Daniela Bertacchi, Fabio Zucca
We investigate the asymptotical behaviour of the transition probabilities of the simple random walk on the 2-comb. In particular we obtain space-time uniform asymptotical estimates which show the lack of symmetry of this walk better than local limit estimates. Our results also point out the impossibility of getting Jones-type non-Gaussian estimates.
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We prove optimal constant over root $n$ upper bounds for the maximal probabilities of $n$th convolution powers of discrete uniform distributions.
Exact probabilities for three and four dice in the balanced uniform model of 3-sided dice
本文计算了三面骰子平衡均匀模型下，随机三颗和四颗独立骰子形成的锦标赛各同构类的精确概率。
Persistence probabilities for MA(1) sequences with uniform innovations
本文研究均匀新息下MA(1)过程的持续性概率，按分布位置与耦合参数划分区域，并在所有区域内显式求出生成函数，部分区域可借助组合量显式表达。
We establish the unimodality and the asymptotic strong unimodality of the ordinary multinomials and give their smallest mode leading to the expression of the maximal probability of convolution powers of the discrete uniform distribution. We conclude giving the generating functions of the sequence of generalized ordinary multinomials and for an extension of the sequence of maximal probabilities for convolution power of discrete uniform distribution.
Uniform asymptotic estimates for ruin probabilities of a multidimensional risk model with cadlag returns and multivariate heavy tailed claims
本文研究多维更新风险模型，在重尾、弱相依及分量任意相依的索赔条件下，给出了有限时间及全时间范围内折现总索赔进入稀有集概率的一致渐近估计，并应用于破产概率估计。
Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability π. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that {(X_n,S_n),n\geq0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(X_n,S_n),n\geq0}. To be more precise, for given b\geq0, define the stopping time τ=τ(b)=inf{n:S_n>b}. When a drift μof the random walk S_n is 0, we derive a one-term Ed…
Hitting probabilities and uniformly $S$-transient subgraphs
该研究围绕图上随机游走的击中概率展开，提出$S$-瞬态概念，在热核满足双侧高斯估计等条件下证明相关概率的上下界，还给出调和测度等相关概率的双侧估计。
Comparison results are obtained for the inclusion probabilities in some unequal probability sampling plans without replacement. For either successive sampling or Hájek's rejective sampling, the larger the sample size, the more uniform the inclusion probabilities in the sense of majorization. In particular, the inclusion probabilities are more uniform than the drawing probabilities. For the same sample size, and given the same set of drawing probabilities, the inclusion probabilities are more uniform for rejective sampling than for successive sampling. This last result confirms a conjecture of …
We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.
This paper studies transition probabilities from a Borel subset of a Polish space to a product of two Borel subsets of Polish spaces. For such transition probabilities it introduces and studies the property of semi-uniform Feller continuity. This paper provides several equivalent definitions of semi-uniform Feller continuity and establishes its preservation under integration. The motivation for this study came from the theory of Markov decision processes with incomplete information, and this paper provides fundamental results useful for this theory.
We compute the Green's function on the double cover of ${\mathbb Z}^2$, branched over a vertex or a face. We use this result to compute the local statistics of the "trunk" of the uniform spanning tree on the square lattice, i.e., the limiting probabilities of cylinder events conditional on the path connecting far away points passing through a specified edge. We also show how to compute the local statistics of large-scale triple points of the uniform spanning tree, where the trunk branches. The method reduces the problem to a dimer system with isolated monomers, and we compute the inverse Kaste…
Exact cospectrality probabilities for uniform random matrices
本文研究了均匀随机矩阵在正交矩阵共轭作用下的共谱概率，并给出了精确公式及数论波动的影响。
We show that certain topologically defined uniform spanning tree probabilities for graphs embedded in an annulus can be computed as linear combinations of Pfaffians of matrices involving the line-bundle Green's function, where the coefficients count cover-inclusive Dyck tilings of skew Young diagrams.
We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.
