This course focuses on Modeling, quantification, and analysis of uncertainty by teaching random variables, simple random processes and their probability distributions, Markov processes, limit theorems, elements of statistical inference, and decision making under uncertainty. This course extends the discrete probability learned in the discrete math class. It focuses on actual applications, and places little emphasis on proofs. A problem set based on identifying tumors using MRI (Magnetic...
Topic: probability
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Topic: Probability
Title from cover
Topic: PROBABILITY.
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76
Oct 6, 2015
10/15
by
Tovey, Craig A.
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Title from cover
Topic: PROBABILITY.
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7.0
Mar 1, 2022
03/22
by
Pfeiffer, Paul E
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xiii, 403 pages 25 cm
Topics: Probabilities, Probability, Probabilités, probability, Wahrscheinlichkeitsrechnung
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Nov 18, 2015
11/15
by
F. Smarandache
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In this paper one generalizes the classical probability and imprecise probability to the notion of “neutrosophic probability” in order to be able to model Heisenberg’s Uncertainty Principle of a particle’s behavior, Schrödinger’s Cat Theory, and the state of bosons which do not obey Pauli’s Exclusion Principle (in quantum physics). Neutrosophic probability is close related to neutrosophic logic and neutrosophic set, and etymologically derived from “neutrosophy”.
Topics: imprecise probability, neutrosophic probability, neutrosophic logic
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71
Nov 18, 2015
11/15
by
Florentin Smarandache
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In this paper one generalizes the classical probability and imprecise probability to the notion of “neutrosophic probability” in order to be able to model Heisenberg’s Uncertainty Principle of a particle’s behavior, Schrödinger’s Cat Theory, and the state of bosons which do not obey Pauli’s Exclusion Principle (in quantum physics). Neutrosophic probability is close related to neutrosophic logic and neutrosophic set, and etymologically derived from “neutrosophy”.
Topics: imprecise probability, neutrosophic probability, neutrosophic logic
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162
Nov 14, 2013
11/13
by
Erik Demaine
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Topics: Maths, Statistics and Probability, Probability, Mathematics
Source: http://www.flooved.com/reader/1568
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1 vol. (VIII-410 p.) ; 24 cm
Topics: Probabilities, Probability, Probabilités, probability, Processus stochastiques
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4.0
Jun 29, 2018
06/18
by
Marc Arnaudon; Michel Bonnefont; Aldéric Joulin
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We continue our investigation of the intertwining relations for Markov semigroups and extend the results of [9] to multi-dimensional diffusions. In particular these formulae entail new functional inequalities of Brascamp-Lieb type for log-concave distributions and beyond. Our results are illustrated by some classical and less classical examples.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1602.03836
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Jun 29, 2018
06/18
by
Michael Damron; Xuan Wang
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The Euclidean first-passage percolation (FPP) model of Howard and Newman is a rotationally invariant model of FPP which is built on a graph whose vertices are the points of homogeneous Poisson point process. It was shown that one has (stretched) exponential concentration of the passage time $T_n$ from $0$ to $n\mathbf{e}_1$ about its mean on scale $\sqrt{n}$, and this was used to show the bound $\mu n \leq \mathbb{E}T_n \leq \mu n + C\sqrt{n} (\log n)^a$ for $a,C>0$ on the discrepancy...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1605.06665
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Jun 30, 2018
06/18
by
Weining Kang; Kavita Ramanan
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Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection and the so-called submartingale problem. We introduce a general formulation of the submartingale problem for (obliquely) reflected diffusions in domains with piecewise C^2 boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the submartingale problem is equivalent to existence and uniqueness in law...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1412.0729
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Jun 30, 2018
06/18
by
Mohammud Foondun; Eulalia Nualart
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Consider the following equation $$\partial_t u_t(x)=\frac{1}{2}\partial _{xx}u_t(x)+\lambda \sigma(u_t(x))\dot{W}(t,\,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if $\lambda$ is large enough. But if $\lambda$ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1412.2343
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Jun 30, 2018
06/18
by
Andreas Milias-Argeitis; Mustafa Khammash
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We address the problem of Lyapunov function construction for a class of continuous-time Markov chains with affine transition rates, typically encountered in stochastic chemical kinetics. Following an optimization approach, we take advantage of existing bounds from the Foster-Lyapunov stability theory to obtain functions that enable us to estimate the region of high stationary probability, as well as provide upper bounds on moments of the chain. Our method can be used to study the stationary...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1412.7770
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Jun 28, 2018
06/18
by
Hanlin Yang
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We study a general class of quadratic BSDEs with terminal value in Lp for p>1. We prove a Lp-type estimate, existence, comparison theorem, uniqueness and stability result. In order to construct a solution, we use a combination of the localization procedure developed by Briand and Hu [6] and the monotone stability result. We point out that our existence result relies on rather weak assumptions on the generators. By additionally assuming monotonicity and convexity, we deduce the comparison...
