The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant, and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of independent and identically distributed random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of independent and identically distributed exponential random variables.

We study a skew Ornstein–Uhlenbeck process with zero being a sticky reflecting boundary, which is defined as the weak solution to a stochastic differential equation (SDE) system involving local time. The main results obtained include: (i) the existence and uniqueness of solutions to the SDE system, (ii) the scale function and speed measure, and (iii) the distributional properties regarding the transition density and the first hitting times. On the application side, we apply the process to interest rate modeling and obtain the explicit pricing formula for zero-coupon bonds. Numerical examples illustrate the impacts on bond yields of skewness and stickiness parameters.

The asymptotic behavior of the Jaccard index in G(n, p), the classical Erdös–Rényi random graph model, is studied as n goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in G(n, p) is shown to be asymptotically normal under an additional mild condition that $np\to\infty$ and $n^2(1-p)\to\infty$.