In this paper we consider some generalizations of the classical d-dimensional Brownian risk model. This contribution derives some non-asymptotic bounds for simultaneous ruin probabilities of interest. In addition, we obtain non-asymptotic bounds also for the case of general trend functions and convolutions of our original risk model.
Correlations in uniform spanning trees: a fermionic approach
本文通过费米子高斯自由场建立了均匀生成树中边包含概率的对应关系，实现了有限图度数联合分布的显式计算及其标度极限。
In this paper we study the asymptotic decay of finite time ruin probabilities for an insurance company that faces heavy-tailed claims, uses predictable investment strategies and makes investments in risky assets whose prices evolve according to quite general semimartingales. We show that the ruin problem corresponds to determining hitting probabilities for the solution to a randomly perturbed stochastic integral equation. We derive a large deviation result for the hitting probabilities that holds uniformly over a family of semimartingales and show that this result gives the asymptotic decay of…
A Sharp Gaussian Tail Bound for Sums of Uniforms
基于独立均匀随机变量和的高斯尾界，证明其尾概率受高斯尾约束，并确定最优常数。
Suppose that S is an asymptotically stable random walk with norming sequence c_{n} and that T_{x} is the time that S first enters (x,\inf), where x\ge 0. The asymptotic behaviour of P(T_0=n) has been described in a recent paper of Vatutin and Wachtel, \cite{vw}, and here we build on that result to give three estimates for P(T_{x}=n), which hold uniformly as n\to\inf in the regions x=o(c_{n}), x=O(c_{n}), and x/c_{n}\to\inf, respectively.
In this paper we consider an irreducible random walk on the integer lattice $\mathbb{Z}$ that is in the domain of normal attraction of a strictly stable process with index $α\in (1, 2)$ and obtain the asymptotic form of the distribution of the hitting time of the origin and that of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in the natural domain of the space and time variables.
We study simple random walk on the uniform spanning tree on Z^2 . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on Z^2 is 16/13 almost surely.
Transition probabilities of step-reinforced random walks
本文研究了群上广义步强化随机游走，统一了现有模型，推导了任意α<1的转移概率上界，并证明在欧几里得空间中，d≥3时游走瞬态，解决 Cayley 树上大象随机游走指数衰减问题。
In this work limit probabilities of first-order properties of the random $s$-uniform hypergraph in the binomial model $G^{s}(n,p)$ are studied. We give a complete discription of all positive $α$ such that $G^{s}(n,n^{-α})$ obeys Zero-One Law. Moreover, for any rational $ρ\geq 1/(s-1)$ we prove the existence of a strictly balanced $s$-uniform hypergraph with the density $ρ$.
Uniform analyticity of local observables in FK-percolation and analyticity of the Ising spontaneous magnetisation
论文证明了在FK渗流模型中，局部事件的概率在渗流参数p下均匀解析，并满足指数增长界，从而证明了Potts模型磁化率在合适参数范围内的解析性，包括伊辛模型在三维及以上所有超临界区域的解析性。
Uniform probability in cosmology
本文探讨宇宙学测度问题与无限样本空间均匀概率悖论的联系，通过分析Kolmogorov等人的概率论基础工作，指出当前障碍的确切起源，并以多元主义立场表明宇宙学概率分配仍有进展空间。
We investigate the asymptotic of ruin probabilities when the company combines the life- and non-life insurance businesses and invests its reserve into a risky asset with stochastic volatility and drift driven by a two-state Markov process. Using the technique of the implicit renewal theory we obtain the rate of convergence to zero of the ruin probabilities.
We investigate asymptotic behaviour of probabilities of large deviations for normalized combinatorial sums. We find a zone in which these probabilities are equivalent to the tail of the standard normal law. Our conditions are similar to the classical Bernstein condition. The range of the zone of the normal convergence can be of power order.