Topics: Mathematics, Probability
Source: http://arxiv.org/abs/1506.08146
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Jun 30, 2018
06/18
by
Thomas Löbbe
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For two lacunary sequences $(M_{n,1})_{n\geq 2},(M_{n,2})_{n\geq 0}$ and suitable functions $f$ we introduce random matrix ensembles with \begin{equation*} X_{n,n'}=f(M_{n+n',1}x_1,M_{|n-n'|,2}x_2). \end{equation*} We prove weak convergence of the mean empirical eigenvalue distribution towards the semicircle law under some further number theoretic properties of the sequence $(M_{n,1})_{n\geq 1}$. Furthermore we give examples to show that even in this particular class of random matrix ensembles...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1408.2218
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4.0
Jun 30, 2018
06/18
by
Thomas Cass; Marcel Ogrodnik
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We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms [25] and prove a better-than-exponential tail estimate for the accummulated local p-variation functional, which has been introduced and studied in [17]. We comment on the significance of these estimates to a range of currently-studied problems, including the recent results of Chevyrev and Lyons in [18].
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1411.5189
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4.0
Jun 30, 2018
06/18
by
Mikolaj J. Kasprzak; Andrew B. Duncan; Sebastian J. Vollmer
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In (Barbour, 1990) foundations for diffusion approximation via Stein's method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein's method. A semigroup argument is used therein to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in (Barbour, 1990), the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on D[0,1] growing slower than a cubic,...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1702.03130
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4.0
Jun 30, 2018
06/18
by
Jinlong Wei; Jinqiao Duan; Guangying Lv
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We consider a transport-diffusion equation with L\'{e}vy noises and H\"{o}lder continuous coefficients. By using the heat kernel estimates, we derive the Schauder estimates for the mild solutions. Moreover, when the transport term vanishes and $p=2$, we show that the H\"{o}lder index in space variable is optimal.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1704.05582
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Jun 30, 2018
06/18
by
Ingo Steinwart
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We establish a Karhunen-Lo`eve expansion for generic centered, second order stochastic processes, which does not rely on topological assumptions. We further investigate in which norms the expansion converges and derive exact average rates of convergence for these norms. For Gaussian processes we additionally prove certain sharpness results in terms of the norm. Moreover, we show that the generic Karhunen-Lo`eve expansion can in some situations be used to construct reproducing kernel Hilbert...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1403.1040
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Jun 30, 2018
06/18
by
Maria Vlasiou; Bert Zwart
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We consider a model describing the waiting time of a server alternating between two service points. This model is described by a Lindley-type equation. We are interested in the time-dependent behaviour of this system and derive explicit expressions for its time-dependent waiting-time distribution, the correlation between waiting times, and the distribution of the cycle length. Since our model is closely related to Lindley's recursion, we compare our results to those derived for Lindley's...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1404.5587
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3.0
Jun 30, 2018
06/18
by
Hua-Huai Chern; Hsien-Kuei Hwang; Tsung-Hsi Tsai
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A detailed study is made of the number of occupied seats in an unfriendly seating scheme with two rows of seats. An unusual identity is derived for the probability generating function, which is itself an asymptotic expansion. The identity implies particularly a local limit theorem with optimal convergence rate. Our approach relies on the resolution of Riccati equations.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1406.0614
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3.0
Jun 30, 2018
06/18
by
Patrizia Berti; Luca Pratelli; Pietro Rigo
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Let $L$ be a convex cone of real random variables on the probability space $(\Omega,\mathcal{A},P_0)$. The existence of a probability $P$ on $\mathcal{A}$ such that $$ P \sim P_0,\quad E_P \abs{X} < \infty\, \text{ and } \, E_P(X) \leq 0\, \text{ for all }X \in L $$ is investigated. Two results are provided. In the first, $P$ is a finitely additive probability, while $P$ is $\sigma$-additive in the second. If $L$ is a linear space then $-X\in L$ whenever $X\in L$, so that $E_P(X)\leq 0$...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1402.3570
