Probability Measure Research Articles

Studies on the Marchenko–Pastur Law

In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical probability that are defined by means of the exponential kernel. Within the context of CSK families, this article presents certain features of the Marchenko–Pastur law based on the Fermi convolution and the t-deformed free convolution. The Marchenko–Pastur law holds significant theoretical and practical implications in various fields, particularly in the analysis of random matrices and their applications in statistics, signal processing, and machine learning. In the specific context of CSK families, our study of the Marchenko–Pastur law is summarized as follows: Let K+(μ)={Qmμ(dx);m∈(m0μ,m+μ)} be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Qmμ•s the Fermi convolution power of order s>0 of the measure Qmμ. We prove that if Qmμ•s∈K+(μ), then μ is of the Marchenko–Pastur type law. The same result is obtained if we replace the Fermi convolution • with the t-deformed free convolution t.

Some Skorohod-type results

AbstractLet S be a metric space, $$g:S\rightarrow \mathbb {R}$$ g : S → R a Borel function, and $$(\mu _n:n\ge 0)$$ ( μ n : n ≥ 0 ) a sequence of tight probability measures on $$\mathcal {B}(S)$$ B ( S ) . If $$\mu _n=\mu _0$$ μ n = μ 0 on $$\sigma (g)$$ σ ( g ) , there are S-valued random variables $$X_n$$ X n , all defined on the same probability space, such that $$X_n\sim \mu _n$$ X n ∼ μ n and $$g(X_n)=g(X_0)$$ g ( X n ) = g ( X 0 ) for all $$n\ge 0$$ n ≥ 0 . Moreover, $$X_n\overset{a.s.}{\longrightarrow }X_0$$ X n ⟶ a . s . X 0 if and only if $$E_{\mu _n}(f\mid g)\,\overset{\mu _0-a.s.}{\longrightarrow }\,E_{\mu _0}(f\mid g)$$ E μ n ( f ∣ g ) ⟶ μ 0 - a . s . E μ 0 ( f ∣ g ) for each $$f\in C_b(S)$$ f ∈ C b ( S ) . This result, proved in Pratelli and Rigo (J Theoret Probab 36:372-389, 2023) , is the starting point of this paper. Three types of contributions are provided. First, $$\sigma (g)$$ σ ( g ) is replaced by an arbitrary sub-$$\sigma $$ σ -field $$\mathcal {G}\subset \mathcal {B}(S)$$ G ⊂ B ( S ) . Second, the result is applied to some specific frameworks, including equivalence couplings, total variation distances, and the decomposition of cadlag processes with finhite activity. Third, following Hansen et al. (Tempered Bayesian analysis, Unpublished manuscript, 2024), the result is extended to models and kernels. This extension has a fairly natural interpretation in terms of decision theory, mass transportation and statistics.

Higher-order moments of spline chaos expansion

Spline chaos expansion, referred to as SCE, is a finite series representation of an output random variable in terms of measure-consistent orthonormal splines in input random variables and deterministic coefficients. This paper reports new results from an assessment of SCE’s approximation quality in calculating higher-order moments, if they exist, of the output random variable. A novel mathematical proof is provided to demonstrate that the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the largest element size decreases. Complementary numerical analyses have been conducted, producing results consistent with theoretical findings. A collection of simple yet relevant examples is presented to grade the approximation quality of SCE with that of the well-known polynomial chaos expansion (PCE). The results from these examples indicate that higher-order moments calculated using SCE converge for all cases considered in this study. In contrast, the moments of PCE of an order larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables. Moreover, when both SCE- and PCE-generated moments converge, the convergence rate of the former is markedly faster than the latter in the presence of nonsmooth functions or unbounded domains of input random variables.

Uniformly ergodic probability measures

Uniformly ergodic probability measures

Hilbert Curve Projection Distance for Distribution Comparison.

Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions with low complexity. In particular, we first project two high-dimensional probability distributions using Hilbert curve to obtain a coupling between them, and then calculate the transport distance between these two distributions in the original space, according to the coupling. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. Furthermore, we demonstrate that the modified empirical HCP distance with the Lp cost in the d-dimensional space converges to its population counterpart at a rate of no more than O(n-1/2max{d,p}). To suppress the curse-of-dimensionality, we also develop two variants of the HCP distance using (learnable) subspace projections. Experiments on both synthetic and real-world data show that our HCP distance works as an effective surrogate of the Wasserstein distance with low complexity and overcomes the drawbacks of the sliced Wasserstein distance.

Direct Measurement of Charge Transfer Probability during Photodissociation of Few-keV OD+ Beam.

We have measured the photodissociation of few-keV OD+ molecular ions into either D+ + O or O+ + D final products. The three-dimensional momentum imaging measurements of the light and massive fragments in coincidence were enabled by using an upgraded two-detector setup. In this work, we show that absorption of a single 790 or 395 nm photon excites the OD+ from its electronic ground state to the B state, which dissociates to the O+(4S) + D dissociation limit. To reach the other nearly degenerate dissociation limit, D+ + O(3P), a unimolecular charge transfer, B to X , transition is required following the same photoexcitation. The measured branching ratio of these dissociation channels is a direct measure of the charge transfer transition probability. This measured probability as a function of energy above the dissociation limit agrees well with our calculations.

