Search ResultsShowing 1-20 of 158
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arXiv:2502.00794 (Published 2025-02-02)
Some Properties of The Finitely Additive Vector Integral
We prove some results concerning the finitely additive, vector integral of Bochner and Pettis and their representation over a countably additive probability space. Applications to convergence of vector valued martingales and to the non compact Choquet theorem are provided.
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arXiv:2501.05061 (Published 2025-01-09)
Properties of the one-component Coulomb gas on a sphere with two macroscopic external charges
Comments: 38 pages, 5 figuresThe one-component Coulomb gas on the sphere, consisting on $N$ unit charges interacting via a logarithmic potential, and in the presence of two external charges each of strength proportional to $N$, is considered. There are two spherical caps naturally associated with the external charges, giving rise to two distinct phases depending on them not overlapping (post-critical) or overlapping (pre-critical). The equilibrium measure in the post-critical phase is known from earlier work. We determine the equilibrium measure in the pre-critical phase using a particular conformal map, with the parameters therein specified in terms of a root of a certain fourth order polynomial. This is used to determine the exact form of the electrostatic energy for the pre-critical phase. Using a duality relation from random matrix theory, the partition function for the Coulomb gas at the inverse temperature $\beta = 2$ can be expanded for large $N$ in the post-critical phase, and in a scaling region of the post and pre-critical boundary. For the pre-critical phase, the duality identity implies a relation between two electrostatic energies, one for the present sphere system, and the other for a certain constrained log-gas relating to the Jacobi unitary ensemble.
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arXiv:2412.10485 (Published 2024-12-13)
Catene ideali con numero fissato di auto-intersezioni
Comments: MS thesis, Simone Franchini, Sapienza Universita di Roma, 85 pages, 21 figures, year: 2011, language: ItalianCategories: cond-mat.stat-mech, math.PRKeywords: numero fissato di auto-intersezioni, catene ideali, random walks, study ideal chains, propertiesTags: dissertationIn this thesis we study in detail the self-intersection properties of Random Walks. Although notoriously hard to tackle, these properties are crucially related to the excluded-volume effect and other central features of real polymers. Our main purpose will be to study ideal chains (Random Walks) on lattice where the ratio between the number of self-intersections and the total length is fixed to some number.
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A Novel Unit Distribution Named As Median Based Unit Rayleigh (MBUR): Properties and Estimations
Comments: It is a new distribution with some of its properties and methods of estimationThe importance of continuously emerging new distribution is a mandate to understand the world and environment surrounding us. In this paper, the author will discuss a new distribution defined on the interval (0,1) as regards the methodology of deducing its PDF, some of its properties and related functions. A simulation and real data analysis will be highlighted.
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arXiv:2409.04798 (Published 2024-09-07)
Weighted Sub-fractional Brownian Motion Process: Properties and Generalizations
Comments: 53 pages, 12 figures, 1 tableCategories: math.PRIn this paper, we present several path properties, simulations, inferences, and generalizations of the weighted sub-fractional Brownian motion. A primary focus is on the derivation of the covariance function $R_{f,b}(s,t)$ for the weighted sub-fractional Brownian motion, defined as: \begin{equation*} R_{f,b}(s,t) = \frac{1}{1-b} \int_{0}^{s \wedge t} f(r) \left[(s-r)^{b} + (t-r)^{b} - (t+s-2r)^{b}\right] dr, \end{equation*} where $f:\mathbb{R}_{+} \to \mathbb{R}_{+}$ is a measurable function and $b\in [0,1)\cup(1,2]$. This covariance function $R_{f,b}(s,t)$ is used to define the centered Gaussian process $\zeta_{t,f,b}$, which is the weighted sub-fractional Brownian motion. Furthermore, if there is a positive constant $c$ and $a \in (-1,\infty)$ such that $0 \leq f(u) \leq c u^{a}$ on $[0,T]$ for some $T>0$. Then, for $b \in (0,1)$, $\zeta_{t,f,b}$ exhibits infinite variation and zero quadratic variation, making it a non-semi-martingale. On the other hand, for $b \in (1,2]$, $\zeta_{t,f,b}$ is a continuous process of finite variation and thus a semi-martingale and for $b=0$ the process $\zeta_{t,f,0}$ is a square integrable continuous martingale. We also provide inferential studies using maximum likelihood estimation and perform simulations comparing various numerical methods for their efficiency in computing the finite-dimensional distributions of $\zeta_{t,f,b}$. Additionally, we extend the weighted sub-fractional Brownian motion to $\mathbb{R}^d$ by defining new covariance structures for measurable, bounded sets in $\mathbb{R}^d$. Finally, we define a stochastic integral with respect to $\zeta_{t,f,b}$ and introduce both the weighted sub-fractional Ornstein-Uhlenbeck process and the geometric weighted sub-fractional Brownian motion.
