An Erd\"os--R\'ev\'esz type law of the iterated logarithm for order statistics of a stationary Gaussian process

Let $\{X(t):t\in\mathbb R_+\}$ be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, $\mathbb E X(t) = 0$, $\mathbb E X^2(t) = 1$ and correlation function satisfying (i) $r(t) = 1 - C|t|^{\alpha} + o(|t|^{\alpha})$ as $t\to 0$ for some $0\le\alpha\le 2, C>0$, (ii) $\sup_{t\ge s}|r(t)|0$ and (iii) $r(t) = O(t^{-\lambda})$ as $t\to\infty$ for some $\lambda>0$. For any $n\ge 1$, consider $n$ mutually independent copies of $X$ and denote by $\{X_{r:n}(t):t\ge 0\}$ the $r$th smallest order statistics process, $1\le r\le n$. We provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\mathbb P(\mathscr E_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text{i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb P(\sup_{s\in[0,1]}X_{r:n}(s)>f_p(t))=\mathscr C(t\log^{1-p} t)^{-1}$, $\mathscr C>0$, $\mathbb P(\mathscr E_{f_p})= 1_{\{p\ge 0\}}$. Consequently, with $\xi_p (t) = \sup\{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}$, for $p\ge 0$, $\lim_{t\to\infty}\xi_p(t)=\infty$ and $\limsup_{t\to\infty}(\xi_p(t)-t)=0$ a.s.. Complementary, we prove an Erd\"os-R\'ev\'esz type law of the iterated logarithm lower bound on $\xi_p(t)$, i.e., $\liminf_{t\to\infty}(\xi_p(t)-t)/h_p(t) = -1$ a.s., $p>1$, $\liminf_{t\to\infty}\log(\xi_p(t)/t)/(h_p(t)/t) = -1$ a.s., $p\in(0,1]$, where $h_p(t)=(1/z_p(t))p\log\log t$

On the unitarity of higher-dervative and nonlocal theories

We consider two simple models of higher-derivative and nonlocal quantu systems.It is shown that, contrary to some claims found in literature, they can be made unitary.Comment: 8 pages, no figure

The Efficacy of Renal Replacement Therapy for Rewarming of Patients in Severe Accidental Hypothermia-Systematic Review of the Literature.

Renal replacement therapy (RRT) can be used to rewarm patients in deep hypothermia. However, there is still no clear evidence for the effectiveness of RRT in this group of patients. This systematic review aims to summarize the rewarming rates during RRT in patients in severe hypothermia, below or equal to 32 °C. This systematic review was registered in the PROSPERO International Prospective Register of Systematic Reviews (identifier CRD42021232821). We searched Embase, Medline, and Cochrane databases using the keywords hypothermia, renal replacement therapy, hemodialysis, hemofiltration, hemodiafiltration, and their abbreviations. The search included only articles in English with no time limit, up until 30 June 2021. From the 795 revised articles, 18 studies including 21 patients, were selected for the final assessment and data extraction. The mean rate of rewarming calculated for all studies combined was 1.9 °C/h (95% CI 1.5-2.3) and did not differ between continuous (2.0 °C/h; 95% CI 0.9-3.0) and intermittent (1.9 °C/h; 95% CI 1.5-2.3) methods (p > 0.9). Based on the reviewed literature, it is currently not possible to provide high-quality recommendations for RRT use in specific groups of patients in accidental hypothermia. While RRT appears to be a viable rewarming strategy, the choice of rewarming method should always be determined by the specific clinical circumstances, the available resources, and the current resuscitation guidelines

Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. The setting considered is that of a centered Gaussian process $X\equiv\{X(t):t\in\mathbb R\}$ with stationary increments and variance function $\sigma^2_X(\cdot)$, equipped with a deterministic drift $c>0$, reflected at 0: $$Q_X^{(c)}(t)=\sup_{-\infty<s\le t}(X(t)-X(s)-c(t-s)).$$ We study the resulting stationary workload process $Q^{(c)}_X\equiv\{Q_X^{(c)}(t):t\ge0\}$ in the limiting regimes $c\to 0$ (heavy traffic) and $c\to\infty$ (light traffic). The primary contribution is that we show for both limiting regimes that, under mild regularity conditions on the variance function, there exists a normalizing function $\delta(c)$ such that $Q^{(c)}_X(\delta(c)\cdot)/\sigma_X(\delta(c))$ converges to a non-trivial limit in $C[0,\infty)$

