Mostrando recursos 1 - 20 de 1.803

  1. Addendum and corrigendum to “Randomized urn models revisited using stochastic approximation”

    Laruelle, Sophie; Pagès, Gilles
    This is a short addendum and corrigendum to the paper “Randomized Urn Models revisited using Stochastic Approximation” published in Annals of Applied Probability.

  2. Corrigendum Weak approximations for Wiener functionals [Ann. Appl. Probab. (2013) 23 1660–1691]

    Leão, Dorival; Ohashi, Alberto

  3. Addendum to “Optimal stopping under model uncertainty: Randomized stopping times approach”

    Belomestny, Denis; Krätschmer, Volker

  4. Evolving voter model on dense random graphs

    Basu, Riddhipratim; Sly, Allan
    In this paper, we examine a variant of the voter model on a dynamically changing network where agents have the option of changing their friends rather than changing their opinions. We analyse, in the context of dense random graphs, two models considered in Durrett et al. [Proc. Natl. Acad. Sci. USA 109 (2012) 3682–3687]. When an edge with two agents holding different opinion is updated, with probability $\frac{\beta }{n}$, one agent performs a voter model step and changes its opinion to copy the other, and with probability $1-\frac{\beta }{n}$, the edge between them is broken and reconnected to a new...

  5. Invariance principles for operator-scaling Gaussian random fields

    Biermé, Hermine; Durieu, Olivier; Wang, Yizao
    Recently, Hammond and Sheffield [Probab. Theory Related Fields 157 (2013) 691–719] introduced a model of correlated one-dimensional random walks that scale to fractional Brownian motions with long-range dependence. In this paper, we consider a natural generalization of this model to dimension $d\geq2$. We define a $\mathbb{Z}^{d}$-indexed random field with dependence relations governed by an underlying random graph with vertices $\mathbb{Z}^{d}$, and we study the scaling limits of the partial sums of the random field over rectangular sets. An interesting phenomenon appears: depending on how fast the rectangular sets increase along different directions, different random fields arise in the limit. In...

  6. Logarithmic tails of sums of products of positive random variables bounded by one

    Kołodziejek, Bartosz
    In this paper, we show under weak assumptions that for $R\stackrel{d}{=}1+M_{1}+M_{1}M_{2}+\cdots$, where $\mathbb{P}(M\in[0,1])=1$ and $M_{i}$ are independent copies of $M$, we have $\ln\mathbb{P}(R>x)\sim Cx\ln\mathbb{P}(M>1-1/x)$ as $x\to\infty$. The constant $C$ is given explicitly and its value depends on the rate of convergence of $\ln\mathbb{P}(M>1-1/x)$. Random variable $R$ satisfies the stochastic equation $R\stackrel{d}{=}1+MR$ with $M$ and $R$ independent, thus this result fits into the study of tails of iterated random equations, or more specifically, perpetuities.

  7. From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step

    Baar, Martina; Bovier, Anton; Champagnat, Nicolas
    We consider a model for Darwinian evolution in an asexual population with a large but nonconstant populations size characterized by a natural birth rate, a logistic death rate modeling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population ($K\to\infty$) size, rare mutations ($u\to0$) and small mutational effects ($\sigma\to0$), proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, for example, by Champagnat and Méléard, we take the three limits simultaneously, that is, $u=u_{K}$ and $\sigma=\sigma_{K}$, tend...

  8. The asymptotic variance of the giant component of configuration model random graphs

    Ball, Frank; Neal, Peter
    For a supercritical configuration model random graph, it is well known that, subject to mild conditions, there exists a unique giant component, whose size $R_{n}$ is $O(n)$, where $n$ is the total number of vertices in the random graph. Moreover, there exists $0<\rho \leq 1$ such that $R_{n}/n\stackrel{p}{\longrightarrow}\rho$ as $n\rightarrow \infty$. We show that for a sequence of well behaved configuration model random graphs with a deterministic degree sequence satisfying $0<\rho <1$; there exists $\sigma^{2}>0$, such that $\operatorname{var}(\sqrt{n}(R_{n}/n-\rho))\rightarrow \sigma^{2}$ as $n\rightarrow \infty$. Moreover, an explicit, easy to compute, formula is given for $\sigma^{2}$. This provides a key stepping stone for...