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4.0
Jun 30, 2018
06/18
by
Georgiy Shevchenko; Lauri Viitasaari
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We study integral representations of random variables with respect to general H\"older continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distribution. If in addition the random variable is a final value of an adapted H\"older continuous process, then it can be represented as a...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1404.7518
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Jun 30, 2018
06/18
by
Noufel Frikha; Lorick Huang
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We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in [Frikha2013]. This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte Carlo linear estimator (introduced in [Talay & Tubaro 1990] and deeply studied in [Pag{\`e}s 2007]) to the framework of stochastic optimization by means of stochastic approximation algorithm. We notably apply the method to the estimation of the...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1409.4748
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Jun 29, 2018
06/18
by
Ibrahima Dramé; Etienne Pardoux; Ahmadou Bamba Sow
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We study the exploration (or height) process of a continuous time non-binary Galton-Watson random tree, in the subcritical, critical and supercritical cases. Thus we consider the branching process in continuous time (Z_{t})_{t\geq 0}, which describes the number of offspring alive at time t. We then renormalize our branching process and exploration process, and take the weak limit as the size of the population tends to infinity. Finally we deduce a Ray-Knight representation.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1602.02007
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Jun 29, 2018
06/18
by
Komla Domelevo; Stefanie Petermichl
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We prove a sharp weighted $L^p$ estimate of $Y^{\ast}$ with respect to $X$. Here $Y$ and $X$ are uniformly integrable cadlag Hilbert space valued martingales and $Y$ differentially subordinate to $X$ via the square bracket process. The proof is via an iterated stopping procedure and self similarity argument known as 'sparse domination'. We point out that in this generality, the special case $Y = X$ addresses a question raised by Bonami--Lepingle. The proof via sparse domination for the latter...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1607.06319
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Jun 29, 2018
06/18
by
Gauthier Dierickx; Uwe Einmahl
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We provide an improved version of the Darling-Erd\"os theorem for sums of i.i.d. random variables with mean zero and finite variance. We extend this result to multidimensional random vectors. Our proof is based on a new strong invariance principle in this setting which has other applications as well such as an integral test refinement of the multidimensional Hartman-Wintner LIL. We also identify a borderline situation where one has weak convergence to a shifted version of the standard...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1608.04549
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3.0
Jun 29, 2018
06/18
by
Mathias Beiglboeck; Nicolas Juillet
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A classical result of Strassen asserts that given probabilities $\mu, \nu$ on the real line which are in convex order, there exists a \emph{martingale coupling} with these marginals, i.e.\ a random vector $(X_1,X_2)$ such that $X_1\sim \mu, X_2\sim \nu$ and $E[X_2|X_1]=X_1$. Remarkably, it is a non trivial problem to construct particular solutions to this problem. In this article, we introduce a family of such martingale couplings, each of which admits several characterizations in terms of...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1609.03340
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Jun 29, 2018
06/18
by
Damjan Škulj
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Coherent lower previsions are general models of uncertainty in probability distributions. They are often approximated by some less general models, such as coherent lower probabilities or in terms of some finite set of constraints. The amount of error induced by the approximations has been often neglected in the literature, despite the fact that it can be quite substantial. The aim of this paper is to provide a general method for estimating the exact degree of error for given approximations of...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1609.05661
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Jun 29, 2018
06/18
by
Yuichi Shiozawa
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We determine the decay rate of the bottom crossing probability for symmetric jump processes under the condition on heat kernel estimates. Our results are applicable to symmetric stable-like processes and stable-subordinated diffusion processes on a class of (unbounded) fractals and fractal-like spaces.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1609.06812
