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Abstract: Abstract In this paper, we study the asymptotic properties of the maximum likelihood estimator of the drift parameter in the fractional Ornstein–Uhlenbeck process. Using the change of measure method and asymptotic analysis technique, we establish an exponential nonuniform Berry–Esseen bound for the maximum likelihood estimator. As an application, the optimal uniform Berry–Esseen bound and Cramér-type moderate deviation are obtained. PubDate: 2022-08-13
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Abstract: Abstract Let \( X_{n} \) be \( n\times N \) random complex matrices, and let \(R_{n}\) and \(T_{n}\) be non-random complex matrices with dimensions \(n\times N\) and \(n\times n\) , respectively. We assume that the entries of \( X_{n} \) are normalized independent random variables satisfying the Lindeberg condition, \( T_{n} \) are nonnegative definite Hermitian matrices and commutative with \(R_nR_n^*\) , i.e., \(T_{n}R_{n}R_{n}^{*}= R_{n}R_{n}^{*}T_{n} \) . The general information-plus-noise-type matrices are defined by \(C_{n}=\frac{1}{N}T_{n}^{\frac{1}{2}} \left( R_{n} +X_{n}\right) \left( R_{n}+X_{n}\right) ^{*}T_{n}^{\frac{1}{2}} \) . In this paper, we establish the limiting spectral distribution of the large-dimensional general information-plus-noise-type matrices \(C_{n}\) . Specifically, we show that as n and N tend to infinity proportionally, the empirical distribution of the eigenvalues of \(C_{n}\) converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform. PubDate: 2022-08-11
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Abstract: Abstract The asymptotic behavior of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with possibly nonempty boundary. We obtain rates of convergence for empirical measures to the invariant measure of the subordinated process under the Wasserstein distance. The results, established for more general subordinated processes than (arXiv:2107.11568), generalize the recent ones in Wang (Stoch Process Appl 144:271–287, 2022) and are shown to be sharp by a typical example. The proof is motivated by the aforementioned works. PubDate: 2022-08-10
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Abstract: Abstract We establish the existence of solutions to a class of nonlinear stochastic differential equations of reaction–diffusion type in an infinite-dimensional space, with diffusion corresponding to a given transition kernel. The solution obtained is the scaling limit of a sequence of interacting particle systems and satisfies the martingale problem corresponding to the target differential equation. PubDate: 2022-08-08
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Abstract: Abstract We consider simple random walk on a realization of an Erdős–Rényi graph with n vertices and edge probability \(p_n\) . We assume that \(n p^2_n/(\log \mathrm{n})^{16 \xi } \rightarrow \infty \) for some \(\xi >1\) defined below. This in particular implies that the graph is asymptotically almost surely (a.a.s.) connected. We show a central limit theorem for the average starting hitting time, i.e., the expected time it takes the random walker on average to first hit a vertex j when starting in a fixed vertex i. The average is taken with respect to \(\pi _j\) , the invariant measure of the random walk. PubDate: 2022-08-08
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Abstract: Abstract We consider random walks on the cone of \(m \times m\) positive definite matrices, where the underlying random matrices have orthogonally invariant distributions on the cone and the Riemannian metric is the measure of distance on the cone. By applying results of Khare and Rajaratnam (Ann Probab 45:4101–4111, 2017), we obtain inequalities of Hoffmann-Jørgensen type for such random walks on the cone. In the case of the Wishart distribution \(W_m(a,I_m)\) , with index parameter a and matrix parameter \(I_m\) , the identity matrix, we derive explicit and computable bounds for each term appearing in the Hoffmann-Jørgensen inequalities. PubDate: 2022-07-28
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Abstract: Abstract We establish several Schur-convexity type results under fixed variance for weighted sums of independent gamma random variables and obtain nonasymptotic bounds on their Rényi entropies. In particular, this pertains to the recent results by Bartczak–Nayar–Zwara as well as Bobkov–Naumov–Ulyanov, offering simple proofs of the former and extending the latter. PubDate: 2022-07-27
