Non-degenerate Distribution Research Articles

Stationary measures and the continuous-state branching process conditioned on extinction

Abstract We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

Estimating the number of common trends in large [formula omitted] and [formula omitted] factor models via canonical correlations analysis

Asymptotic results for canonical correlations are derived when the analysis is performed between levels and cumulated levels of N time series of length T, generated by a factor model with s common stochastic trends. For T→∞ and fixed N and s, the largest s squared canonical correlations are shown to converge to a non-degenerate limit distribution while the remaining N−s converge in probability to 0. Furthermore, if s grows at most linearly in N, the largest s squared canonical correlations are shown to converge in probability to 1 as (T,N)seq→∞. This feature allows one to estimate the number of common trends as the integer with largest decrease in adjacent squared canonical correlations. The maximal gap equals 1 in the limit and this criterion is shown to be consistent. A Monte Carlo simulation study illustrates the findings.

Schooling, Skill Demand, and Differential Fertility in the Process of Structural Transformation

Demography and structural transformation are interrelated, and depend critically on education. At the turn of the twentieth century, US parents began having fewer children while increasing educational investment per child. This quantity-quality tradeoff facilitated job reallocation from the low-skilled agricultural sector to the high-skilled nonagricultural sector. This transformation is examined in a heterogeneous agent model with a nondegenerate human capital distribution, focusing on how fertility and education decisions affect structural transformation. The result shows that the quantity-quality decisions account for up to approximately one-third of the decline in the agricultural employment share. (JEL E24, I21, I26, J11, J13, J24, N31)

Sequentially mixed search and equilibrium price dispersion

Many markets feature sequentially mixed search (SMS), which is search directed toward a price distribution followed by noisy matching with multiple offers. I construct a tractable model of SMS, establish existence of a unique equilibrium, and prove that the equilibrium is constrained efficient. Although individuals on each side of the market are homogeneous and search is directed, the equilibrium has a non-degenerate and continuous distribution of posted prices on a connected support. Despite the possibility of multiple offers, the highest price is lower than the joint value of a match so that both sides of the market obtain strictly positive surpluses of trade at all equilibrium prices. This feature captures the common practice that searchers restrict search below a price bound. Moreover, I show that an increase in the meeting efficiency widens price dispersion, which helps explain the puzzling fact that the increased use of the Internet has not reduced price dispersion.

The long-run redistributive effects of monetary policy

Using a general equilibrium search-theoretic model of money, I study the long-run distributional effects of monetary policy. In my model, heterogeneous agents trade bilaterally in a frictional market and save using cash and illiquid short-term nominal government bonds. Wealth effects generate slow adjustments in agents’ portfolios following their trading activity in decentralized markets, giving rise to a persistent and non-degenerate distribution of assets. The model reproduces the distribution of asset levels and portfolios across households observed in the data. I show that, as wealth inequality increases the incidence of inefficiencies in decentralized trading, policies that improve the ability to self-insure against idiosyncratic shocks are welfare-improving and redistribute resources towards agents that are relatively poor and more liquidity constrained.

Angle Sums of Random Polytopes

For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in Rd. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.

Autoencoding Galaxy Spectra. II. Redshift Invariance and Outlier Detection

We present an unsupervised outlier detection method for galaxy spectra based on the spectrum autoencoder architecture spender, which reliably captures spectral features and provides highly realistic reconstructions for SDSS galaxy spectra. We interpret the sample density in the autoencoder latent space as a probability distribution, and identify outliers as low-probability objects with a normalizing flow. However, we found that the latent-space position is not, as expected from the architecture, redshift invariant, which introduces stochasticity into the latent space and the outlier detection method. We solve this problem by adding two novel loss terms during training, which explicitly link latent-space distances to data-space distances, preserving locality in the autoencoding process. Minimizing the additional losses leads to a redshift-invariant, nondegenerate latent-space distribution with clear separations between common and anomalous data. We inspect the spectra with the lowest probability and find them to include blends with foreground stars, extremely reddened galaxies, galaxy pairs and triples, and stars that are misclassified as galaxies. We release the newly trained spender model and the latent-space probability for the entire SDSS-I galaxy sample to aid further investigations.

Approximately Hadamard Matrices and Riesz Bases in Random Frames

Abstract An $n \times n$ matrix with $\pm 1$ entries that acts on ${\mathbb {R}}^{n}$ as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools, we construct matrices with $\pm 1$ entries that act as approximate scaled isometries in ${\mathbb {R}}^{n}$ for all $n \in {\mathbb {N}}$. More precisely, the matrices we construct have condition numbers bounded by a constant independent of $n$. Using this construction, we establish a phase transition for the probability that a random frame contains a Riesz basis. Namely, we show that a random frame in ${\mathbb {R}}^{n}$ formed by $N$ vectors with independent identically distributed coordinate having a nondegenerate symmetric distribution contains many Riesz bases with high probability provided that $N \ge \exp (Cn)$. On the other hand, we prove that if the entries are sub-Gaussian, then a random frame fails to contain a Riesz basis with probability close to $1$ whenever $N \le \exp (cn)$, where $c<C$ are constants depending on the distribution of the entries.