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Jun 29, 2018
06/18
by
Balazs Rath
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We modify the definition of Aldous' multiplicative coalescent process and introduce the multiplicative coalescent with linear deletion (MCLD). A state of this process is a square-summable decreasing sequence of cluster sizes. Pairs of clusters merge with a rate equal to the product of their sizes and clusters are deleted with a rate linearly proportional to their size. We prove that the MCLD is a Feller process. This result is a key ingredient in the description of scaling limits of the...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1610.00021
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Jun 29, 2018
06/18
by
Antoine-Marie Bogso; Patrice Takam Soh
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We introduce the notion of weak decreasing stochastic (WDS) ordering for real-valued processes with negative means, which, to our knowledge, has not been studied before. Thanks to Madan-Yor's argument, it follows that the WDS ordering is a necessary and sufficient condition for a process with negative mean to be embeddable in a standard Brownian motion by the Cox and Hobson extension of the Az\'ema-Yor algorithm. Since the decreasing stochastic order is stronger than the WDS order, then, for...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1610.02190
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3.0
Jun 29, 2018
06/18
by
Andrey Pilipenko
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We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1611.02841
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Jun 29, 2018
06/18
by
Julien Bureaux; Nathanaël Enriquez
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An equivalence is proven between the Riemann Hypothesis and the speed of convergence to 1/zeta(2) of the probability that two independent random variables following the same geometric distribution are coprime integers, when the parameter of the distribution goes to 0.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1612.03700
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3.0
Jun 30, 2018
06/18
by
Zhen-Qing Chen; Lidan Wang
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In this paper, we study inverse local time at 0 of one-dimensional reflected diffusions on $[0, \infty)$, and establish a comparison principle for inverse local times. Applications to Green function estimates for non-local operators are given.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1702.04095
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3.0
Jun 30, 2018
06/18
by
Christophe Sabot; Xiaolin Zeng
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It is well-known that the first hitting time of 0 by a negatively drifted one dimensional Brownian motion starting at positive initial position has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3-dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1704.05394
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Jun 28, 2018
06/18
by
Mohamed Bouali
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We investigate the product of $n$ complex non-Hermitian, independent random matrices, each of size $N_i\times N_{i+1}$ $(i=1,...,n)$, with independent identically distributed Cauchy entries (Cauchy-Lorentz matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Cauchy-Lorentz matrix, but with weight given by a Meijer G-function depending on $n$ and $N_i$.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1512.08179
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Jun 29, 2018
06/18
by
Changqing Liu
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Concentration inequalities, which prove to be very useful in a variety of fields, provide fairly tight bounds for large deviation probability while central limit theorem (CLT) describes the asymptotic distribution around the mean (within scope of $\sqrt{n}$ order). Harris (1963) conjectured that for a supercritical branching random walk (BRW) of i.i.d offspring and iid displacement, population's positions in $nth$ generation approach to Gaussian distribution --- central limit theorem. This...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1604.00056
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4.0
Jun 29, 2018
06/18
by
Dariusz Buraczewski; Piotr Dyszewski; Konrad Kolesko
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We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a~sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure $\mu$ fluctuates. For any given point $x$, denoting by $B_n(x)$ the ball of radius $2^{-n}$ centered around $x$, we present optimal lower and upper estimates of $\mu(B_n(x))$ as $n \to \infty$.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1604.03328
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Jun 29, 2018
06/18
by
Hirofumi Osada; Hideki Tanemura
