Abstract: Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov’s theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov’s theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich–Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a necessary step to further research in probability theory and its applications in the context of (deficient) topological measures and corresponding nonlinear functionals. PubDate: 2021-04-24

Abstract: In this paper, we study the asymptotic properties of drift parameter estimations in reflected Ornstein–Uhlenbeck process, and establish their moderate deviations in both cases with one-sided barrier and two-sided barriers. The main methods consist of regenerative process techniques and the strong Markov property, as well as moderate deviations for martingales. PubDate: 2021-04-19

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: 2021-04-02

Abstract: We consider the existence and Hölder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional Brownian motion, where \(k=(k_1,k_2,\ldots , k_d)\) . Moreover, we show a limit theorem for the critical case with \(H=\frac{2}{3}\) and \(d=1\) , which was conjectured by Jung and Markowsky [7]. PubDate: 2021-04-01

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: 2021-03-31

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: 2021-03-30

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: 2021-03-29

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: 2021-03-29

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: 2021-03-26

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: 2021-03-24

Abstract: We consider decoupling inequalities for random variables taking values in a Banach space X. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not affect the decoupling properties (in particular, it does not affect the constants involved). As a special case, we deal with one-sided moment inequalities for decoupled dyadic (i.e., Paley–Walsh) martingales and show that Burkholder–Davis–Gundy-type inequalities for stochastic integrals of X-valued processes can be obtained from decoupling inequalities for X-valued dyadic martingales. PubDate: 2021-03-14

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: 2021-03-11

Abstract: In this paper, we explore the generalized mixed fractional Brownian motion in the set-indexed framework and generalize several recent results from Miao et al. (Lecture Notes and Math, Springer, New York, 2008), Zili (J. Appl. Math. Stoch. Anal. 30:1–9, 2006) and Thale (Appl. Math. Sci. 3(28):1885–1901, 2009). We present the characterization of generalized mixed set-indexed fractional Brownian motion (gmsifBM) by flows, and we extend some selected aspects to the gmsifBM for the following issues: stationary increments, self-similarity, long-range dependence, Hölder continuity, differentiability, Hausdorff dimension, etc. PubDate: 2021-03-11

Abstract: In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters \(H<\frac{1}{2}.\) Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument. PubDate: 2021-03-08

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: 2021-03-05

Abstract: A theta function for an arbitrary connected and simply connected compact simple Lie group is defined as an infinite determinant that is naturally related to the transformation of a family of independent Gaussian random variables associated with a pinned Brownian motion in the Lie group. From this definition of a theta function, the equality of the product and the sum expressions for a theta function is obtained. This equality for an arbitrary connected and simply connected compact simple Lie group is known as a Macdonald identity which generalizes the Jacobi triple product for the elliptic theta function associated with su(2). PubDate: 2021-03-01 DOI: 10.1007/s10959-019-00977-y

Abstract: We consider marked point processes on the d-dimensional Euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We construct absolutely summable Hamiltonians in terms of hyperedge potentials in the sense of Georgii et al. (Probab Theory Relat Fields 153(3–4):643–670, 2012), which are useful in models of stochastic geometry. These potentials allow for weak non-localities and are a natural generalization of the usual physical multi-body potentials, which are strictly local. Our proof relies on regrouping arguments, which use the possibility of controlled non-localities in the class of hyperedge potentials. As an illustration, we also provide such representations for the Widom–Rowlinson model under independent spin-flip time evolution. With this work, we aim to draw a link between the abstract theory of point processes in infinite volume, the study of measures under transformations and statistical mechanics of systems of point particles. PubDate: 2021-03-01 DOI: 10.1007/s10959-019-00960-7

Abstract: We consider a branching random walk \(S_nX(t)\) on a supercritical random Galton–Watson tree. We compute the Hausdorff and packing dimensions of the level set \(E(\alpha )\) of infinite branches in the boundary of tree endowed with random metric along which the average of \(S_n X(t)/n\) have a given limit point. PubDate: 2021-03-01 DOI: 10.1007/s10959-019-00984-z

Abstract: We study the Doob’s h-transform of the two-dimensional simple random walk with respect to its potential kernel, which can be thought of as the two-dimensional simple random walk conditioned on never hitting the origin. We derive an explicit formula for the Green’s function of this random walk and also prove a quantitative result on the speed of convergence of the (conditional) entrance measure to the harmonic measure (for the conditioned walk) on a finite set. PubDate: 2021-03-01 DOI: 10.1007/s10959-019-00963-4

Abstract: Let \(\{X_n\}_0^{\infty }\) be a supercritical branching process with immigration with offspring distribution \(\{p_j\}_0^{\infty }\) and immigration distribution \(\{h_i\}_0^{\infty }.\) Throughout this paper, we assume that \(p_0=0, p_j\ne 1\) for any \(j\ge 1\) , \(1<m=\sum _{j=0}^{\infty } jp_j<\infty ,\) and \(h_0<1\) , \(0<a=\sum _{j=0}^{\infty } jh_j<\infty .\) We first show that \(Y_n=m^{-n}(X_n-\frac{m^{n+1}-1}{m-1}a)\) is a martingale and converges to a random variable Y. Secondly, we study the rates of convergence to 0 as \(n\rightarrow \infty \) of $$\begin{aligned} P(\left Y_n-Y\right >\varepsilon ), \ \ P\left( \left \frac{X_{n+1}}{X_n}-m\right >\varepsilon \Bigg Y\ge \alpha \right) \end{aligned}$$ for \(\varepsilon >0\) and \(\alpha >0\) under various moment conditions on \(\{p_j\}_0^{\infty }\) and \(\{h_i\}_0^{\infty }.\) It is shown that the rates are always supergeometric under a finite moment generating function hypothesis. PubDate: 2021-03-01 DOI: 10.1007/s10959-019-00968-z