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arXiv:2408.13600 (Published 2024-08-24)
Some properties on the reversibility and the linear response theory of Langevin dynamics
Comments: 40 pagesLinear response theory is a fundamental framework studying the macroscopic response of a physical system to an external perturbation. This paper focuses on the rigorous mathematical justification of linear response theory for Langevin dynamics. We give some equivalent characterizations for reversible overdamped/underdamped Langevin dynamics, which is the unperturbed reference system. Then we clarify sufficient conditions for the smoothness and exponential convergence to the invariant measure for the overdamped case. We also clarify those sufficient conditions for the underdamped case, which corresponds to hypoellipticity and hypocoercivity. Based on these, the asymptotic dependence of the response function on the small perturbation is proved in both finite and infinite time horizons. As applications, Green-Kubo relations and linear response theory for a generalized Langevin dynamics are also proved in a rigorous fashion.
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arXiv:2406.08043 (Published 2024-06-12)
Some Properties of the Plaquette Random-Cluster Model
Comments: 25 pagesWe show that the $i$-dimensional plaqutte random-cluster model with coefficients in $\mathbb{Z}_q$ is dual to a $(d-i)$-dimensional plaquette random cluster model. In addition, we explore boundary conditions, infinite volume limits, and uniqueness for these models. For previously known results, we provide new proofs that rely more on the tools of algebraic topology.
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arXiv:2403.15275 (Published 2024-03-22)
Some properties of stable snakes
Comments: 35 pagesCategories: math.PRWe prove some technical results relating to the Brownian snake on a stable L\'evy tree. This includes some estimates on the range of the snake, estimates on its occupation measure around its minimum and also a proof of the fact that the snake and the height function of the associated tree have no common increase points.
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Maxentropy completion and properties of some partially defined Stationary Markov chains
Comments: Section 4 modified, misprints correctedCategories: math.PRWe consider a stationary Markovian evolution with values on a disjointly partitioned set space $I\sqcup {\cal E}$. The evolution is visible (in the sense of knowing the transition probabilities) on the states in $I$ but not for the states in ${\cal E}$. One only knows some partial information on the transition probabilities on ${\cal E}$, the input and output transition probabilities and some constraints of the transition probabilities on ${\cal E}$. Under some conditions we supply the transition probabilities on ${\cal E}$ that satisfies the maximum entropy principle.
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arXiv:2403.08805 (Published 2024-02-06)
Properties of Shannon and Rényi entropies of the Poisson distribution as the functions of intensity parameter
Comments: 23 pages, 8 figuresWe consider two types of entropy, namely, Shannon and R\'{e}nyi entropies of the Poisson distribution, and establish their properties as the functions of intensity parameter. More precisely, we prove that both entropies increase with intensity. While for Shannon entropy the proof is comparatively simple, for R\'{e}nyi entropy, which depends on additional parameter $\alpha>0$, we can characterize it as nontrivial. The proof is based on application of Karamata's inequality to the terms of Poisson distribution.
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arXiv:2312.04951 (Published 2023-12-08)
On some characterizations of probability distributions based on maxima or minima of some families of dependent random variables
Subjects: 62M10Most of the characterizations of probability distributions are based on properties of functions of possibly independent random variables. We investigate characterizations of probability distributions through properties of minima or maxima of max-independent, min-independent and quasi-independent random variables generalizing the results from independent random variables of Kotlarski (1978), Prakasa Rao (1992) and Klebanov (1973).
