scholarly journals The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem

2019 ◽  
Vol 51 (03) ◽  
pp. 835-864 ◽  
Author(s):  
Yuqing Pan ◽  
Konstantin A. Borovkov
Keyword(s):  
Large Deviation ◽  
Risk Process ◽  
Boundary Crossing ◽  
Moment Condition ◽  
Two Dimensional ◽  
Exact Asymptotics ◽  
Negative Component ◽  
Positive Orthant ◽  
Mean Vector ◽  
Target Sets

AbstractFor a multivariate random walk with independent and identically distributed jumps satisfying the Cramér moment condition and having mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results of Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a series of papers by Borovkov and Mogulskii from around 2000 with new auxiliary constructions, enabling us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a ‘corner’ at the ‘most probable hitting point’. We also discuss how our results can be extended to the case of more general target sets.

Large-deviation approximations to the distribution of scan statistics

1991 ◽  
Vol 23 (04) ◽  
pp. 751-771 ◽  
Author(s):  
Clive R. Loader
Keyword(s):  
Large Deviation ◽  
Two Dimensions ◽  
Unit Interval ◽  
Scan Statistics ◽  
Boundary Crossing ◽  
Two Dimensional ◽  
Likelihood Ratio Criterion ◽  
Scan Statistic ◽  
Window Width ◽  
Ratio Criterion

Suppose a Poisson process is observed on the unit interval. The scan statistic is defined as the maximum number of events observed as a window of fixed width is moved across the interval, and the distribution under homogeneity has been widely studied. Frequently, we may not wish to specify the window width in advance but to consider scan statistics with varying window widths. We propose a modification of the scan statistic based on a likelihood ratio criterion. This leads to a boundary-crossing problem for a two-dimensional random field, which we approximate using a large-deviation scaling under homogeneity. Similar results are obtained for Poisson processes observed in two dimensions. Numerical computations and simulations are used to illustrate the accuracy of the approximations.

Large-deviation approximations to the distribution of scan statistics

1991 ◽  
Vol 23 (4) ◽  
pp. 751-771 ◽  
Author(s):  
Clive R. Loader
Keyword(s):  
Large Deviation ◽  
Two Dimensions ◽  
Unit Interval ◽  
Scan Statistics ◽  
Boundary Crossing ◽  
Two Dimensional ◽  
Likelihood Ratio Criterion ◽  
Scan Statistic ◽  
Window Width ◽  
Ratio Criterion

Suppose a Poisson process is observed on the unit interval. The scan statistic is defined as the maximum number of events observed as a window of fixed width is moved across the interval, and the distribution under homogeneity has been widely studied. Frequently, we may not wish to specify the window width in advance but to consider scan statistics with varying window widths. We propose a modification of the scan statistic based on a likelihood ratio criterion. This leads to a boundary-crossing problem for a two-dimensional random field, which we approximate using a large-deviation scaling under homogeneity. Similar results are obtained for Poisson processes observed in two dimensions. Numerical computations and simulations are used to illustrate the accuracy of the approximations.

scholarly journals Tails in generalized Jackson networks with subexponential service-time distributions

2005 ◽  
Vol 42 (2) ◽  
pp. 513-530 ◽  
Author(s):  
François Baccelli ◽  
Serguei Foss ◽  
Marc Lelarge
Keyword(s):  
Closed Form ◽  
Service Time ◽  
Large Deviation ◽  
Exact Asymptotics ◽  
Fluid Limits ◽  
Jackson Networks ◽  
Single Station ◽  
Service Times

We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

scholarly journals Two-dimensional stochastic modeling for predicting bankruptcy for manufacturing companies

2021 ◽  
Vol 54 (2) ◽  
pp. 123-129
Author(s):  
James C. Fu ◽  
Winnie H. W. Fu
Keyword(s):  
Brownian Motion ◽  
Real Data ◽  
Bankruptcy Prediction ◽  
Boundary Crossing ◽  
Time Interval ◽  
Two Dimensional ◽  
Manufacturing Companies ◽  
Boundary Crossing Probability ◽  
Data Set ◽  
Crossing Probability

Increasing accuracy of the model prediction on business bankruptcy helps reduce substantial losses for owners, creditors, investors and workers, and, further, minimize an economic and social problem frequently. In this study, we propose a stochastic model of financial working capital and cashflow as a two-dimensional Brownian motion X(t) = (X1(t),X2(t)) on the business bankruptcy prediction. The probability of bankruptcy occurring in a time interval [0,T] is defined by the boundary crossing probability of the two-dimensional Brownian motion entering a predetermined threshold domain. Mathematically, we extend the result in Fu and Wu (2016) on the boundary crossing probability of a high dimensional Brownian motion to an unbounded convex hull. The proposed model is applied to a real data set of companies in US and the numerical results show the proposed method performs well.

scholarly journals Construction of two-dimensional topological field theories with non-invertible symmetries

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Tzu-Chen Huang ◽  
Ying-Hsuan Lin ◽  
Sahand Seifnashri
Keyword(s):  
Topological Field Theories ◽  
Hilbert Spaces ◽  
Topological Defect ◽  
Field Theories ◽  
Boundary Crossing ◽  
Two Dimensional ◽  
Fusion Categories ◽  
Explicit Constructions ◽  
Crossing Symmetry ◽  
Topological Field

Abstract We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup ℋ3 fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.

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