ne1$) and $P^1_n$ are asymptotically singular. Measures $P^r_n$ are constructed, following Sinai's approach, as conditional distributions $Q_z^r(\cdot |{\varPi}_n)$, where $Q_z^r$ are suitable product measures on the space ${\varPi}=\bigcup_n{\varPi}_n$, depending on an auxiliary "free" parameter $z=(z_1,z_2)$. The transition from $({\varPi},Q_z^r)$ to $({\varPi}_n,P_n^r)$ is based on the asymptotics of the probability $Q_z^r({\varPi}_n)$, furnished by a certain two-dimensional local limit theorem. The proofs involve subtle analytical tools including the Möbius inversion formula and properties of zeroes of the Riemann zeta function.
It is known that random convex polygonal lines on $\mathbb{Z}_+^2$ (with the endpoints fixed at $0=(0,0)$ and $n=(n_1,n_2)\to\infty$) have a limit shape with respect to the uniform probability measure, identified as the parabola arc $\sqrt{c\myp(1-x_1)}+\sqrt{x_2}=\sqrt{c}$, where $n_2/n_1\to c$. The present paper is concerned with the inverse problem of the limit shape. We show that for any strictly convex, $C^3$-smooth arc $γ\subset\mathbb{R}_+^2$ starting at the origin, there is a probability measure $P_n^γ$ on convex polygonal lines, under which the curve $γ$ is their limit shape.
A Bayesian multiple change-point model is proposed to analyse violations of air quality standards by pollutants such as nitrogen oxides (NO2 and NO) and carbon monoxide (CO). The model is built on the assumption that the occurrence of threshold exceedances may be described by a non-homogeneous Poisson process with a step rate function. Unlike earlier approaches, our model is not restricted by a predetermined number of change-points, nor does it involve any covariates. Possible short-range correlations in the exceedance data (e.g., due to chemical and meteorological factors) are removed via declusterisation. The unknown rate function is estimated using a reversible jump MCMC sampling algorithm adapted from Green (1995), which allows for transitions between parameter subspaces of varying dimension. This technique is applied to the 17-year (1993-2009) daily NO2, NO and CO concentration data in the City of Leeds, UK. The results are validated by running the MCMC estimator on simulated data replicated via a posterior estimate of the rate function. The findings are interpreted and discussed in relation to some known traffic control actions. The proposed methodology may be useful in the air quality management context by providing quantitative objective means to measure the efficacy of pollution control programmes.