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A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some of its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, statistical parameter estimation, and relations to Sibuya distributions and Yule–Simon distributions are discussed. This toy model is relevant in several applications including reliability, species sampling problems, record values breaking and random walks with disasters.

The present study explored acute responses of heart rate (HR) variability (HRV) metrics, incorporating the nonlinear index alpha 1 of detrended fluctuation analysis (DFAa1) during passive recovery, providing information about correlation properties of HR time series during the regulation of recovery processes. Recreationally active female (n = 13) and male (n = 13) participants participated. In a first session, a graded exercise test was conducted to determine peak HR (HRPEAK) and peak oxygen consumption (VO2PEAK). In a second and third session, participants completed an endurance training with moderate intensity (MOD) on a treadmill and an exergaming training with vigorous intensity (VIG), randomized and counterbalanced. Before and up to 45 min after the respective exercise sessions, RR‐interval and hemodynamic measurements (peripheral systolic, SBP; diastolic blood pressure, DBP; and pulse wave velocity, PWV) were conducted. Internal load analysis of MOD versus VIG revealed significant differences and appropriate prescription of intensity domains during exercise (%HRPEAK: ∼66% vs. 86% and %VO2PEAK: ∼48% vs. 66%). The present data showed significant main effects of time, intensity, and their interaction for all RR‐interval outcomes, PWV, and SBP. DFAa1 demonstrated a stronger correlated reorganization and overcompensation after VIG, with higher values and therefore increased correlation properties throughout the recovery process. The present data suggest that VIG transiently delays the recovery of cardiac parasympathetic activity and the normalization of correlation properties of HR time series. Regarding acute early and delayed recovery processes, higher correlation properties may reflect more order (less complexity) and interaction of involved physiological subsystems, supporting the assumption of increased systemic control to process the demands of higher exercise intensity. Highlights Higher exercise intensity results in delayed reorganization, transiently delays parasympathetic reactivation, and decreases values of linear HRV metrics during passive recovery. The present study adds to the current body of research by showing that DFAa1, as a nonlinear metric of HRV, exhibits a stronger correlated reorganization and overcompensation after a vigorous compared to a moderate exercise bout. Higher correlation properties may indicate more order and interaction of the involved control processes managing recovery and suggests a stronger systemic control to process the demands of higher exercise intensities. Further research is needed to verify the potential of using standardized postexercise linear and nonlinear HRV metrics as a monitoring tool for individual cardiac reactivation patterns.

The probability h(n, m) that the block counting process of the Bolthausen-Sznitman n-coalescent ever visits the state m is analyzed. It is shown that the asymptotic hitting probabilities h(m) = lim n→∞ h(n, m), m ∈ N, exist and an integral formula for h(m) is provided. The proof is based on generating functions and exploits a certain convolution property of the Bolthausen-Sznitman coalescent. It follows that h(m) ∼ 1/log m as m → ∞. An application to linear recursions is indicated.

This study aimed to investigate whether a single bout of Self-myofascial Release (SMR) has a beneficial effect on peripheral and central Blood Pressure (BP) and different parameters of arterial stiffness. Twenty nine healthy male recreational athletes (26.1 ± 2.9 years, BMI 23.4 ± 1.5 kg/m 2 ) completed an instructed SMR using a foam roller. Peripheral and central BP and different parameters of arterial stiffness were measured noninvasively before SMR and at different time points (t1, t15, t30) during a subsequent 30-min recovery phase. There was a significant decrease in both systolic (t15, −2.36 ± 4.45 mmHg, p = 0.05; t30, −4.01 ± 4.47 mmHg, p = 0.003) and diastolic (t30, −2.45 ± 5.45 mmHg, p = 0.025) peripheral pressure during the recovery phase after SMR. Regarding central BP, only systolic pressure showed a significant decrease (t30, −3.64 ± 5.83 mmHg, p = 0.003). Mean arterial pressure (t15, −1.91 ± 3.36, p = 0.03; t30, −3.05 ± 2.88 mmHg, p < 0.001), augmentation pressure (t30, −1.60 ± 2.40 mmHg, p = 0.009), peripheral resistance (t30, −0.09 ± 0.10 s * mmHg/ml, p < 0.001), and stiffness index β 0 (t30, −0.33 ± 0.55, p = 0.021) were significantly reduced after SMR. No significant changes were determined for reflection coefficient, augmentation index, cardiac output, and heart rate, respectively. SMR showed effects on peripheral and central BP and different parameters of arterial stiffness in healthy young adults. A single bout of self-myofascial release confers favourable cardiovascular benefits. Self-myofascial release induces mechanical stress to the vascular system. Effects on vascular function is comparable to the effects after a bout of exercise.

