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

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

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

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

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

Abstract: We analyze a random graph model with preferential attachment rule and edge-step functions that govern the growth rate of the vertex set, and study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function f is a monotone regularly varying function at infinity, the degree sequence of graphs associated with it obeys a (generalized) power-law distribution whose exponent belongs to (1, 2] and is related to the index of regular variation of f at infinity whenever said index is greater than \(-1\) . When the regular variation index is less than or equal to \(-1\) , we show that the empirical degree distribution vanishes for any fixed degree. PubDate: 2021-03-01

Abstract: For a continuous-time catalytic branching random walk (CBRW) on \({\mathbb {Z}}\) , with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include regular variation of the jump distribution tail for underlying random walk and the well-known \(L\log L\) condition for the offspring numbers. In our classification, given in Bulinskaya (Theory Probab Appl 59(4):545–566, 2015), the analysis refers to supercritical CBRW. The principal result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a non-trivial law. An explicit formula is provided for this normalization, and nonlinear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with “heavy” tails, in contrast to the known behavior in case of “light” tails of the random walk jumps. The new tools such as “many-to-few lemma” and spinal decomposition appear ineffective here. The approach developed in this paper combines the techniques of renewal theory, Laplace transform, nonlinear integral equations and large deviations theory for random sums of random variables. PubDate: 2021-03-01

Abstract: We introduce a class of absorption mechanisms and study the behavior of real-valued centered random walks with finite variance that do not get absorbed. Our main results serve as a toolkit which allows obtaining persistence and scaling limit results for many different examples in this class. Further, our results reveal new connections between results in Kemperman (The passage problem for a stationary Markov chain. Statistical research monographs, The University of Chicago Press, Chicago, 1961) and Vysotsky (Stoch Processes Appl 125(5):1886–1910, 2015). PubDate: 2021-03-01

Abstract: Let X be some homogeneous additive functional of a skew Bessel process Y. In this note, we compute the asymptotics of the first passage time of X to some fixed level b, and study the position of Y when X exits a bounded interval [a, b]. As a by-product, we obtain the probability that X reaches the level b before the level a. Our results extend some previous works on additive functionals of Brownian motion by Isozaki and Kotani for the persistence problem and by Lachal for the exit time problem. PubDate: 2021-03-01

Abstract: This paper establishes complete convergence for weighted sums and the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables \(\{X,X_n,n\ge 1\}\) with general normalizing constants under a moment condition that \(ER(X)<\infty \) , where \(R(\cdot )\) is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Stat Probab Lett 92:45–52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijn conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944–1946, 1995) on the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrative examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game. PubDate: 2021-03-01

Abstract: We introduce a stochastic integral with respect to cylindrical Lévy processes with finite p-th weak moment for \(p\in [1,2]\) . The space of integrands consists of p-summing operators between Banach spaces of martingale type p. We apply the developed integration theory to establish the existence of a solution for a stochastic evolution equation driven by a cylindrical Lévy process. PubDate: 2021-03-01

Abstract: In this paper, we prove the Girsanov theorem for G-Brownian motion without the non-degenerate condition. The proof is based on the perturbation method in the nonlinear setting by constructing a product space of the G-expectation space and a linear space that contains a standard Brownian motion. The estimates for exponential martingales of G-Brownian motion are important for our arguments. PubDate: 2021-03-01

Abstract: Let \(X=\{X(t)\in {{\mathbb {R}}}^d, t\in {{\mathbb {R}}}^N\}\) be a centered space–time anisotropic Gaussian random field with stationary increments, whose components are independent but may not be identically distributed. Under certain mild conditions, we determine the exact Hausdorff measure function for the range \(X([0,1]^N)\) . Our result extends those in Talagrand (Ann Probab 23:767–775, 1995) for fractional Brownian motion and Luan and Xiao (J Fourier Anal Appl 18:118–145, 2012) for time-anisotropic and space-isotropic Gaussian random fields. PubDate: 2021-03-01

Abstract: We study the motion of the hypersurface \((\gamma _t)_{t\ge 0}\) evolving according to the mean curvature perturbed by \(\dot{w}^Q\) , the formal time derivative of the Q-Wiener process \({w}^Q\) , in a two-dimensional bounded domain. Namely, we consider the equation describing the evolution of \(\gamma _t\) as a stochastic partial differential equation (SPDE) with a multiplicative noise in the Stratonovich sense, whose inward velocity V is determined by \(V=\kappa \,+\,G \circ \dot{w}^Q\) , where \(\kappa \) is the mean curvature and G is a function determined from \(\gamma _t\) . Already known results in which the noise depends on only the time variable are not applicable to our equation. To construct a local solution of the equation describing \(\gamma _t\) , we derive a certain second-order quasilinear SPDE with respect to the signed distance function determined from \(\gamma _0\) . Then we construct the local solution making use of probabilistic tools and the classical Banach fixed point theorem on suitable Sobolev spaces. PubDate: 2021-03-01

Abstract: We prove existence and uniqueness of solutions to Fokker–Planck equations associated with Markov operators multiplicatively perturbed by degenerate time-inhomogeneous coefficients. Precise conditions on the time-inhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be either globally bounded or bounded away from zero. The approach is based on constructing random time-changes and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces. PubDate: 2021-03-01

Abstract: In this paper, we study a spectrally negative Lévy process that is reflected at its drawdown level whenever a drawdown time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are also obtained. The results are expressed in terms of scale functions for the spectrally negative Lévy process. PubDate: 2021-03-01