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Abstract: Abstract We consider the spectral gap of a uniformly chosen random \((d_1,d_2)\) -biregular bipartite graph G with \( V_1 =n, V_2 =m\) , where \(d_1,d_2\) could possibly grow with n and m. Let A be the adjacency matrix of G. Under the assumption that \(d_1\ge d_2\) and \(d_2=O(n^{2/3}),\) we show that \(\lambda _2(A)=O(\sqrt{d_1})\) with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random d-regular digraph is \(O(\sqrt{d})\) for \(1\le d\le n/2\) with high probability. Assuming \(d_2\) is fixed and \(d_1=O(n^2)\) , we further prove that for a random \((d_1,d_2)\) -biregular bipartite graph, \( \lambda _i^2(A)-d_1 =O(\sqrt{d_1})\) for all \(2\le i\le n+m-1\) with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random d-regular graphs and several new switching operations we define for random bipartite biregular graphs. PubDate: 2022-07-15
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Abstract: Abstract The spectral heat content is investigated for time-changed killed Brownian motions on \(C^{1,1}\) open sets, where the time change is given by either a subordinator or an inverse subordinator, with the underlying Laplace exponent being regularly varying at \(\infty \) with index \(\beta \in (0,1)\) . In the case of inverse subordinators, the asymptotic limit of the spectral heat content in small time is shown to involve a probabilistic term depending only on \(\beta \in (0,1)\) . In contrast, in the case of subordinators, this universality holds only when \(\beta \in (\frac{1}{2}, 1)\) . PubDate: 2022-07-15
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Abstract: Abstract In this paper we prove a support theorem for a class of Itô–Volterra equations related to the fractional Brownian motion. The simplified method developed by Millet and Sanz-Solé plays an important role. PubDate: 2022-07-01
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Abstract: Abstract In this paper, we propose two numerical methods for solving certain kinds of mean-field backward stochastic differential equations: first-order numerical scheme and Crank–Nicolson numerical scheme. Then, we study \(L^p\) -error estimates for the proposed schemes. We prove that the two schemes are of second-order convergence in solving for \(Y_t\) in \(L^p\) norm; the first-order scheme is of first-order convergence and the Crank–Nicolson scheme is of second-order convergence in solving \(Z_t\) in \(L^p\) norm. PubDate: 2022-06-30
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Abstract: Abstract We establish a novel convergent iteration framework for a weak approximation of general switching diffusion. The key theoretical basis of the proposed approach is a restriction of the maximum number of switching so as to untangle and compensate a challenging system of weakly coupled partial differential equations to a collection of independent partial differential equations, for which a variety of accurate and efficient numerical methods are available. Upper and lower bounding functions for the solutions are constructed using the iterative approximate solutions. We provide a rigorous convergence analysis for the iterative approximate solutions, as well as for the upper and lower bounding functions. Numerical results are provided to examine our theoretical findings and the effectiveness of the proposed framework. PubDate: 2022-06-24
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Abstract: Abstract Consider the inhomogeneous Erdős-Rényi random graph (ERRG) on n vertices for which each pair \(i,j\in \{1,\ldots ,n\}\) , \(i\ne j,\) is connected independently by an edge with probability \(r_n(\frac{i-1}{n},\frac{j-1}{n})\) , where \((r_n)_{n\in \mathbb {N}}\) is a sequence of graphons converging to a reference graphon r. As a generalisation of the celebrated large deviation principle (LDP) for ERRGs by Chatterjee and Varadhan (Eur J Comb 32:1000–1017, 2011), Dhara and Sen (Large deviation for uniform graphs with given degrees, 2020. arXiv:1904.07666) proved an LDP for a sequence of such graphs under the assumption that r is bounded away from 0 and 1, and with a rate function in the form of a lower semi-continuous envelope. We further extend the results by Dhara and Sen. We relax the conditions on the reference graphon to \((\log r, \log (1- r))\in L^1([0,1]^2)\) . We also show that, under this condition, their rate function equals a different, more tractable rate function. We then apply these results to the large deviation principle for the largest eigenvalue of inhomogeneous ERRGs and weaken the conditions for part of the analysis of the rate function by Chakrabarty et al. (Large deviation principle for the maximal eigenvalue of inhomogeneous Erdoös-Rényi random graphs, 2020. arXiv:2008.08367). PubDate: 2022-06-14
