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Abstract: Abstract The Half-Plane Half-Comb walk is a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We prove that the probability that this walk returns to the origin in 2N steps is asymptotically equal to \(2/(\pi N).\) As a consequence, we prove strong laws and a limit distribution for the local time. PubDate: 2022-06-01

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Abstract: Abstract We study a process of generating random positive integer weight sequences \(\{ W_n \}\) where the gaps between the weights \(\{ X_n = W_n - W_{n-1} \}\) are i.i.d. positive integer-valued random variables. The main result of the paper is that if the gap distribution has a moment generating function with large enough radius of convergence, then the weight sequence is almost surely asymptotically m-complete for every \(m\ge 2\) , i.e. every large enough multiple of the greatest common divisor (gcd) of gap values can be written as a sum of m distinct weights for any fixed \(m \ge 2\) . Under the weaker assumption of finite \(\frac{1}{2}\) -moment for the gap distribution, we also show the simpler result that, almost surely, the resulting weight sequence is asymptotically complete, i.e. all large enough multiples of the gcd of the possible gap values can be written as a sum of distinct weights. PubDate: 2022-06-01

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Abstract: Abstract The aim of this paper is to develop tractable large deviation approximations for the empirical measure of a small noise diffusion. The starting point is the Freidlin–Wentzell theory, which shows how to approximate via a large deviation principle the invariant distribution of such a diffusion. The rate function of the invariant measure is formulated in terms of quasipotentials, quantities that measure the difficulty of a transition from the neighborhood of one metastable set to another. The theory provides an intuitive and useful approximation for the invariant measure, and along the way many useful related results (e.g., transition rates between metastable states) are also developed. With the specific goal of design of Monte Carlo schemes in mind, we prove large deviation limits for integrals with respect to the empirical measure, where the process is considered over a time interval whose length grows as the noise decreases to zero. In particular, we show how the first and second moments of these integrals can be expressed in terms of quasipotentials. When the dynamics of the process depend on parameters, these approximations can be used for algorithm design, and applications of this sort will appear elsewhere. The use of a small noise limit is well motivated, since in this limit good sampling of the state space becomes most challenging. The proof exploits a regenerative structure, and a number of new techniques are needed to turn large deviation estimates over a regenerative cycle into estimates for the empirical measure and its moments. PubDate: 2022-06-01

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Abstract: Abstract In this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes. PubDate: 2022-06-01

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Abstract: We build on the formalism developed in Ernst et al. (First order covariance inequalities via Stein’s method, 2019) to propose new representations of solutions to Stein equations. We provide new uniform and nonuniform bounds on these solutions (a.k.a. Stein factors). We use these representations to obtain representations for differences between expectations in terms of solutions to the Stein equations. We apply these to compute abstract Stein-type bounds on Kolmogorov, total variation and Wasserstein distances between arbitrary distributions. We apply our results to several illustrative examples and compare our results with current literature on the same topic, whenever possible. In all occurrences our results are competitive. PubDate: 2022-06-01

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Abstract: Abstract We develop Stein’s method for \(\alpha \) -stable approximation with \(\alpha \in (0,1]\) , continuing the recent line of research by Xu (Ann Appl Probab 29(1):458–504, 2019) and Chen et al. (J Theor Probab, 2018. https://doi.org/10.1007/s10959-020-01004-1) in the case \(\alpha \in (1,2)\) . The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, the proof strategy is significantly different from that in the integrable case. We rely on fine regularity estimates of the solution to Stein’s equation established in this paper. PubDate: 2022-06-01

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Abstract: Abstract The goal of this paper is to approximate two kinds of McKean–Vlasov stochastic differential equations (SDEs) with irregular coefficients via weakly interacting particle systems. More precisely, propagation of chaos and convergence rate of Euler–Maruyama scheme associated with the consequent weakly interacting particle systems are investigated for McKean–Vlasov SDEs, where (1) the diffusion terms are Hölder continuous by taking advantage of Yamada–Watanabe’s approximation approach and (2) the drifts are Hölder continuous by freezing distributions followed by invoking Zvonkin’s transformation trick. PubDate: 2022-06-01

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Abstract: Abstract We give two asymptotic results for the empirical distance covariance on separable metric spaces without any iid assumption on the samples. In particular, we show the almost sure convergence of the empirical distance covariance for any measure with finite first moments, provided that the samples form a strictly stationary and ergodic process. We further give a result concerning the asymptotic distribution of the empirical distance covariance under the assumption of absolute regularity of the samples and extend these results to certain types of pseudometric spaces. In the process, we derive a general theorem concerning the asymptotic distribution of degenerate V-statistics of order 2 under a strong mixing condition. PubDate: 2022-06-01

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Abstract: Abstract We construct graphs (trees of bounded degree) on which the contact process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and \(\lambda _c({\mathbb {Z}})\) , the critical rate of the one-dimensional contact process. We exhibit both graphs in which the process at this target critical value survives (locally) and graphs where it dies out (globally). PubDate: 2022-06-01

