Probabilities Of Deviations Research Articles

Asymptotic local probabilities of large deviations for branching process in random environment with geometric distribution of descendants

Abstract We consider the branching process Z n = X n , 1 + ⋯ + X n Z n − 1 $ Z_{n} =X_{n, 1} + \dotsb +X_{nZ_{n-1}} $ , in random environments η , where η is a sequence of independent identically distributedvariables, for fixed η the random variables X i, j areindependent, have the geometric distribution. We suppose that the associated random walk S n = ξ 1 + ⋯ + ξ n $ S_n = \xi_1 + \dotsb + \xi_n $ has positive meanμ,0 < h<h +satisfies the right-hand Cramer’s condition E exp(h ξ i ) < ∞ for, some h +. Under theseassumptions, we find the asymptotic representation for local probabilities P (Z n =⌊exp(θ n)⌋) for θ ∈ [θ 1, θ 2]⊂</given−names><x> </x><surname>(μ;μ +) and someμ +.

Large deviations for subcomplex counts and Betti numbers in multiparameter simplicial complexes

We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical Erdős–Rényi graph. We investigate appearance of “unusual” topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices of the multiparameter simplicial complexes. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

Accuracy of Estimation of the Vector of Queue Lengths for Open Jackson Networks

The accuracy of approximation of the vector of queue lengths for an open Jackson network with regenerative input flow and unreliable servers is estimated. A theorem on the accuracy of approximation of the vector of queue lengths in open Jackson networks is put forward, i.e., an estimate for the probability of deviations of the norm of the difference between the process of queue lengths and the constructed reflected Brownian motion is obtained. As a corollary, an estimate of the Wasserstein distance is given.

Resting state network mapping in individuals using deep learning.

Resting state functional MRI (RS-fMRI) is currently used in numerous clinical and research settings. The localization of resting state networks (RSNs) has been utilized in applications ranging from group analysis of neurodegenerative diseases to individual network mapping for pre-surgical planning of tumor resections. Reproducibility of these results has been shown to require a substantial amount of high-quality data, which is not often available in clinical or research settings. In this work, we report voxelwise mapping of a standard set of RSNs using a novel deep 3D convolutional neural network (3DCNN). The 3DCNN was trained on publicly available functional MRI data acquired in n = 2010 healthy participants. After training, maps that represent the probability of a voxel belonging to a particular RSN were generated for each participant, and then used to calculate mean and standard deviation (STD) probability maps, which are made publicly available. Further, we compared our results to previously published resting state and task-based functional mappings. Our results indicate this method can be applied in individual subjects and is highly resistant to both noisy data and fewer RS-fMRI time points than are typically acquired. Further, our results show core regions within each network that exhibit high average probability and low STD. The 3DCNN algorithm can generate individual RSN localization maps, which are necessary for clinical applications. The similarity between 3DCNN mapping results and task-based fMRI responses supports the association of specific functional tasks with RSNs.

Moderate deviation probabilities for empirical distribution of the branching random walk

Given a super-critical branching random walk {Zn}n≥0 on R, let Zn(A) be the number of particles located in some Borel set A⊂R at generation n. Under some mild conditions, it's well-known (e.g., [4]) that Zn(nA)/Zn(R) converges almost surely to ν(A) as n→∞, where ν(⋅) is the standard Gaussian measure on R. Large deviation probabilities of Zn(nA)/Zn(R) have been well studied (see [10] and [27]). In this work, we investigate its moderate deviation probabilities, i.e. the convergence rate ofP(Zn(nA)/Zn(R)≥ν(A)+Δn) as n→∞, where {Δn}n≥0 is some positive sequence tending to zero.

On probabilities of large deviations of combinatorial sums of independent random variables satisfying Linnik’s condition

We derive new results on asymptotic behaviour for probabilities of large deviations of combinatorial sums of independent random variables satisfying Linnik’s condition. We find zones in which these probabilities are equivalent to the tail of the standard normal law. The author earlier obtained such results under Bernstein’s condition. The truncations method is applied in proofs of results.