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In this note we review recent results on existence and uniqueness of solutions of infinite-dimensional stochastic differential equations describing interacting Brownian motions on $\R^d$.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1605.04417
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Jun 29, 2018
06/18
by
Stephen B. Connor; Richard Pymar
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We introduce a natural extension of the exclusion process to hypergraphs and prove an upper bound for its mixing time. In particular we show the existence of a constant $C$ such that for any connected hypergraph $G$ within some class (which includes regular hypergraphs), the $\varepsilon$-mixing time of the exclusion process on $G$ with any feasible number of particles can be upper-bounded by $CT_{\text{EX}(2,G)}\log(|V|/\varepsilon)$, where $|V|$ is the number of vertices in $G$ and...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1606.02703
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Jun 29, 2018
06/18
by
Tobias Stüwe; Andrea Barth
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This work considers weak approximations of stochastic partial differential equations (SPDEs) driven by L\'evy noise. The SPDEs at hand are parabolic with additive noise processes. A weak-convergence rate for the corresponding Galerkin approximation is derived. The convergence result is derived by use of the Malliavin derivative rather then the common approach via the Kolmogorov backward equation.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1603.02422
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Jun 29, 2018
06/18
by
Peter Eichelsbacher; Thomas Kriecherbauer; Katharina Schüler
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We prove precise deviations results in the sense of Cram\'er and Petrov for the upper tail of the distribution of the maximal value for a special class of determinantal point processes that play an important role in random matrix theory. Here we cover all three regimes of moderate, large and superlarge deviations for which we determine the leading order description of the tail probabilities. As a corollary of our results we identify the region within the regime of moderate deviations for which...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1603.04328
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Jun 29, 2018
06/18
by
Dirk Blömker; Philipp Wacker; Thomas Wanner
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We study the maximum norm behavior of $L^2$-normalized random Fourier cosine series with a prescribed large wave number. Precise bounds of this type are an important technical tool in estimates for spinodal decomposition, the celebrated phase separation phenomenon in metal alloys. We derive rigorous asymptotic results as the wave number converges to infinity, and shed light on the behavior of the maximum norm for medium range wave numbers through numerical simulations. Finally, we develop a...
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1603.04300
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4.0
Jun 29, 2018
06/18
by
Jiang Zhou; Lan Wu
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For a refracted spectrally negative Levy process, we find some new and fantastic formulas for its q-potential measures without killing. Unlike previous results, which are written in terms of the known q-scale functions, our formulas are free of the q-scale functions. This makes our results become extremely important since it is likely that our formulas also hold for a general refracted Levy process.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1603.04944
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Jun 29, 2018
06/18
by
Servet Martinez
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We give a closed form of the discrete-time evolution of a recombination transformation in population genetics. This decomposition allows to define a Markov chain in a natural way. We describe the geometric decay rate to the limit distribution, and the quasi-stationary behavior when conditioned to the event that the chain does not hit the limit distribution.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1603.07201
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5.0
Jun 29, 2018
06/18
by
Assia Boumahdaf
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We consider an $M/M/1$ queueing system with impatient customers with multiple and single vacations. It is assumed that customers are impatient whenever the state of the server. We derive the probability generating functions of the number of customers in the system and we obtain some performance measures.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1604.02449
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Jun 29, 2018
06/18
by
Hao Wu
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We derive the alternating arm exponents of critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions $(\ominus\oplus)$, $(\ominus\text{free})$ and $(\text{free}\text{free})$, and the alternating interior arm exponents.
Topics: Probability, Mathematics
Source: http://arxiv.org/abs/1605.00985