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Abstract: Abstract This paper contributes to the study of stochastic processes of the class \((\Sigma )\) . First, we extend the notion of the above-mentioned class to càdlàg semi-martingales, whose finite variation part is considered càdlàg instead of continuous. Thus, we present some properties and propose a method to characterize such stochastic processes. Second, we investigate continuous processes of the class \((\Sigma )\) . More precisely, we derive a series of new characterization results. In addition, we construct solutions for skew Brownian motion equations using continuous stochastic processes of the class \((\Sigma )\) . PubDate: 2022-06-01

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Abstract: Abstract Consider \(n\) nodes \(\{X_i\}_{1 \le i \le n}\) independently distributed in the unit square \(S,\) each according to a density \(f\) , and let \(K_n\) be the complete graph formed by joining each pair of nodes by a straight line segment. For every edge \(e\) in \(K_n\) , we associate a weight \(w(e)\) that may depend on the individual locations of the endvertices of \(e\) and is not necessarily a power of the Euclidean length of \(e.\) Denoting \(\mathrm{TSP}_n\) to be the minimum weight of a spanning cycle of \(K_n\) corresponding to the travelling salesman problem (TSP) and assuming an equivalence condition on the weight function \(w(\cdot ),\) we prove that \(\mathrm{TSP}_n\) appropriately scaled and centred converges to zero almost surely and in mean as \(n \rightarrow \infty .\) We also obtain upper and lower bound deviation estimates for \(\mathrm{TSP}_n.\) PubDate: 2022-06-01

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Abstract: Abstract This paper considers random processes of the form \(X_{n+1}=a_nX_n+b_n\pmod p\) where p is odd, \(X_0=0\) , \((a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots \) are i.i.d., and \(a_n\) and \(b_n\) are independent with \(P(a_n=2)=P(a_n=(p+1)/2)=1/2\) and \(P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3\) . This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order \((\log p)^2\) steps suffice for \(X_n\) to be close to uniformly distributed on the integers mod p for all odd p while order \((\log p)^2\) steps are necessary for \(X_n\) to be close to uniformly distributed on the integers mod p. PubDate: 2022-06-01

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Abstract: Abstract We investigate the asymptotic behaviour of the difference between the tails of a self-decomposable distribution with a two-sided regularly varying density on the real line and its Lévy measure. Moreover, we study the second-order asymptotic behaviour of the tail of the t-th convolution power of a self-decomposable distribution with a two-sided regularly varying density. PubDate: 2022-06-01

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Abstract: Abstract In this paper, based on the theory of regularly varying functions we study central limit theorems for the weighted sum \(S_n=\sum _{j=1}^{m_n}c_{nj}X_{nj}\) , where \((X_{nj};1\le j \le m_n,n\ge 1)\) is a Hilbert-space-valued identically distributed martingale difference array and \((c_{nj};1\le j \le m_n,n\ge 1)\) is an array of real numbers. As an application, we present a central limit theorem for moving average processes of martingale differences. PubDate: 2022-06-01

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Abstract: Abstract We consider a stationary sequence \((X_n)\) constructed by a multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the multiple integral is non-Gaussian and infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter \(\beta \in (0,1)\) quantifying the conservativity of the system. This parameter \(\beta \) together with the order of the integral determines the decay rate of the covariance of \((X_n)\) . The goal of the paper is to establish limit theorems for the partial sum process of \((X_n)\) . We obtain a central limit theorem with Brownian motion as limit when the covariance decays fast enough, as well as a non-central limit theorem with fractional Brownian motion or Rosenblatt process as limit when the covariance decays slowly enough. PubDate: 2022-06-01

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Abstract: Abstract We prove local convergence results for rerooted conditioned multi-type Galton–Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine. PubDate: 2022-06-01

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Abstract: Abstract We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to the domain of geometric partial attraction of a semistable law with index \(\alpha \in (1/2,1]\) . In the process we obtain local limit theorems for both finite and infinite mean, that is, for the whole range \(\alpha \in (0,2)\) . We also derive the asymptotics of the renewal function for \(\alpha \in (0,1]\) . PubDate: 2022-06-01

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Abstract: Abstract In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space X is linearly isomorphic to a p-uniformly smooth space ( \(1<p\le 2\) ) if and only if an Azuma-type inequality holds for X-valued martingales. This can be viewed as a generalization of Pinelis’ work on an Azuma inequality for martingales with values in 2-uniformly smooth spaces. Secondly, an Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double indexed dyadic martingales and De la Peña-type inequalities for conditionally symmetric martingales, will also be discussed. PubDate: 2022-06-01

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Abstract: Abstract We study the local convergence of critical Galton–Watson trees and Lévy trees under various conditionings. Assuming a very general monotonicity property on the measurable functions of critical random trees, we show that random trees conditioned to have large function values always converge locally to immortal trees. We also derive a very general ratio limit property for measurable functions of critical random trees satisfying the monotonicity property. Finally we study the local convergence of critical continuous-state branching processes, and prove a similar result. PubDate: 2022-06-01

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Abstract: Abstract Stephenson (2018) established annealed local convergence of Boltzmann planar maps conditioned to be large. The present work uses results on rerooted multi-type branching trees to prove a quenched version of this limit. PubDate: 2022-06-01