  9. Quickest detection problems for Bessel processes

    Johnson, Peter; Peskir, Goran
    Consider the motion of a Brownian particle that initially takes place in a two-dimensional plane and then after some random/unobservable time continues in the three-dimensional space. Given that only the distance of the particle to the origin is being observed, the problem is to detect the time at which the particle departs from the plane as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion of the particle in the plane. The solution is expressed in terms of a stopping time that...

  10. Optimal consumption under habit formation in markets with transaction costs and random endowments

    Yu, Xiang
    This paper studies the optimal consumption via the habit formation preference in markets with transaction costs and unbounded random endowments. To model the proportional transaction costs, we adopt the Kabanov’s multi-asset framework with a cash account. At the terminal time $T$, the investor can receive unbounded random endowments for which we propose a new definition of acceptable portfolios based on the strictly consistent price system (SCPS). We prove a type of super-hedging theorem using the acceptable portfolios which enables us to obtain the consumption budget constraint condition under market frictions. Following similar ideas in [Ann. Appl. Probab. 25 (2015) 1383–1419]...

  11. Scaling limit of the corrector in stochastic homogenization

    Mourrat, Jean-Christophe; Nolen, James
    In the homogenization of divergence-form equations with random coefficients, a central role is played by the corrector. We focus on a discrete space setting and on dimension $3$ and more. Under a minor smoothness assumption on the law of the random coefficients, we identify the scaling limit of the corrector, which is akin to a Gaussian free field. This completes the argument started in [Ann. Probab. 44 (2016) 3207–3233].

  12. The rounding of the phase transition for disordered pinning with stretched exponential tails

    Lacoin, Hubert
    The presence of frozen-in or quenched disorder in a system can often modify the nature of its phase transition. A particular instance of this phenomenon is the so-called rounding effect: it has been shown in many cases that the free energy curve of the disordered system at its critical point is smoother than that of the homogeneous one. In particular some disordered systems do not allow first-order transitions. We study this phenomenon for the pinning of a renewal with stretched-exponential tails on a defect line (the distribution $K$ of the renewal increments satisfies $K(n)\sim c_{K}\exp(-n^{\zeta})$, $\zeta\in(0,1)$) which has a first...

  13. Equivalence of ensembles for large vehicle-sharing models

    Fricker, Christine; Tibi, Danielle
    For a class of large closed Jackson networks submitted to capacity constraints, asymptotic independence of the nodes in normal traffic phase is proved at stationarity under mild assumptions, using a local limit theorem. The limiting distributions of the queues are explicit. In the Statistical Mechanics terminology, the equivalence of ensembles—canonical and grand canonical—is proved for specific marginals. The framework includes the case of networks with two types of nodes: single server/finite capacity nodes and infinite servers/infinite capacity nodes, that can be taken as basic models for bike-sharing systems. The effect of local saturation is modeled by generalized blocking and rerouting...

  14. A piecewise deterministic scaling limit of lifted Metropolis–Hastings in the Curie–Weiss model

    Bierkens, Joris; Roberts, Gareth
    In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410–414] a nonreversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis–Hastings (LMH). A scaling limit of the magnetization process in the Curie–Weiss model is derived for LMH, as well as for Metropolis–Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals $n^{1/2}$ for LMH, which should be compared to $n$ for MH. At the critical temperature, the required jump rate equals $n^{3/4}$ for LMH and $n^{3/2}$ for MH, in agreement with experimental results of Turitsyn, Chertkov and...