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Abstract: Abstract Given a supercritical branching random walk \(\{Z_n\}_{n\ge 0}\) on \({\mathbb {R}}\) , let \(Z_n(A)\) be the number of particles located in a set \(A\subset {\mathbb {R}}\) at generation n. It is known from Biggins (J Appl Probab 14:630–636, 1977) that under some mild conditions, for \(\theta \in [0,1)\) , \(n^{-1}\log Z_n([\theta x^* n,\infty ))\) converges almost surely to \(\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*)\) as \(n\rightarrow \infty \) , where \(x^*\) is the speed of the maximal position of \(\{Z_n\}_{n\ge 0}\) and \(I(\cdot )\) is the large deviation rate function of the underlying random walk. In this work, we investigate its lower deviation probabilities, in other words, the convergence rates of $$\begin{aligned} {\mathbb {P}}\left( Z_n([\theta x^* n,\infty ))<e^{an}\right) \end{aligned}$$ as \(n\rightarrow \infty \) , where \(a\in [0,\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*))\) . Our results complete those in Chen and He (Ann Institut Henri Poincare Probab Stat 56:2507–2539, 2020), Gantert and Höfelsauer (Electron Commun Probab 23(34):1–12, 2018) and Öz (Latin Am J Probab Math Stat 17:711–731, 2020). PubDate: 2022-06-07
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Abstract: Abstract Let \(X_{1, N}\geqslant X_{2, N} \geqslant \cdots \geqslant X_{N , N}\) be the order statistics of independent identically distributed random variables \(X_k\) ( \(1\leqslant k \leqslant N\) ). For fixed natural K and a nonnegative bounded deterministic function \(G_N\) on \(\mathbb {R}^N\) satisfying mild conditions of Lebesgue’s measurability, we obtain the following bound for the expectations: $$\begin{aligned}&\mathbb {E}G_N \big (X_{1,N},X_{2,N},\ldots ,X_{K,N}, X_{K+1,N},\ldots , X_{N,N} \big ) \\&\quad \leqslant T \cdot \mathbb {E}G_N \big (X_{1,N}^{(1)},X_{1,N}^{(2)},\ldots ,X_{1,N}^{(K)}, X _{K+1,N},\ldots , X _{N,N} \big ) +\vartheta _T \end{aligned}$$ for any \(T \geqslant T_0(K)\) and any \(N \geqslant N_0(T)\) large enough; here constants \(\vartheta _T> 0\) tend to zero as T approaches infinity; \(X _{1,N}^{(i)}\) ( \(1\leqslant i\leqslant K\) ) are mutually independent copies of the maximum \(X_{1,N}\) ; and each \(X _{1,N}^{(i)}\) is also independent of the sample \(\{X_k\}_{1 \leqslant k\leqslant N}\) . With \(G_N\) as relevant indicator functions and \(N \rightarrow \infty \) , these bounds are applied to study \(\mathrm{o}\) - and \(\mathrm{O}\) -type asymptotic properties of the following functions on order statistics: (Appl-1) the numbers of observations near the Kth extremes \(X_{K,N}\) and (Appl-2) the sums of negative powers of spacings \(X_{K,N}-X_{i,N}\) ( \(K+1 \leqslant i \leqslant N\) ). PubDate: 2022-06-05 DOI: 10.1007/s10959-022-01179-9
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Abstract: Abstract In this paper, we study geometric properties of the unique infinite cluster \(\Gamma ^{u,T}\) in a sufficiently supercritical finitary random interlacements \(\mathcal {FI}^{u,T}\) in \({\mathbb {Z}}^d, \ d\ge 3\) . We prove that the chemical distance in \(\Gamma ^{u,T}\) is, with stretched exponentially high probability, of the same order as the Euclidean distance in \({\mathbb {Z}}^d\) . This also implies a shape theorem parallel to those for percolation and regular random interlacements. We also prove local uniqueness of \(\mathcal {FI}^{u,T}\) , which says that any two large clusters in \(\mathcal {FI}^{u,T}\) “close to each other" will be connected within the same order of their diameters except a stretched exponentially small probability. PubDate: 2022-06-03 DOI: 10.1007/s10959-022-01182-0
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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 DOI: 10.1007/s10959-020-01065-2
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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 DOI: 10.1007/s10959-021-01091-8
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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 DOI: 10.1007/s10959-020-01072-3
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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 DOI: 10.1007/s10959-021-01087-4
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