Killed distribution dependent SDE for nonlinear Dirichlet problem

To characterize nonlinear Dirichlet problems in an open domain, we investigate killed distribution dependent SDEs. By constructing the coupling by projection and using the Zvonkin/Girsanov transforms, the well-posedness is proved for three different situations:1) monotone case with distribution dependent noise (possibly degenerate);2) singular case with non-degenerate distribution dependent noise;3) singular case with non-degenerate distribution independent noise. In the first two cases the domain is $ C^2 $ smooth such that the Lipschitz continuity in initial distributions is also derived, and in the last case the domain is arbitrary.

Branching random walk in a random time-independent environment

ABSTRACT In a lattice population model, particles move randomly from one site to another as independent random walks, split into two offspring, or die. If duplication and mortality rates are equal and take the same value over all lattice sites, the resulting model is a critical branching random walk (characterized by a mean total number of offspring equal to ). There exists an asymptotical statistical equilibrium, also called steady state. In contrast, when duplication and mortality rates take independent random values drawn from a common nondegenerate distribution (so that the difference between duplication and mortality rates has nonzero variance), then the steady state no longer exists. Simultaneously, at all lattice sites, if the difference between duplication and mortality rates takes strictly positive values with strictly positive probability, the total number of particles grows exponentially. The lattice includes large connected sets where the duplication rate exceeds the mortality rate by a positive constant amount, and these connected sets provide the growth of the total population. This is the supercritical regime of branching processes. On the other hand, if the difference between duplication and mortality rates is almost surely negative or null except when it is almost surely zero, then the total number of particles vanishes asymptotically. The steady state can be reached only if the difference between duplication and mortality rates is almost surely zero.

A mathematical analysis of EDAs with distance-based exponential models

Estimation of Distribution Algorithms have been successfully used to solve permutation-based Combinatorial Optimization Problems. In this case, the algorithms use probabilistic models specifically designed for codifying probability distributions over permutation spaces. One class of these probability models are distance-based exponential models, and one example of this class is the Mallows model. In spite of its practical success, the theoretical analysis of Estimation of Distribution Algorithms for permutation-based Combinatorial Optimization Problems has not been developed as extensively as it has been for binary problems. With this motivation, this paper presents a first mathematical analysis of the convergence behavior of Estimation of Distribution Algorithms based on Mallows models. The model removes the randomness of the algorithm in order to associate a dynamical system to it. Several scenarios of increasing complexity with different fitness functions and initial probability distributions are analyzed. The obtained results show: a) the strong dependence of the final results on the initial population, and b) the possibility to converge to non-degenerate distributions even in very simple scenarios, which has not been reported before in the literature.

On the pure theory of wage dispersion

We study an equilibrium model of the labor market with identical firms and homogeneous workers, and with search and on-the-job search. Jobs are dynamic contracts that allow firms to match the worker's outside offers or let the job be terminated. For a non-degenerate distribution of wage offers to arise in the environment, it is necessary and sufficient that (i) there is a positive cost of job turnover, in terminating an existing job or posting a new one; and (ii) there is asymmetric information regarding the worker's outside offers. The model is calibrated to the U.S. labor market to match observed worker flows and the observed mean-min ratio in wages earned. The calibrated model predicts a unimodal distribution for both wages offered and wages earned.

Equilibrium Bid-Price Dispersion

Competitive bidding by homogeneous agents in a first-price auction can yield a non-degenerate bid price distribution. This price dispersion is the unique equilibrium in a setting where bidders “pay to play.” Ex ante, bidders decide simultaneously on whether to play or not. Ex post, those who play submit their bid simultaneously not knowing who else is in the market. The price-dispersion result is applied to high-frequency bidding in limit-order markets. The parsimonious model fits the bid-price dispersion for S&P 500 stocks remarkably well.

Tests of multivariate copula exchangeability based on Lévy measures

AbstractThis paper introduces tests for the symmetry of the copula of random vector. The proposed statistics are based on the copula characteristic function and the weight function that appears naturally in their definition are assumed to belong to the general family of Lévy measures. The proposed test statistics are rank‐based and expresses as weighted ‐norms computed from a vector of empirical copula characteristic functions. Their nondegenerate asymptotic distributions under the null hypothesis and general alternatives, as well as the validity of a multiplier bootstrap for the computation of p‐values, are derived using nonstandard arguments. Extended Monte–Carlo experiments show that the new tests hold their size well and are powerful against a wide range of alternatives, and appear to be more powerful than a Cramér–von Mises test based on empirical copulas.