On asymptotic behaviour for probabilities of moderate deviations of combinatorial sums

We investigate an asymptotic behaviour for probabilities of moderate deviations of combinatorial sums of independent random variables having moments of order p > 2. We find zones in which these probabilities are equivalent to the tail of the standard normal law. The width of the zone is a function from the logarithm of a combinatorial variant for Lyapunov’s ratio. The author earlier obtained similar results under Bernstein’s and Linnik’s conditions. The truncations method is used in proofs of the results.

МЕТОДИКА ВНУТРІШНЬОГО КОНТРОЛЮ РОЗРАХУНКІВ ПІДПРИЄМСТВА ЗА ПОДАТКАМИ ТА ІНШИМИ ОБОВ’ЯЗКОВИМИ ПЛАТЕЖАМИ

The article identifies the main methodical approaches to identifying and solving tasks of the enterprise's internal control over the implementation of fiscal payments. It was established that the main criterion for the effectiveness of measures of the internal control system of the business entity in the context of assessing compliance with the requirements of fiscal legislation is to ensure the optimal use of its resources when solving the relevant tasks. It was determined that appropriate measures should be taken at each of the stages of conducting relevant payments, namely: formalization of tax liabilities, their recognition and repayment. The results of the research made it possible to identify the following measures related to the implementation of company’s payments: determination of the object of taxation, its assessment, selection of the tax rate stipulated by the law, calculation of the tax liability, filling of tax reporting forms, as well as repayment of recognized tax liabilities. At the same time, the influence of a number of factors on them should be taken into account, including regulatory and legal regulation, type of enterprise activity, geographical location, organizational structure, taxation system, functional structure, scope of activity, accounting policy, qualification of accounting service employees and nature of applied information technologies. A number of procedures are proposed to address these challenges, including inspection of documents and records, inspection of assets, observation, external confirmation, recalculation, re-execution and analytical procedures. The study of the content of the relevant procedures gives reason to state that their target direction should relate to the identification of circumstances, the existence of which may indicate a high probability of deviations. Measures carried out by the internal control system in relation to the assessment of the correctness of the definition, completeness and timeliness of the payment of tax obligations give grounds to recognize its preliminary nature, since they are implemented before the official recognition of the relevant parameters by submitting tax reporting.

Deviation probabilities for arithmetic progressions and irregular discrete structures

Deviation probabilities for arithmetic progressions and irregular discrete structures

Large deviations for values of $L$-functions attached to cusp forms in the level aspect

We study the distribution of extreme values of automorphic $L$-functions in a family of holomorphic cusp forms with prime levels. First, we prove an asymptotic formula for a certain density function closely related to this value-distribution. Then we apply it to estimate the probabilities of large deviations for values of automorphic $L$-functions.

Nonasymptotic Convergence Rates for the Plug-in Estimation of Risk Measures

Let ρ be a general law-invariant convex risk measure, for instance, the average value at risk, and let X be a financial loss, that is, a real random variable. In practice, either the true distribution μ of X is unknown, or the numerical computation of [Formula: see text] is not possible. In both cases, either relying on historical data or using a Monte Carlo approach, one can resort to an independent and identically distributed sample of μ to approximate [Formula: see text] by the finite sample estimator [Formula: see text] (μNdenotes the empirical measure of μ). In this article, we investigate convergence rates of [Formula: see text] to [Formula: see text]. We provide nonasymptotic convergence rates for both the deviation probability and the expectation of the estimation error. The sharpness of these convergence rates is analyzed. Our framework further allows for hedging, and the convergence rates we obtain depend on neither the dimension of the underlying assets nor the number of options available for trading.Funding: Daniel Bartl is grateful for financial support through the Vienna Science and Technology Fund [Grant MA16-021] and the Austrian Science Fund [Grants ESP-31 and P34743]. Ludovic Tangpi is supported by the National Science Foundation [Grant DMS-2005832] and CAREER award [Grant DMS-2143861].

Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants

Abstract We consider local probabilities of lower deviations for branching process Z n = X n , 1 + ⋯ + X n , Z n − 1 ${{Z}_{n}}={{X}_{n,1}}+\cdots +{{X}_{n,{{Z}_{n-1}}}}$ in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables X i,j are geometric ones.We suppose that the associated random walk S n = ξ 1 + ⋯ + ξ n ${{S}_{n}}={{\xi }_{1}}+\cdots +{{\xi }_{n}}$ has positive mean μ and satisfies left-hand Cramer’s condition E exp ( h ξ i ) < ∞ if h − < h < 0 $\mathbf{E}\exp \left( h{{\xi }_{i}} \right)<\infty \text{ if }{{h}^{-}}<h<0$ for some h − < − 1. ${{h}^{-}}<-1.$ Under these assumptions, we find the asymptotic representation of local probabilities P ( Z n = ⌊ exp ( θ n ) ⌋ ) for θ ∈ [ θ 1 , θ 2 ] ⊂ ( μ − ; μ ) $\mathbf{P}\left( {{Z}_{n}}=\left\lfloor \exp (\theta n) \right\rfloor \right)\text{ for }\theta \in \left[ {{\theta }_{1}},{{\theta }_{2}} \right]\subset \left( {{\mu }^{-}};\mu \right)$ for some non-negative μ −.

Limit laws for rational continued fractions and value distribution of quantum modular forms

We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in (0, 1] ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallée on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We obtain the Gaussian behaviour of central values of the Esterman function ∑ n ⩾ 1 τ ( n ) e 2 π i n x / n $\sum _{n\geqslant 1} \tau (n) {\rm e}^{2\pi i n x}/\sqrt {n}$ ( x ∈ Q $x\in {\mathbb {Q}}$ ), a problem for which known approaches based on Eisenstein series have been so far ineffective. We give a new proof, based on dynamical systems, that central modular symbols associated with a holomorphic cusp form for S L ( 2 , Z ) $SL(2,{\mathbb {Z}})$ have a Gaussian distribution, and give the first proof of an estimate for their probabilities of large deviations. We also recover a result of Vardi on the convergence of Dedekind sums to a Cauchy law, using dynamical methods.

Forecasting crash risk in U.S. bank returns—The role of credit booms

We propose a new credit variable called probability of aggregate loan trend deviations (pltd) that forecasts crash risk in U.S. bank stock returns. Compared to more classic measures of credit, the main innovation in pltd is that it incorporates a large amount of macroeconomic information in the form of money and interest rate variables known to drive the aggregate loan supply. Positive changes in pltd increase the level of systemic risk in the economy and also trigger lower bank returns and cash flows up to horizons of one year. To disentangle the effect our credit variable has on future bank returns we break down the total variance into individual components and estimate series for news of discount-rates and cash flows. We find that pltd only leads the cash-flow news component without significant effects on the discount-rate news. Finally, we show that pltd is not only useful for predicting the mean of future bank stock returns in-sample and out-of-sample but is also an important driver of the tail of the return distribution proxied by CATFIN.

Clinical and Visual Field Outcomes Following Minimally Invasive Glaucoma Surgery Combined with Cataract Surgery.

PurposeTo determine the change in Humphrey visual field and clinical parameters after minimally invasive glaucoma surgery combined with cataract surgery.Patients and MethodsPatients undergoing minimally invasive glaucoma surgery combined with cataract surgery in a multicenter retrospective case series between 2013 and 2021 with reliable preoperative and 12 to 18 month postoperative visual field measurements were included. Devices included iStent, XEN, and Hydrus. Clinical parameters were compared with a generalized linear model with generalized estimating equations between preoperative and postoperative visits including best corrected visual acuity, intraocular pressure, number of glaucoma medications and visual fields. Visual field metrics included mean deviation (MD), pattern standard deviation (PSD), visual field index (VFI), and Collaborative Initial Glaucoma Treatment Study (CIGTS) score of total deviation probability and pattern deviation probability.ResultsForty-four eyes from 39 patients were included. During the follow up period, visual acuity improved from 0.23±0.17 to 0.10±0.14 logMAR (mean ± standard deviation, p<0.001), number of glaucoma medications was reduced from 2.68±1.06 to 1.46±1.32 (p<0.001), and intraocular pressure decreased from 17.08±4.23 mmHg to 14.92±3.13 mmHg (p=0.003). Differences across devices were negligible. The only significant difference was a greater reduction in number of glaucoma medications in the XEN group (p<0.001). There were no significant changes in the global parameters of VFI, MD, PSD, or CIGTS.ConclusionOverall, minimally invasive glaucoma surgery combined with cataract surgery appears to be effective at stabilizing visual field function, reducing intraocular pressure, reducing number of glaucoma medications, and improving visual acuity over a 12 to 18 month follow-up period across MIGS devices.