Confidence Intervals for Ratio of Percentiles of Birnbaum-Saunders Distributions and Its Application to PM2.5 in Northern Thailand

In the statistical literature, the Birnbaum-Saunders distribution has garnered considerable attention as an important fatigue life distribution for engineers. This distribution finds applications in medicine, engineering, and science. A percentile denotes a threshold below which a designated percentage of scores or data points are situated. In particular cases, percentiles are the preferred choice over mean and variance, especially when dealing with skewed data. This paper investigates confidence intervals for the ratio of percentiles of Birnbaum-Saunders distributions. It discusses the generalized fiducial confidence interval, Bayesian credible interval, and the highest posterior density intervals using a prior distribution with partial information and a proper prior with known hyperparameters. To compare the performance of these confidence intervals, the study employs coverage probability and average length measurements. Monte Carlo simulation via the R package is used to calculate the coverage probability and average length. The results indicate that the highest posterior density interval using a prior distribution with partial information outperforms the other confidence intervals. Finally, the paper presents the results of the simulation study and applies them in the field of environmental sciences.

Phase Retrieval for Probability Measures on the Group ℤ_2^3

Phase Retrieval for Probability Measures on the Group ℤ_2^3

Sampling with flows, diffusion, and autoregressive neural networks from a spin-glass perspective

Recent years witnessed the development of powerful generative models based on flows, diffusion, or autoregressive neural networks, achieving remarkable success in generating data from examples with applications in a broad range of areas. A theoretical analysis of the performance and understanding of the limitations of these methods remain, however, challenging. In this paper, we undertake a step in this direction by analyzing the efficiency of sampling by these methods on a class of problems with a known probability distribution and comparing it with the sampling performance of more traditional methods such as the Monte Carlo Markov chain and Langevin dynamics. We focus on a class of probability distribution widely studied in the statistical physics of disordered systems that relate to spin glasses, statistical inference, and constraint satisfaction problems. We leverage the fact that sampling via flow-based, diffusion-based, or autoregressive networks methods can be equivalently mapped to the analysis of a Bayes optimal denoising of a modified probability measure. Our findings demonstrate that these methods encounter difficulties in sampling stemming from the presence of a first-order phase transition along the algorithm's denoising path. Our conclusions go both ways: We identify regions of parameters where these methods are unable to sample efficiently, while that is possible using standard Monte Carlo or Langevin approaches. We also identify regions where the opposite happens: standard approaches are inefficient while the discussed generative methods work well.

Retransmission performance in a stochastic geometric cellular network model

Suppose sender–receiver transmission links in a downlink network at a given data rate are subject to fading, path loss, and inter-cell interference, and that transmissions either pass, suffer loss, or incur retransmission delay. We introduce a method to obtain the average activity level of the system required for handling the buffered work and from this derive the resulting coverage probability and key performance measures. The technique involves a family of stationary buffer distributions which is used to solve iteratively a nonlinear balance equation for the unknown busy-link probability and then identify throughput, loss probability, and delay. The results allow for a straightforward numerical investigation of performance indicators, are in special cases explicit and may be easily used to study the trade-off between reliability, latency, and data rate.

A Joint Limit Theorem for Epstein and Hurwitz Zeta-Functions

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C2 defined by means of the Epstein ζ(s;Q) and Hurwitz ζ(s,α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter α are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function.

Stochastic optimization problems with nonlinear dependence on a probability measure via the Wasserstein metric

Nonlinear dependence on a probability measure has recently been encountered with increasing intensity in stochastic optimization. This type of dependence corresponds to many situations in applications; it can appear in problems static (one-stage), dynamic with finite (multi-stage) or infinite horizon, and single- and multi-objective ones. Moreover, the nonlinear dependence can appear not only in the objective functions but also in the constraint sets. In this paper, we will consider static one-objective problems in which the nonlinear dependence appears in the objective function and may also appear in the constraint sets. In detail, we consider “deterministic” constraint sets, whose dependence on the probability measure is nonlinear, constraint sets determined by second-order stochastic dominance, and sets given by mean-risk problems. The last mentioned instance means that the constraint set corresponds to solutions which guarantee acceptable values of both criteria. To obtain relevant assertions, we employ the stability results given by the Wasserstein metric, based on the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {{\\mathcal {L}}}_{1} $$\\end{document} norm. We mainly focus on the case in which a solution has to be obtained on the basis of the data and of investigating a relationship between the original problem and its empirical version.