Due to physiological and anatomical sex differences, there are variations in the training response, and the recovery periods following exercise may be different. High-intensity interval training (HIIT) protocols are well-suited to differentially investigate the course of recovery. This study was conducted to determine sex-specific differences in the recovery following HIIT intervals interspersed with recovery phases of different lengths. Methods: Well-trained cyclists and triathletes (n = 11 females, n = 11 males) participated in this study. There were no significant sex differences in maximal heart rate (HR), relative peak power to body mass and fat-free mass, training volume, and VO2max-percentiles (females: 91.8 ± 5.5 %, males: 94.6 ± 5.4 %). A 30 s Wingate test was performed four times, separated by different active recovery periods (1, 3, or 10 min). Lactate, HR, oxygen uptake, and subjective rating of exertion and recovery were determined. Results: For the recovery time of three and ten minutes, men showed significantly higher lactate concentrations (p = 0.04, p = 0.004). Contrary, HR recovery and subjective recovery were significant slower in women than in men. Conclusion: During HIIT, women may be more resistant to fatigue and have a greater ability to recover metabolically, but have a slower HR and subjective recovery.

Due to physiological and morphological differences, younger and older athletes may recover differently from training loads. High-intensity interval training (HIIT) protocols are useful for studying the progression of recovery. It was the objective of this study to determine age differences in performance and recovery following different HIIT protocols. Methods: 12 younger (24.5 ± 3.7 years) and 12 older (47.3 ± 8.6 years) well-trained cyclists and triathletes took part in this study. Between the age groups there were no significant differences in relative peak power to fat-free mass, maximal heart rate (HR), training volume, and VO2max-percentiles (%). Participants performed different HIIT protocols consisting of 4 × 30 s Wingate tests with different active rest intervals (1, 3, or 10 min). Peak and average power, lactate, HR, respiratory exchange ratio (RER), subjective rating of perceived exertion (RPE), and recovery (Total Quality Recovery scale, TQR) were assessed. Results: During the different HIIT protocols, metabolic, cardiovascular, and subjective recovery were similar between the two groups. No significant differences were found in average lactate concentration, peak and average power, fatigue (%), %HRmax, RER, RPE, and TQR values between the groups (p > 0.05). Conclusion: The findings of this study indicate that recovery following HIIT does not differ between the two age groups. Furthermore, older and younger participants displayed similar lactate kinetics after the intermittent exercise protocols.

An extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer N let (X N (r)) r be a Markov chain with the same finite state space S and transition matrix Π N =I+d N B N , where I is the unit matrix, Q a generator matrix, (B N ) N a sequence of matrices, lim N℩∞ c N = lim N→∞d N =0 and lim N→∞ c N ∕d N =0. Suppose that the limits P≔lim m→∞(I+d N Q) m and G≔lim N→∞ P B N P exist. If the sequence of initial distributions P X N (0) converges weakly to some probability measure μ, then the finite-dimensional distributions of (X N ([t∕c N )) t≥0 converge to those of the Markov process (X t ) t≥0 with initial distribution μ, transition matrix Pe t G and lim N→∞(I+d N Q+c N B N )[t∕c N ]

In addition to the features of the two-parameter Chinese restaurant process (CRP), the restaurant under consideration has a cocktail bar and hence allows for a wider range of (bar and table) occupancy mechanisms. The model depends on three real parameters, $\\alpha$ , $\\theta_1$ , and $\\theta_2$ , fulfilling certain conditions. Results known for the two-parameter CRP are carried over to this model. We study the number of customers at the cocktail bar, the number of customers at each table, and the number of occupied tables after n customers have entered the restaurant. For $\\alpha>0$ the number of occupied tables, properly scaled, is asymptotically three-parameter Mittag–Leffler distributed as n tends to infinity. We provide representations for the two- and three-parameter Mittag–Leffler distribution leading to efficient random number generators for these distributions. The proofs draw heavily from methods known for exchangeable random partitions, martingale methods known for generalized Pólya urns, and results known for the two-parameter CRP.

We consider a class of haploid population models with nonoverlapping generations and fixed population size N assuming that the family sizes within a generation are exchangeable random variables. A weak convergence criterion is established for a properly scaled ancestral process as N → ∞. It results in a full classification of the coalescent generators in the case of exchangeable reproduction. In general the coalescent process allows for simultaneous multiple mergers of ancestral lines.

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)-coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1, b)-coalescent by exploiting the method of sequential approximations.