Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson's integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT

A comparison of airway interventions and gastrostomy tube placement in infants with Robin sequence

The purpose of this study was to evaluate feeding impairment following non-operative or operative management of airway obstruction in a large series of infants with Robin sequence (RS) by rate of G-tube placement. A retrospective study was conducted at Boston Children's Hospital including 225 patients (47.1% female) with RS treated between 1976 and 2018. Subjects were grouped by intervention required for successful management of airway obstruction: non-operative only (n = 120), tongue–lip adhesion (TLA, n = 75), mandibular distraction osteogenesis (MDO, n = 21), or tracheostomy (n = 9). The operative group had a higher rate of G-tube placement (58.1%) than the non-operative group (28.3%, P < 0.0001). Subjects in the TLA and tracheostomy groups had higher odds of G-tube placement than subjects in the MDO group: odds ratio (OR) 5.5 (95% confidence interval (CI) 1.8–17.3, P = 0.004) and OR 27.0 (95% CI 3.2–293.4, P = 0.007), respectively. Syndromic patients and those with gastrointestinal anomalies also had higher odds of G-tube placement: OR 3.5 (95% CI 1.7–7.2, P = 0.001) and OR 5.9 (95% CI 1.6–21.0, P = 0.007), respectively. Infants with RS who require an airway operation and those with a syndromic diagnosis or gastrointestinal anomalies are more likely to require placement of a G-tube. Of the operative groups, MDO was associated with the lowest G-tube rate, compared to TLA and tracheostomy

Two-loop O(GF2MH4){\rm O}\left(G_F^2M_H^4\right) corrections to the fermionic decay rates of the Higgs boson

We calculate the dominant ${\rm O}\left(G_F^2M_H^4\right)$ two-loop electroweak corrections to the fermi\-onic decay widths of a heavy Higgs boson in the Standard Model. Use of the Goldstone-boson equivalence theorem reduces the problem to one involving only the physical Higgs boson $H$ and the Goldstone bosons $w^\pm$ and $z$ of the unbroken theory. The two-loop corrections are opposite in sign to the one-loop electroweak corrections, exceed the one-loop corrections in magnitude for $M_H>1114\ {\rm GeV}$, and increase in relative magnitude as $M_H^2$ for larger values of $M_H$. We conclude that the perturbation expansion in powers of $G_FM_H^2$ breaks down for $M_H\approx 1100\ {\rm GeV}$. We discuss briefly the QCD and the complete one-loop electroweak corrections to $H\rightarrow b\bar{b}, \,t\bar{t}$, and comment on the validity of the equivalence theorem. Finally we note how a very heavy Higgs boson could be described in a phenomenological manner.Comment: 24 pages, RevTeX file, 4 figures in a separate compressed uuencoded Postscript file or available by mail on request. Fig. 1 not included see Figs. 1, 2 in Phys. Rev. D 48, 1061 (1993

Hyoid bone morphology in patients with isolated robin sequence – A case-control study utilizing 3D morphable models

Background: Abnormalities of the hyoid bone are associated with impairment of oropharyngeal functions including feeding, swallowing, and breathing. Few studies have characterized anatomic abnormalities of the hyoid in patients with Robin sequence (RS), e.g. a less mineralized and voluminous hyoid. The purpose of this study was to compare normal hyoid bone morphology and hyoid bone morphology in children with isolated RS. Methods: Three-dimensional (3D) reconstructions of the hyoid bone were obtained from CT-imaging of children with RS and unaffected controls. A 3D morphable model was constructed using Principal Component Analysis (PCA). Partial least squares – Discriminant Analysis (PLS-DA) and multivariate analysis of variance (MANOVA) were used to characterize and compare hyoid shape differences between patients with RS and an age-matched control group. Results: The study included 23 subjects with RS (mean age 9.8 ± 10.3 months) and 46 age-matched control samples. A less voluminous hyoid was observed for the RS group with a larger lateral divergence of the greater horns compared to controls (MANOVA, p-value<0.001). The first shape variable from the PLS-DA model showed a significant correlation for the observed variance between the two groups (Spearman R = −0.56, p-value<0.001). The control samples and 151 CT-scans of subjects up to age 4 years were used to create a 3D morphable model of normal hyoid shape variation (n = 197, mean age 22.1 ± 13.1 months). For the normal 3D morphable model, a high degree of allometric shape variation was observed along the first principal component. Conclusions: The 3D morphable models provide a comprehensive and quantitative description of variation in normal hyoid bone morphology, and allow detection of distinct differences between patients with isolated RS and controls