K. R. Parthasarathy
By using a quantum probabilistic approach we obtain a description of the extreme points of the convex set of all joint probability distributions on the product of two standard Borel spaces with fixed marginal distributions.
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Max-semistable extreme value laws for autoregressive processes with Cantor-like marginals
本文研究边缘分布类似康托尔函数的自动回归过程，证明其边缘分布属于极大半稳定分布的吸引域，并通过引入极值指数得到该过程的极值律，同时揭示其与确定性动力系统极值的联系。
We introduce the concept of an extremely negatively dependent (END) sequence of random variables with a given common marginal distribution. The END structure, as a new benchmark for negative dependence, is comparable to comonotonicity and independence. We show that an END sequence always exists for any given marginal distributions with a finite mean and we provide a probabilistic construction. Through such a construction, the partial sum of identically distributed but dependent random variables is controlled by a random variable that depends only on the marginal distribution of the sequence. T…
We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study. Finally, we present some applications to the theory of random quantum states in quan…
Rigidity of infinite exchangeable sequences with Gaussian marginals
研究具有高斯边缘分布的无限可交换序列，证明前四个坐标的联合高斯性足以迫使整个序列为高斯过程，并给出在导向测度几乎必然无限可分条件下的两点判据。
For the directed edge preferential attachment network growth model studied by Bollobas et al. (2003) and Krapivsky and Redner (2001), we prove that the joint distribution of in-degree and out-degree has jointly regularly varying tails. Typically the marginal tails of the in-degree distribution and the out-degree distribution have different regular variation indices and so the joint regular variation is non-standard. Only marginal regular variation has been previously established for this distribution in the cases where the marginal tail indices are different.
A marginal problem asks whether a given family of marginal distributions for some set of random variables arises from some joint distribution of these variables. Here we point out that the existence of such a joint distribution imposes non-trivial conditions already on the level of Shannon entropies of the given marginals. These entropic inequalities are necessary (but not sufficient) criteria for the existence of a joint distribution. For every marginal problem, a list of such Shannon-type entropic inequalities can be calculated by Fourier-Motzkin elimination, and we offer a software interfac…
Extreme Values in Closed Networks
本文研究了闭合网络中最大队列长度分布的极限特性，通过概率表示和大偏差率函数推导出显式结果，并提出了一种新的方法计算全局特性。
Moderately Heavy Extreme Values under Extreme Long Range Dependence
本文研究边际尾部为次指数且属于Gumbel最大值吸引域的平稳序列，针对极端长程依赖导致的极值由多个大值引起并呈分形聚集的现象，建立了具有非Gumbel极限对象的函数型极值极限定理。
An Explicit Description of Extreme Points of the Set of Couplings with Given Marginals: with Application to Minimum-Entropy Coupling Problems
该研究刻画了给定边缘分布的耦合集的极点结构并给出枚举算法，证明严格Schur-凹函数的最小值在极点处取得，得到含香农、Rényi、Tsallis熵的最小熵耦合精确解，并推广至多边缘情形。
In this paper, we continue Voiculescu's recent work on the analogous extreme value theory in the context of bi-free probability theory. We derive various equivalent conditions for a bivariate distribution function to be bi-freely max-infinitely divisible. A bi-freely max-infinitely divisible distribution function can be expressed in terms of its marginals and a special form of copulas. Such a distribution function is shown to be also max-infinitely divisible in the classical sense. In addition, we characterize the set of bi-free extreme value distribution functions. A distribution function of …
Moments of random quantum marginals via Weingarten calculus
本文利用 Weingarten 演算研究固定谱随机厄米算子边际的混合矩，为可区分粒子、玻色子和费米子系统给出完全确定边际联合分布的矩公式，并分析了两可区分粒子情形下边际的渐近行为。
We study the problem of maximizing a spectral risk measure of a given output function which depends on several underlying variables, whose individual distributions are known but whose joint distribution is not. We establish and exploit an equivalence between this problem and a multi-marginal optimal transport problem. We use this reformulation to establish explicit, closed form solutions when the underlying variables are one dimensional, for a large class of output functions. For higher dimensional underlying variables, we identify conditions on the output function and marginal distributions u…
Sets that Support a Joint Distribution
该论文研究测度空间上联合分布支撑集的存在性问题，针对离散与连续边缘分布情形给出解答，将离散情形推广至图的流问题，对连续正则集给出简洁刻画。
This paper provides a characterization of all possible dependency structures between two stochastically ordered random variables. The answer is given in terms of copulas that are compatible with the stochastic order and the marginal distributions. The extremal values for Kendall's $τ$ and Spearman's $ρ$ for all these copulas are given in closed form. We also find an explicit form for the joint distribution with the maximal entropy. A multivariate extension and a generalization to random elements in partially ordered spaces are also provided.
There are interesting extensions of the problem of determining a joint probability with known marginals. On the one hand, one may impose size constraints on the joint probabilities. On the other, one may impose additional constraints like the expected values of known random variables. If we think of the marginal probabilities as demands or supplies, and of the joint probability as the fraction of the supplies to be shipped from the production sites to the demand sites, instead of joint probabilities we can think of transportation policies. Clearly, fixing the cost of a transportation policy is…
Last Hitting Time Distributions for Solvable Diffusions
本文针对一维时齐可解扩散过程，建立了计算末次击中时间分布及其与有限时间范围内过程值联合分布的完整分析框架，推导了相关公式并给出应用实例。
On Joint Marginal Expected Shortfall and Associated Contribution Risk Measures
本文提出新型系统性风险测度联合边际期望短缺（JMES）及两种贡献测度，研究其性质与比较条件，并通过数值示例和股市风险传染实证验证其性能。
In this paper, joint asymptotics of powered maxima for a triangular array of bivariate powered Gaussian random vectors are considered. Under the Hüsler-Reiss condition, limiting distributions of powered maxima are derived. Furthermore, the second-order expansions of the joint distributions of powered maxima are established under the refined Hüsler-Reiss condition.
We study the problem of maximizing the probability that (i) an electric component or financial institution $X$ does not default before another component or institution $Y$ and (ii) that $X$ and $Y$ default jointly within the class of all random variables $X,Y$ with given univariate continuous distribution functions $F$ and $G$, respectively, and show that the maximization problems correspond to finding copulas maximizing the mass of the endograph $Γ^\leq(T)$ and the graph $Γ(T)$ of $T=G \circ F^-$, respectively. After providing simple, copula-based proofs for the existence of copulas attaining…
Replication of financial derivatives under extreme market models given marginals
本文研究极端市场模型下金融衍生品的复制问题，通过多期向量鞅最优传输理论，证明了在给定边际分布时存在复制衍生品收益的对冲投资组合，并解决其对偶可达性问题。
This paper focuses on the extreme-value problem for Shannon entropy of the joint distribution with given marginals. It is proved that the minimum-entropy coupling must be of order-preserving, while the maximum-entropy coupling coincides with the independent one. Note that in this sense, we interpret entropy as a measure of system disorder.
Tree-structured Markov random fields with Poisson marginal distributions
该研究提出一类边缘为同均值泊松分布且与依赖结构解耦的树结构马尔可夫随机场，具备高效采样与解析计算优势，并分析了其变量和的分布、单变量贡献及随机序特性。
On random classical marginal problems with applications to quantum information theory
本文研究随机经典边缘问题，通过图编码问题，分析联合分布存在的概率，受Fine定理启发，详细研究CHSH和Bell-Wigner场景中的局部与非信号多面体体积比。
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