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arXiv:2311.15184 (Published 2023-11-26)
Factorial moments of the Poisson distribution of order $k$
Comments: 6 pagesThe factorial moments of the standard Poisson distribution are well known. The present note presents an explicit combinatorial sum for the factorial moments of the Poisson distribution of order $k$. Unlike the standard Poisson distribution (the case $k=1$), for $k>1$ the structure of the factorial moments is much more complicated. Some properties of the factorial moments of the Poisson distribution of order $k$ are elucidated in this note.
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arXiv:2310.00827 (Published 2023-10-02)
Analytical proofs for the properties of the probability mass function of the Poisson distribution of order $k$
Comments: 19 pages, 4 figuresCategories: math.PRThe Poisson distribution of order $k$ is a special case of a compound Poisson distribution. For $k=1$ it is the standard Poisson distribution. Our main result is a proof that for sufficiently small values of the rate parameter $\lambda$, the probability mass function (pmf) decreases monotonically for all $n\ge k$ (it is known that the pmf increases strictly for $1\le n \le k$, for fixed $k\ge2$ and all $\lambda>0$). The second main result is a partial proof that the difference (mean $-$ mode) does not exceed $k$. The term `partial proof' signifies that the derivation is conditional on an assumption which, although plausible and supported by numerical evidence, is as yet not proved. This note also presents new inequalities, and sharper bounds for some published inequalities, for the Poisson distribution of order $k$.
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arXiv:2309.10415 (Published 2023-09-19)
The Properties of Fractional Gaussian Process and Their Applications
Categories: math.PRThe process $(G_t)_{t\in[0,T]}$ is referred to as a fractional Gaussian process if the first-order partial derivative of the difference between its covariance function and that of the fractional Brownian motion $(B^H_t)_{t\in[0,T ]}$ is a normalized bounded variation function. We quantify the relation between the associated reproducing kernel Hilbert space of $(G)$ and that of $(B^H)$. Seven types of Gaussian processes with non-stationary increments in the literature belong to it. In the context of applications, we demonstrate that the Gladyshev's theorem holds for this process, and we provide Berry-Ess\'{e}en upper bounds associated with the statistical estimations of the ergodic fractional Ornstein-Uhlenbeck process driven by it. The second application partially builds upon the idea introduced in \cite{BBES 23}, where they assume that $(G)$ has stationary increments. Additionally, we briefly discuss a variant of this process where the covariance structure is not entirely linked to that of the fractional Brownian motion.
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arXiv:2309.08475 (Published 2023-09-15)
Doeblin Coefficients and Related Measures
Comments: 36 pagesDoeblin coefficients are a classical tool for analyzing the ergodicity and exponential convergence rates of Markov chains. Propelled by recent works on contraction coefficients of strong data processing inequalities, we investigate whether Doeblin coefficients also exhibit some of the notable properties of canonical contraction coefficients. In this paper, we present several new structural and geometric properties of Doeblin coefficients. Specifically, we show that Doeblin coefficients form a multi-way divergence, exhibit tensorization, and possess an extremal trace characterization. We then show that they also have extremal coupling and simultaneously maximal coupling characterizations. By leveraging these characterizations, we demonstrate that Doeblin coefficients act as a nice generalization of the well-known total variation (TV) distance to a multi-way divergence, enabling us to measure the "distance" between multiple distributions rather than just two. We then prove that Doeblin coefficients exhibit contraction properties over Bayesian networks similar to other canonical contraction coefficients. We additionally derive some other results and discuss an application of Doeblin coefficients to distribution fusion. Finally, in a complementary vein, we introduce and discuss three new quantities: max-Doeblin coefficient, max-DeGroot distance, and min-DeGroot distance. The max-Doeblin coefficient shares a connection with the concept of maximal leakage in information security; we explore its properties and provide a coupling characterization. On the other hand, the max-DeGroot and min-DeGroot measures extend the concept of DeGroot distance to multiple distributions.