24-2 SITA Standard versus 24-2 SITA Faster in Perimetry-Naive Normal Subjects

To compare the Swedish Interactive Thresholding Algorithm (SITA) Standard (SS) and SITA Faster (SFR) strategies in normal individuals undergoing standard automated perimetry (SAP) for the first time. Randomized, comparative, observational case series. Seventy-four perimetry-naive healthy individuals. All individuals underwent SAP 24-2 testing with the Humphrey Field Analyzer III (model 850 Zeiss) using the SS and SFR strategies. One eye of each individual was tested. Test order between strategies was randomized, and an interval of 15 minutes was allowed between the tests. The following variables were compared: test time, foveal threshold, false-positive errors, number of unreliable tests, mean deviation (MD), visual field index (VFI), pattern standard deviation (PSD), glaucoma hemifield test (GHT), and number of depressed points deviating at P < 5%, P < 2%, P < 1%, and P<0.5% on the total and pattern deviation probability maps. Specificity of the SS and SFR strategies were compared using Anderson's criteria for abnormal visual fields. The SFR tests were 60.4% shorter in time compared with SS (P < 0.001) and were associated with a significantly lower PSD (1.75 ± 0.80 decibel [dB] vs. 2.15 ± 1.25 dB; P= 0.016). There were no significant differences regarding the MD, VFI, foveal threshold, GHT, and number of points depressed at P < 5%, P < 2%, P<1%, and P < 0.5% on the total deviation and pattern deviation probability maps between SS and SFR. When all exams were analyzed and any of Anderson's criteria was applied, the specificity was 68% with SFR and 61% with SS (P= 0.250). The specificities observed with SFR and SS when only the first or second exams were analyzed were also similar (63% vs. 64% and 72% vs. 58%, respectively, P > 0.05). The SS and SFR were associated with similar specificities in perimetry-naive individuals. The SFR did not increase the number of depressed points in the total and pattern deviation probability maps. Ophthalmologists should be aware that both strategies are associated with disturbingly high false-positive rates in perimetry-naive individuals. Proprietary or commercial disclosure may be found after the references.

While stability lasts: A stochastic model of noncustodial stablecoins

AbstractThe “Black Thursday” crisis in cryptocurrency markets demonstrated deleveraging risks in over‐collateralized noncustodial stablecoins. We develop a stochastic model that helps explain deleveraging crises in these over‐collateralized systems. In our model, the stablecoin supply is decided by speculators who optimize the profitability of a leveraged position while incorporating the forward‐looking cost of collateral liquidations, which involves the endogenous price of the stablecoin. We formally characterize regimes that are interpreted as stable and unstable for the stablecoin. We prove bounds on quadratic variation (QV) and the probability of large deviations in the stable domain and we demonstrate distinctly greater price variance in the unstable domain. We identify a deflationary deleveraging spiral by means of a submartingale. These deleveraging spirals, which resemble short squeezes, lead to faster collateral drawdown (and potential shortfalls) and are accompanied by higher price variance, as experienced on Black Thursday. We conclude by discussing noncustodial ways in which the issues raised in this paper can be mitigated.

Threat to fixation and vision-related quality of life in early open-angle glaucoma - results from the Glaucoma Intensive Treatment Study.