Matrix Completion with Deterministic Sampling: Theories and Methods.

In some significant applications such as data forecasting, the locations of missing entries cannot obey any non-degenerate distributions, questioning the validity of the prevalent assumption that the missing data is randomly chosen according to some probabilistic model. To break through the limits of random sampling, we explore in this paper the problem of real-valued matrix completion under the setup of deterministic sampling. We propose two conditions, isomeric condition and relative well-conditionedness, for guaranteeing an arbitrary matrix to be recoverable from a sampling of the matrix entries. It is provable that the proposed conditions are weaker than the assumption of uniform sampling and, most importantly, it is also provable that the isomeric condition is necessary for the completions of any partial matrices to be identifiable. Equipped with these new tools, we prove a collection of theorems for missing data recovery as well as convex/nonconvex matrix completion. Among other things, we study in detail a Schatten quasi-norm induced method termed isomeric dictionary pursuit (IsoDP), and we show that IsoDP exhibits some distinct behaviors absent in the traditional bilinear programs.

Regulating gene expression to achieve temporal precision

Cellular response to a stimulus is often triggered by accumulation of an appropriate gene product up to a critical threshold. How cells regulate gene expression to achieve precision in timing of such responses is a question of interest. Earlier work has shown that for a stable gene product, a constant rate of accumulation provides minimum noise in the time of response, provided that initial gene product distribution is degenerate. Here, we show that this strategy is no longer optimal if the initial gene product level is drawn from a non-degenerate distribution. We also discuss biological relevance of this finding.

Unobservable Investments, Trade Efficiency and Search Frictions

Market participants can invest to increase their gains from trade before searching for partners. The entrants' type distribution and trade efficiency are jointly determined and correlated. This paper explores a random search model of ex ante investments and trade efficiency, assuming that the investment amount remains the investor's private information. We show that the ex ante payoffs are equivalent to the equilibrium payoffs when investments are observable. In the unique steady state equilibrium, non-degenerate investment distribution and price distribution emerge simultaneously with ex-ante identical agents. In the basic model, where investing agents have no bargaining power, the extra investments fail to improve social welfare even when the search friction vanishes and the investments become efficient due to the mismatches caused by unobservability. Allocating positive bargaining power to investors alleviates the mismatch problem and improves social welfare.

Faces in random great hypersphere tessellations

The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is introduced and studied. Probabilistic interpretations for such spherical faces are given and their directional distributions are determined. Explicit formulas for the expected $f$-vector, the expected spherical Quermasintegrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as $n\to \infty $ is discussed and compared to the corresponding notions and results in the Euclidean case. The expected statistical dimension and a problem related to intersection probabilities of spherical random polytopes is investigated.

Short-run dynamics in a search-theoretic model of monetary exchange

We study the short-run effects of monetary policy in a search-theoretic monetary model in which agents are subject to idiosyncratic liquidity shocks as well as aggregate monetary shocks. Namely, we analyze the role of the endogenous non-degenerate distribution of liquidity, liquidity constraints, and decentralized trade, for the transmission and propagation of monetary policy shocks. Money is injected through lump-sum transfers which have redistributive and persistent effects on output and prices. We propose a new numerical algorithm in the spirit of Algan et al. (2008) to solve the model. We find that a one-time expansionary monetary policy shock has persistent positive effects on output, prices, and welfare, even in the absence of nominal rigidities. Furthermore, the effects of positive and negative monetary shocks are typically asymmetric. Negative (contractionary) shocks have bigger effects than positive (expansionary) shocks. In addition, in an economy with larger shocks, the responses tend to be disproportionately larger than those in an economy with smaller shocks. Finally, the effectiveness of monetary shocks depends on the steady-state level of inflation. The higher the average level of inflation (money growth), the bigger the impact effect of a shock of a given size but the smaller its cumulative effect. These results are consistent with existing empirical evidence.

Gaining traction: on the convergence of an inner approximation scheme for probability maximization

We analyze an inner approximation scheme for probability maximization. The approach was proposed in Fábián et al. (Acta Polytech Hung 15:105–125, 2018), as an analogue of a classic dual approach in the handling of probabilistic constraints. Even a basic implementation of the maximization scheme proved usable and endured noise in gradient computations without any special effort. Moreover, the speed of convergence was not affected by approximate computation of test points. This robustness was then explained in an idealized setting. Here we work out convergence proofs and efficiency arguments for a nondegenerate normal distribution. The main message of the present paper is that the procedure gains traction as an optimal solution is approached.