Development of the interatrial wall during the ontogenesis of foetuses and children up to one year of age

Background: The foramen ovale, present in foetal interatrial septum, plays an important role during foetal life. During delivery, foramen ovale closes and becomes fossa ovalis, starting the pulmonary circulation. The aim of our study was to describe the growth of the interatrial wall and changes in location of the foramen ovale, and fossa ovalis during the ontogenesis in the human hearts.Materials and methods: The study was performed on post-mortem material obtained from 92 human hearts from 22nd week of foetal life up to 1 year of age, fixed in a 4% formalin solution.Results: The interatrial wall size in the studied development period was greater in the horizontal than in the vertical dimension. During ontogenesis up to 1 year old, the anterior and inferior parts of the interatrial wall increased their shares considerably by 8% and 6%, respectively. The percentage participation of foramen ovale in the interatrial wall construction in the foetal period formed more than 50% of its size and fairly decreased reaching in infants about 39%.Conclusions: Our study demonstrated that during ontogenesis, from the foetal period to infancy, the parts of the interatrial wall increase their dimensions unevenly. The foramen ovale growth is smaller, compared to the rest of the interatrial wall development. On the basis of our data we can assume that the foramen ovale centre tends to be found in the postero-inferior quadrant of the interatrial wall (foetuses) and in postero-superior quadrant of the interatrial wall — in infants

Prognosis of Hypothermic Patients Undergoing ECLS Rewarming-Do Alterations in Biochemical Parameters Matter?

While ECLS is a highly invasive procedure, the identification of patients with a potentially good prognosis is of high importance. The aim of this study was to analyse changes in the acid-base balance parameters and lactate kinetics during the early stages of ECLS rewarming to determine predictors of clinical outcome. This single-centre retrospective study was conducted at the Severe Hypothermia Treatment Centre at John Paul II Hospital in Krakow, Poland. Patients ≥18 years old who had a core temperature (Tc) &lt; 30 °C and were rewarmed with ECLS between December 2013 and August 2018 were included. Acid-base balance parameters were measured at ECLS implantation, at Tc 30 °C, and at 2 and 4 h after Tc 30 °C. The alteration in blood lactate kinetics was calculated as the percent change in serum lactate concentration relative to the baseline. We included 50 patients, of which 36 (72%) were in cardiac arrest. The mean age was 56 ± 15 years old, and the mean Tc was 24.5 ± 12.6 °C. Twenty-one patients (42%) died. Lactate concentrations in the survivors group were significantly lower than in the non-survivors at all time points. In the survivors group, the mean lactate concentration decreased -2.42 ± 4.49 mmol/L from time of ECLS implantation until 4 h after reaching Tc 30 °C, while in the non-survivors' group (p = 0.024), it increased 1.44 ± 6.41 mmol/L. Our results indicate that high lactate concentration is associated with a poor prognosis for hypothermic patients undergoing ECLS rewarming. A decreased value of lactate kinetics at 4 h after reaching 30 °C is also associated with a poor prognosis

On the infimum attained by a reflected L\'evy process

This paper considers a L\'evy-driven queue (i.e., a L\'evy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided L\'evy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of \prob{M(T_u)> u} (for different classes of functions $T_u$ and $u$ large); here we have to distinguish between heavy-tailed and light-tailed scenarios

Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime

In this paper we derive a technique of obtaining limit theorems for suprema of L\'evy processes from their random walk counterparts. For each $a>0$, let $\{Y^{(a)}_n:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X^{(a)}_t:t\ge 0\}$ be a L\'evy processes such that $X_1^{(a)}\stackrel{d}{=} Y_1^{(a)}$, $\mathbb E X_1^{(a)}<0$ and $\mathbb E X_1^{(a)}\uparrow0$ as $a\downarrow0$. Let $S^{(a)}_n=\sum_{k=1}^n Y^{(a)}_k$. Then, under some mild assumptions, $\Delta(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iff\Delta(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$, for some random variable $R$ and some function $\Delta(\cdot)$. We utilize this result to present a number of limit theorems for suprema of L\'evy processes in the heavy-traffic regime