To determine the effect of glaucomatous visual field (VF) damage close to the point of fixation, called threat-to-fixation (TTF), on vision-related quality of life (VRQoL) in open-angle glaucoma. A total of 239 patients from the Glaucoma Intensive Treatment Study (GITS) were included in this analysis. The second VF of patients with newly diagnosed primarily early glaucoma was evaluated for the presence or absence of TTF. TTF was defined as VF loss including one or more of the four innermost test points depressed at p< 1% in the total deviation probability map of Humphrey 24-2 SITA Standard visual fields. VRQoL was evaluated using Rasch-analysed National Eye Institute Visual Function Questionnaire 25 (NEI VFQ-25) scores. The correlation between VRQoL and TTF was evaluated using uni- and multivariable regression analyses. TTF was present in at least one eye in 115 patients (48%); located in the superior hemifield alone in 47% (54 of 115), in the inferior hemifield alone in 23% (27 of 115), and in 30% (34 of 115) in both hemifields. The median Rasch-calibrated NEI VFQ-25 scores were identical when comparing patients with TTF (VRQoL score 66, 95% CI: 23-100) and those with no-TTF (VRQoL score 66, 95% CI: 21-100) (p= 0.925). Neither the presence of TTF (R2 = -0.004, p= 0.968) nor the location of TTF (R2 = 0.023, p= 0.103) was significantly correlated to Rasch-calibrated NEI VFQ-25 scores. The presence of TTF did not influence VRQoL, as measured by the NEI-VFQ-25, in this relatively large group of patients with mainly early glaucoma.

Comparison between frequency-doubling technology perimetry and standard automated perimetry in early glaucoma

This study aimed to find out the significance of the difference between frequency-doubling technology perimetry (FDT) and standard automated perimetry (SAP) in terms of the detected visual field (VF) damage, and evaluate associated factors to SAP–FDT difference in early glaucoma. Glaucoma patients in early stage (MD better than − 6.0 decibel, 96 eyes) were included in this cross-sectional study. We subtracted mean deviation (MD) and pattern standard deviation (PSD) of FDT from those of SAP, respectively. Additionally, we counted significantly depressed points of P < 5% and P < 1% on the pattern deviation probability plot of both FDT and SAP and defined eyes with significant SAP–FDT difference when the number of abnormal points were greater than 4 points on FDT. We measured lamina cribrosa depth (LCD) and lamina cribrosa curvature index (LCCI) for structural parameters of the optic nerve head from images using enhanced depth imaging of the optical coherence tomography (OCT). Peripapillary vessel density (VD) and presence of microvasculature dropout (MvD), the complete loss of choriocapillaris in localized regions of parapapillary atrophy, was evaluated using deep layer map of OCT angiography (OCT-A) for vascular parameters. Peripheral nasal step (PNS) group had an isolated glaucomatous VF defect within nasal periphery outside 10° of fixation. Parafoveal scotoma (PFS) group had an isolated glaucomatous VF defect within 12 points of a central 10˚ radius. Eyes with significant SAP–FDT difference showed higher detection of MvD on deep layer map of OCT-A, greater LCD, and greater LCCI (all P < 0.05, respectively). In logistic regression analysis, frequent presence of MvD, less presence of disc hemorrhage, and greater LCD were significantly associated with significant SAP–FDT difference. Sub-analysis was performed in eyes with PNS (50 eyes) and PFS (46 eyes). SAP–FDT difference of MD value showed positive association with peripapillary VD on deep layer of OCT-A, which was significant in eyes with PFS compared to eyes with PNS. SAP–FDT difference of PSD value showed negative association with LCCI and LCD, which was significant in eyes with PNS compared to eyes with PFS. Glaucomatous eyes classified by the difference of the detected VF damage on FDT versus SAP showed different clinical features. Greater SAP–FDT difference was significantly associated with structural parameters such as LCD and LCCI. Less SAP–FDT difference was associated with presence of disc hemorrhage and lower deep layer peripepillary VD. There is possibility to use the difference of SAP and FDT to identify associated risk factors in glaucoma patients.

Lower Deviation Probabilities for Level Sets of the Branching Random Walk

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).