  15. Finite-length analysis on tail probability for Markov chain and application to simple hypothesis testing

    Watanabe, Shun; Hayashi, Masahito
    Using terminologies of information geometry, we derive upper and lower bounds of the tail probability of the sample mean for the Markov chain with finite state space. Employing these bounds, we obtain upper and lower bounds of the minimum error probability of the type-2 error under the exponential constraint for the error probability of the type-1 error in a simple hypothesis testing for a finite-length Markov chain, which yields the Hoeffding-type bound. For these derivations, we derive upper and lower bounds of cumulant generating function for Markov chain with finite state space. As a byproduct, we obtain another simple proof...

  16. On the connection between symmetric $N$-player games and mean field games

    Fischer, Markus
    Mean field games are limit models for symmetric $N$-player games with interaction of mean field type as $N\to \infty $. The limit relation is often understood in the sense that a solution of a mean field game allows to construct approximate Nash equilibria for the corresponding $N$-player games. The opposite direction is of interest, too: When do sequences of Nash equilibria converge to solutions of an associated mean field game? In this direction, rigorous results are mostly available for stationary problems with ergodic costs. Here, we identify limit points of sequences of certain approximate Nash equilibria as solutions to mean...

  17. Chi-square approximation by Stein’s method with application to Pearson’s statistic

    Gaunt, Robert E.; Pickett, Alastair M.; Reinert, Gesine
    This paper concerns the development of Stein’s method for chi-square approximation and its application to problems in statistics. New bounds for the derivatives of the solution of the gamma Stein equation are obtained. These bounds involve both the shape parameter and the order of the derivative. Subsequently, Stein’s method for chi-square approximation is applied to bound the distributional distance between Pearson’s statistic and its limiting chi-square distribution, measured using smooth test functions. In combination with the use of symmetry arguments, Stein’s method yields explicit bounds on this distributional distance of order $n^{-1}$.

  18. Optimal Skorokhod embedding given full marginals and Azéma–Yor peacocks

    Källblad, Sigrid; Tan, Xiaolu; Touzi, Nizar
    We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval $[0,1]$. The problem is related to the study of extremal martingales associated with a peacock (“process increasing in convex order,” by Hirsch, Profeta, Roynette and Yor [Peacocks and Associated Martingales, with Explicit Constructions (2011), Springer, Milan]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordère, Obłój, Spoida and Touzi [Ann. Appl. Probab. 26 (2016) 1–44]. Under...

  19. Reconstruction of a multidimensional scenery with a branching random walk

    Matzinger, Heinrich; Pachon, Angelica; Popov, Serguei
    We consider a $d$-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it observes. We show that up to equivalence the scenery can be reconstructed a.s. from the color record of all particles. To do so, we assume that the scenery has at least $2d+1$ colors which are i.i.d. with uniform probability. This is an improvement in comparison to Popov and Pachon [Stochastics 83 (2011) 107–116], where at each time the particles needed to see a window around their current position, and in Löwe and Matzinger [Ann. Appl. Probab....

  20. Kac’s walk on $n$-sphere mixes in $n\log n$ steps

    Pillai, Natesh S.; Smith, Aaron
    Determining the mixing time of Kac’s random walk on the sphere $\mathrm{S}^{n-1}$ is a long-standing open problem. We show that the total variation mixing time of Kac’s walk on $\mathrm{S}^{n-1}$ is between $\frac{1}{2}n\log(n)$ and $200n\log(n)$ for all $n$ sufficiently large. Our bound is thus optimal up to a constant factor, improving on the best-known upper bound of $O(n^{5}\log(n)^{2})$ due to Jiang [Ann. Appl. Probab. 22 (2012) 1712–1727]. Our main tool is a “non-Markovian” coupling recently introduced by the second author in [Ann. Appl. Probab. 24 (2014) 114–130] for obtaining the convergence rates of certain high dimensional Gibbs samplers in continuous...

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