A necessary and sufficient condition for double coset lumping of Markov chains on groups with an application to the random to top shuffle

Let Q Q be a probability measure on a finite group G G , and let H H be a subgroup of G G . We show that a necessary and sufficient condition for the random walk driven by Q Q on G G to induce a Markov chain on the double coset space H ∖ G / H H\backslash G/H is that Q ( g H ) Q(gH) is constant as g g ranges over any double coset of H H in G G . We obtain this result as a corollary of a more general theorem on the double cosets H ∖ G / K H \backslash G / K for K K an arbitrary subgroup of G G . As an application we study a variation on the r r -top to random shuffle which we show induces an irreducible, recurrent, reversible and ergodic Markov chain on the double cosets of S y m r × S y m n − r \mathrm {Sym}_r \times \mathrm {Sym}_{n-r} in S y m n \mathrm {Sym}_n . The transition matrix of the induced walk has remarkable spectral properties: we find its invariant distribution and its eigenvalues and hence determine its rate of convergence.

Approximation for the invariant measure with applications for jump processes (convergence in total variation distance)

In this paper, we establish an abstract framework for the approximation of the invariant probability measure for a Markov semigroup. Following Pagès and Panloup (2022) we use an Euler scheme with decreasing step (unadjusted Langevin algorithm). Under some contraction property with exponential rate and some regularization properties, we give an estimate of the error in total variation distance. This abstract framework covers the main results in Pagès and Panloup (2022) and Chen et al. (2023). As a specific application we study the convergence in total variation distance to the invariant measure for jump type equations. The main technical difficulty consists in proving the regularization properties — this is done under an ellipticity condition, using Malliavin calculus for jump processes.

"Modeling Diffusive Search by Non-Adaptive Sperm: Empirical and Computational Insights".

During fertilization, mammalian sperm undergo a winnowing selection process that reduces the candidate pool of potential fertilizers from ~106-1011 cells to 101-102 cells (depending on the species). Classical sperm competition theory addresses the positive or 'stabilizing' selection that acts on sperm phenotypes within populations of organisms but does not strictly address the developmental consequences of sperm traits among individual organisms that are under purifying selection during fertilization. It is the latter that is of utmost concern for improving assisted reproductive technologies (ART) because 'low fitness' sperm may be inadvertently used for fertilization during interventions that rely heavily on artificial sperm selection, such as intracytoplasmic sperm injection (ICSI). Importantly, some form of sperm selection is used in nearly all forms of ART (e.g., differential centrifugation, swim-up, or hyaluronan binding assays, etc.). To date, there is no unifying quantitative framework (i.e., theory of sperm selection) that synthesizes causal mechanisms of selection with observed natural variation in individual sperm traits. In this report, we reframe the physiological function of sperm as a collective diffusive search process and develop multi-scale computational models to explore the causal dynamics that constrain sperm 'fitness' during fertilization. Several experimentally useful concepts are developed, including a probabilistic measure of sperm 'fitness' as well as an information theoretic measure of the magnitude of sperm selection, each of which are assessed under systematic increases in microenvironmental selective pressure acting on sperm motility patterns.

Interpreting systems of continuity equations in spaces of probability measures through PDE duality

We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality is, under suitable assumptions, equivalent to gradient flow solutions in case the single/system of equations has this structure. In contrast, we can deal with a quite general system of nonlinear non-local, diffusive or not, system of PDEs without any variational structure.

Cubic Hermite interpolators on the space of probability measures

In this paper, we introduce a novel two‐step modeling method to generalize cubic Hermite interpolators on the space of probability measures . First, we develop new approaches to capture the Riemannian geometric structure of when equipped with Fisher–Rao metric. Furthermore, we develop and detail all numerical tools on , namely, Levi–Civita connection, minimal geodesics, parallel transport, exponential map, and logarithm map. Then, we demonstrate that preliminary analysis results yield significant benefits in constructing an optimal cubic Hermite spline on as a nonlinear Riemannian manifold, precisely where conventional numerical methods fail.

Lower bounds for the trade-off between bias and mean absolute deviation

In nonparametric statistics, rate-optimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with regression function f in a class of β-Hölder smooth functions. Let ’worst-case’ refer to the supremum over all functions f in the Hölder class. It is shown that any estimator with worst-case bias ≲n−β/(2β+1)≕ψn must necessarily also have a worst-case mean absolute deviation that is lower bounded by ≳ψn. To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.

Coercive inequalities on Carnot groups: taming singularities

In the setting of Carnot groups, we propose an approach of taming singularities to get coercive inequalities. To this end, we develop a technique to introduce natural singularities in the energy function U in order to force one of the coercivity conditions. In particular, we explore explicit constructions of probability measures on Carnot groups which secure Poincaré and even Logarithmic Sobolev inequalities. As applications, we get analogues of the Dyson–Ornstein–Uhlenbeck model on the Heisenberg group and obtain results on the discreteness of the spectrum of related Markov generators.

Information Aggregation with Asymmetric Asset Payoffs

ABSTRACTWe study noisy aggregation of dispersed information in financial markets without imposing parametric restrictions on preferences, information, and return distributions. We provide a general characterization of asset returns by means of a risk‐neutral probability measure that features excess weight on tail risks. Moreover, we link excess weight on tail risks to observable moments such as forecast dispersion and accuracy, and argue that it provides a unified explanation for several prominent cross‐sectional return anomalies. Simple calibrations suggest the model can account for a significant fraction of empirical returns to skewness, returns to disagreement, and interaction effects between the two.