The beneficial effects of high-intensity interval training (HIIT) and chlorella vulgaris (CV) on body composition and mitochondrial biogenesis have been shown in some mechanistic studies. This study aimed to determine the effects of CV and/or HIIT on mitochondrial biogenesis, performance and body composition among overweight/obese women. There was a significant reduction in the fat mass (FM) of the CV + HIIT group, as compared with the placebo group (P = 0·005). A marginal significant increase in body water (P = 0·050) and PPAR-γ coactivator-1α (P = 0·050) was also found only in the CV + HIIT group, as compared with the placebo. Relative (P < 0·001) and absolute (P < 0·001) VO2max, as well as Bruce MET (P < 0·001), were significantly increased in the HIIT and HIIT + CV groups. Besides, the synergistic effect of CV and HIIT on the Bruce MET increment was found (interaction P-value = 0·029). No significant changes were observed in BMI, fat-free mass, visceral fat, silent information regulator 1 and fibroblast growth factor-21. In this randomised clinical trial, forty-six overweight/obese women were assigned to four groups including CV + HIIT and HIIT + placebo groups that received three capsules of CV (300 mg capsules, three times a day) or corn starch, in combination with three sessions/week of HIIT. CV and placebo groups only received 900 mg of CV or corn starch, daily, for 8 weeks. Biochemical assessments, performance assessment and body composition were obtained at the beginning and end of the intervention. HIIT may be, therefore, effective in improving mitochondrial biogenesis, performance and body composition in overweight/obese women.

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

Let {Xn, n ≥ 1} be an independent, identically distributed random sequence with each Xn having the general error distribution. In this paper we derive the exact uniform convergence rate of the distribution of the maximum to its extreme value limit.

Let {Xk, k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {wk, k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑k=1nwkXk and the maximum of weighted sums max1≤m≤n∑k=1mwkXk, subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss Xk during the kth year, with a discount or inflation factor wk.

The almost sure convergence of the maximum of a strictly stationary sequence is studied. We show that, if a particular mixing condition, related to the asymptotic independence of maxima defined by O'Brien, is satisfied, then the maximum behaves asymptotically like the quantile F←(1-1/n) whenever the marginal tail distribution decreases quickly enough. A necessary condition for the almost sure convergence of the maximum is derived. Applications are also discussed.

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.

Previous work on the joint asymptotic distribution of the sum and maxima of Gaussian processes is extended here. In particular, it is shown that for a stationary sequence of standard normal random variables with correlation function r, the condition r(n)ln n = o(1) as n → ∞ suffices to establish the asymptotic independence of the sum and maximum.

Let {Xn, n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = EX0Xn. Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

There is a well-known connection between the asymptotic behaviour of the tail of the distribution of the increasing ladder height and the integrated tail of the step distribution of a random walk which either drifts to –∞, or oscillates and whose decreasing ladder height has finite mean. We establish a similar connection in a local sense; this means that in the lattice case we link the asymptotic behaviours of the mass function of the ladder height distribution and of the tail of the step distribution. We deduce the asymptotic behaviour of the mass function of the maximum of the walk, when this is finite, and also treat the non-lattice case.

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

Joint densities concerning in particular the value and time of the maximum over a fixed time interval, or the behavior over intervals determined by some first- and last-passage times, are determined for Brownian motion, the three-dimensional Bessel process and Brownian meander. Simple change of measure formulas permit easy passage from one process to the other. Examples are given.

This note is concerned with N, the time at which a random walk first exits from [0,∞), and M, the maximum of the random walk up to time N. In the case that the random walk has zero mean and finite variance, simple proofs are given of asymptotic estimates for P{M > x}, P{N ≦ ux2|M > x} and P{M ≦ v √n|N > n}.

A general formula is obtained for the variance of the maximum of partial sums of n-exchangeable random variables, derived from a result of Spitzer's. The formula is applied in particular to obtain the variance of the maximum of adjusted rescaled partial sums of normal summands. This is of direct relevance to the Hurst effect.

Let be a family of random walks with For ε↓0 under certain conditions the random walk U(∊)n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M∊ = max {U(∊)n, n ≧ 0}, v0 = min {n : U(∊)n = M∊}, v1 = max {n : U(∊)n = M∊}. The joint limiting distribution of ∊2σ∊–2v0 and ∊σ∊–2M∊ is determined. It is the same as for ∊2σ∊–2v1 and ∊σ–2∊M∊. The marginal ∊σ–2∊M∊ gives Kingman's heavy traffic theorem. Also lim ∊–1P(M∊ = 0) and lim ∊–1P(M∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U(∊)n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.

Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.