S Satheesh, E Sandhya
Non-negative integer-valued semi-selfsimilar processes are introduced. Levy processes in this class are characterized. Its relation to an AR(1) scheme is derived.
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Remark.9 in Bose-Dasgupta-Rubin (2002) review states that when a non-negative integer-valued infinitely divisible law has an atom at unity then its support cannot have any gaps. Here one has two questions. (i) Why there are no gaps and (ii) Can there be gaps if the condition is not satisfied. Our investigation with these questions in mind centers on the implications of having and not having atoms at zero and unity. We give two examples/ constructions, which show that the remark needs modification and we modify it.
Random multiplication versus random sum: auto-regressive-like models with integer-valued random inputs
本文提出基于随机乘法算子的RMINAR整数值时间序列模型，证明其解处处严平稳遍历，并给出四阶段加权最小二乘估计及理论性质。
We study the asymptotic behavior of the weighted least squares estimators of the unknown parameters of bifurcating integer-valued autoregressive processes. Under suitable assumptions on the immigration, we establish the almost sure convergence of our estimators, together with the quadratic strong law and central limit theorems. All our investigation relies on asymptotic results for vector-valued martingales.
We investigate the order of the maximum of the integer-valued Gaussian free field in two dimensions, and show that it grows logarithmically with the size of the box. Our treatment follows closely that of a recent paper by Kharash and Peled on the Fröhlich-Spencer proof of the Berezinskii-Kosterlitz-Thouless transition.
Integer-valued multifractal processes
本文提出整数值多重分形过程的框架，利用细化操作作为标量乘法的离散对应，通过时间变换复合泊松过程构造过程，推导矩标度律并数值验证，将多重分形概念引入点过程理论。
We prove a complete class theorem that characterizes \emph{all} stationary time reversible Markov processes whose finite dimensional marginal distributions (of all orders) are infinitely divisible. Aside from two degenerate cases (iid and constant), in both discrete and continuous time every such process with full support is a branching process with Poisson or Negative Binomial marginal univariate distributions and a specific bivariate distribution at pairs of times. As a corollary, we prove that every nondegenerate stationary integer valued processes constructed by the Markov thinning process…
This paper introduces a Nearly Unstable INteger-valued AutoRegressive Conditional Heteroskedasticity (NU-INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU-INARCH process endowed with a Skorohod topology weakly converges to a Cox-Ingersoll-Ross diffusion. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution u…
In this paper, we proved an exact asymptotically sharp upper bound of the $L^p$ Lebesgue Constant (i.e. the $L^p$ norm of Dirichlet kernel) for $p\ge 2$. As an application, we also verified the implication of a new $\infty $-Rényi entropy power inequality for integer valued random variables.
A short note on super-hedging an arbitrary number of European options with integer-valued strategies
本文研究离散时间下整数价值策略的超对冲定价，针对连续分段仿射终端索赔提出可实施于历史数据的动态规划原理，并给出最优策略。
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。
It is known that the space of boundedly finite integer-valued measures on a complete separable metric space becomes itself a complete separable metric space when endowed with the weak-hash metric. It is also known that convergence under this topology can be characterised in a way that is similar to the weak convergence of totally finite measures. However, the original proofs of these two fundamental results assume that a certain term is monotonic, which is not the case as we give a counterexample. We manage to clarify these original proofs by addressing specifically the parts that rely on this…
Functional calculus and martingale representation formula for integer-valued random measures
本文针对连续鞅与整值随机测度建立非预期微积分，构造泛函导数并推导鞅表示公式，展示其在整值随机测度相关问题中的多项应用。
On certain integral functionals of integer-valued subordinators
本文推导了任意整数值次ordinator指数泛函密度的显式表达式及纯跳跃次ordinator的极限表示，通过分段基函数处理带正漂移情况，推进Lévy泛函分布解析研究并给出逆幂泛函密度的极限级数表示，数值性能良好。
The probability density function (PDF) and cumulative distribution function of the sum of L independent but not necessarily identically distributed Gamma variates, applicable to the output statistics of maximal ratio combining (MRC) receiver operating over Nakagami-m fading channels or in other words to the statistical analysis of the scenario where the sum of squared Nakagami-m distributions are user-of-interest, is presented in closed-form in terms of well-known Meijer's G function and easily computable Fox's H-bar function for integer valued and non-integer valued m fading parameters. Furth…
We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and trigonometric polynomials. Our results do not require any additional assumptions. As an application of our expansions we find necessary and sufficient conditions for the classical Edgeworth expansion. It turns out that there are two possible obstructions for the validity of the Edgeworth expansion of order $r$. First, the distance between the distribution of the u…
Arithmetic properties of multiplicative integer-valued perturbed random walks
本文研究乘性整数值扰动随机游走对应的两类随机集的算术性质，推导了其素数计数和最小公倍数的分布极限定理，丰富了相关随机过程的算术性质研究。
Quantitative bounds on vortex fluctuations in $2d$ Coulomb gas and maximum of the integer-valued Gaussian free field
本文研究二维库仑气体、Villain模型及整值高斯自由场中涡旋对涨落的影响，给出定量上下界，证明涡旋效应与自旋波同阶，并提出高效采样方法及对偶转移结果。
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