Analysis of an Inhomogeneous Random Walk with Spatial Decay of Transition Probabilities and Parameter Renewal per Excursion
本文提出并分析了一种新的二维不均匀随机游走模型，结合空间衰减的转移概率与每次游走的时序重置结构，研究了空间不均匀性对返回原点的影响以及随机环境对转移概率参数的影响。
Last-Coverage Probabilities of Random Walks on Cycle Graphs for Symmetric, Biased, and Time-Dependent Walks
本文针对环图上的对称、有偏和时变随机游走，利用连通弧性质推导最后覆盖概率公式，通过模拟验证结果，并对比不同图结构的最后覆盖分布。
Growing integer partitions with uniform marginals and the equivalence of partition ensembles
本文构造了整数分拆上具有均匀边际的马尔可夫增长过程，分析其跳跃概率的渐近行为以解释极限形状的形成，并利用该过程给出两个随机分拆系综等价的充分条件，进而证明奇偶部分计数渐近独立且极限分布不同。
For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, consider the geodesic $Γ_n$ joining $(0,0)$ and $(n,n)$ for large $n$. It is well known that the transversal fluctuation of $Γ_n$ around the line $x=y$ is $n^{2/3+o(1)}$ with high probability. We obtain the exponent governing the decay of the small ball probability for $Γ_{n}$ and establish that for small $δ$, the probability that $Γ_{n}$ is contained in a strip of width $δn^{2/3}$ around the diagonal is $\exp (-Θ(δ^{-3/2}))$ uniformly in high $n$. We also obtain optimal small deviation estimates for the …
This paper studies the tail probability of weighted sums of the form $\sum_{i=1}^n c_i X_i$, where random variables $X_i$'s are either independent or pairwise quasi-asymptotical independent with heavy tails. Using $h$-insensitive function, the uniform asymptotic equivalence of the tail probabilities of $\sum_{i=1}^n c_iX_i$, $\max_{1\le k\le n}\sum_{i=1}^k c_iX_i$ and $\sum_{i=1}^n c_iX_i^+$ is established, where $X_i$'s are independent and follow the long-tailed distribution, and $c_i$'s take value in a broad interval. Some further uniform asymptotic results for the weighted sums of $X_i$'s w…
Generalized Edgeworth expansions for integer-valued additive functionals of uniformly elliptic Markov chains
本文为一致椭圆非齐次马尔可夫链的整数值加性泛函的部分和概率建立了广义Edgeworth展开，并利用三角多项式的显式公式，将标准Edgeworth展开的存在性与模等分布速率联系起来。
Many problems in computing, service, and manufacturing systems can be modeled via infinite repeating Markov chains with an infinite number of levels and a finite number of phases. Many such chains are quasi-birth-death processes (QBDs) with transitions that are skip-free in level, in that one can only transition between consecutive levels, and unidirectional in phase, in that one can only transition from lower-numbered phases to higher-numbered phases. We present a procedure, which we call Clearing Analysis on Phases (CAP), for determining the limiting probabilities of such Markov chains exact…
Sylvester's four point problem for ball-convex bodies
本文研究球凸体的西尔维斯特四点问题，证明Efron-Buchta恒等式成立，确定平面球凸体中三、四点的西尔维斯特概率，推广Groemer不等式并推导相关等周不等式，还给出单位圆盘下西尔维斯特概率的显式结果及外接圆半径的分布函数。
In this thesis the properties of two kinds of non-uniform random recursive trees are studied. In the first model weights are assigned to each node, thus altering the attachment probabilities. We will call these trees weighted recursive trees. In the second model a different distribution rather than the uniform one is chosen on the symmetric group, namely a riffle shuffle distribution. These trees will be called biased recursive trees. For both of these models the number of branches, the number of leaves, the depth of nodes and some other properties are studied. The focus is on asymptotic resul…
A Refinement of Vapnik--Chervonenkis' Theorem
本文改进了Vapnik--Chervonenkis定理，通过正态近似和Berry--Esseen误差控制，提高了中等偏差下的收敛速率估计。
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