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arXiv:2307.10352 (Published 2023-07-19)
Properties of Discrete Sliced Wasserstein Losses
The Sliced Wasserstein (SW) distance has become a popular alternative to the Wasserstein distance for comparing probability measures. Widespread applications include image processing, domain adaptation and generative modelling, where it is common to optimise some parameters in order to minimise SW, which serves as a loss function between discrete probability measures (since measures admitting densities are numerically unattainable). All these optimisation problems bear the same sub-problem, which is minimising the Sliced Wasserstein energy. In this paper we study the properties of $\mathcal{E}: Y \longmapsto \mathrm{SW}_2^2(\gamma_Y, \gamma_Z)$, i.e. the SW distance between two uniform discrete measures with the same amount of points as a function of the support $Y \in \mathbb{R}^{n \times d}$ of one of the measures. We investigate the regularity and optimisation properties of this energy, as well as its Monte-Carlo approximation $\mathcal{E}_p$ (estimating the expectation in SW using only $p$ samples) and show convergence results on the critical points of $\mathcal{E}_p$ to those of $\mathcal{E}$, as well as an almost-sure uniform convergence. Finally, we show that in a certain sense, Stochastic Gradient Descent methods minimising $\mathcal{E}$ and $\mathcal{E}_p$ converge towards (Clarke) critical points of these energies.
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arXiv:2307.09798 (Published 2023-07-19)
Mixed Poisson process with Max-U-Exp mixing variable -- Working version
Comments: Presented on 49th International Conference Applications of Mathematics in Engineering and Economics, 10 - 16 June 2023, Sozopol, BulgariaCategories: math.PRSubjects: 60J05Keywords: max-u-exp mixing variable, working version, properties, probability density function, work definesTags: conference paperThis work defines and investigates the properties of the Max-U-Exp distribution. The method of moments is applied in order to estimate its parameters. Then, by using the previous general theory about Mixed Poisson processes, developed by Grandel (1997), and Karlis and Xekalaki (2005), and analogously to Jordanova et al. (2023), and Jordanova and Stehlik (2017) we define and investigate the properties of the new random vectors and random variables, which are related with this particular case of a Mixed Poisson process. Exp-Max-U-Exp distribution is defined and thoroughly investigated. It arises in a natural way as a distribution of the inter-arrival times in the Mixed Poisson process with Max-U-Exp mixing variable. The distribution of the renewal moments is called Erlang-Max-U-Exp and is defined via its probability density function. Investigation of its properties follows. Finally, the corresponding Mixed Poisson process with Max-U-Exp mixing variable is defined. Its finite dimensional and conditional distributions are found and their numerical characteristics are determined.
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arXiv:2307.00545 (Published 2023-07-02)
Properties and conjectures regarding discrete renewal sequences
Categories: math.PRIn this work we review and derive some elementary properties of the discrete renewal sequences based on a positive, finite and integer-valued random variable. Our results consider these sequences as dependent on the probability masses of the underlying random variable. In particular we study the minima and the maxima of these sequences and prove that they are attained for indices of the sequences smaller or equal than the support of the underlying random variable. Noting that the minimum itself is a minimum of multi-variate polynomials we conjecture that one universal polynomial envelopes the minimum from below and that it is maximal in some sense and largest in another. We prove this conjecture in a special case.
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arXiv:2305.01297 (Published 2023-05-02)
Properties of Non-Equilibrium Steady States for the non-linear BGK equation on the torus
Comments: 41 pagesWe study the non-linear BGK model in 1d coupled to a spatially varying thermostat. We show existence, local uniqueness and linear stability of a steady state when the linear coupling term is large compared to the non-linear self interaction term. This model possesses a non-explicit spatially dependent non-equilibrium steady state. We are able to successfully use hypocoercivity theory in this case to prove that the linearised operator around this steady state posesses a spectral gap.
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arXiv:2304.01271 (Published 2023-04-03)
Some properties of Markov chains on the free group $\mathbb F_2$
Comments: 14 pages, 2 figuresRandom walks cannot, in general, be pushed forward by quasi-isometries. Tame Markov chains were introduced as a `quasi-isometry invariant' are a generalization of random walks. In this paper, we construct several examples of tame Markov chains on the free group exhibiting `exotic' behaviour; one, where the drift is not well defined and one where the drift is well defined but the Central Limit Theorem does not hold. We show that this is not a failure of the notion of tame Markov chain, but rather that any quasi-isometry invariant theory that generalizes random walks will include examples without well-defined drift.