In this paper, we establish the existence of a square integrable occupation density for two classes of stochastic processes. First, we consider a Gaussian process with an absolutely continuous random drift, and second we handle the case of a (Skorohod) integral with respect to subfractional Brownian motion with Hurst parameter $H>\frac{1}{2}$. The proof of these results uses a general criterion for the existence of a square integrable local time, which is based on the techniques of Malliavin calculus.

In the context of an over-dispersed count time series data on disease incidences, we consider the Geometric integer-valued autoregressive process of order 1 or GINAR(1), which was first introduced by McKenzie (Adv. Appl. Probab. 18 (1986) 679–705) as an analogue of continuous AR(1) process with exponential margin (Adv. Appl. Probab. 12 (1980) 727–745) on the positive support ($\mathbb{R}^{+}$). A strong enthusiasm still persists as it is apparent from Ristić et al. (J. Stat. Plann. Inf. 139 (2009) 2218–2226). Coherent forecasting of Poisson INAR(1) process due to Al-Osh and Alzaid (J. Time Ser. Anal. 8 (1987) 261–275) was studied by Freeland and McCabe (Int. J. Forecast. 20 (2004) 427–434). Here, we study the $h$-step ahead forecasting distribution corresponding to GINAR(1) process in details using probability generating function. Large sample distributions of the conditional least squares estimates of the model parameters are derived. Some numerical study is performed to illustrate the theoretical results.

This paper revisits the Hölder regularity of mild solutions of parabolic stochastic Cauchy problems in Lebesgue spaces $L^{p}(\mathcal{O})$, with $p\geq2$ and $\mathcal{O}\subset\mathbb{R}^{d}$ a bounded domain. We find conditions on $p,\beta$ and $\gamma$ under which the mild solution has almost surely trajectories in $\mathcal{C}^{\beta}([0,T];\mathcal{C}^{\gamma}(\bar{\mathcal{O}}))$. These conditions do not depend on the Cameron–Martin Hilbert space associated with the driving cylindrical noise. The main tool of this study is a regularity result for stochastic convolutions in M-type 2 Banach spaces by Brzeźniak (Stochastics Stochastics Rep. 61 (1997) 245–295).

In this paper, we investigate the $L^{2}$-consistency and the strong consistency of the maximum likelihood estimators (MLE) of the mean and variance of the sub-fractional Brownian motion with drift at discrete observation. By combining the Stein’s method with Malliavin calculus, we obtain the central limit theorem and the Berry–Esséen bounds for these estimators.

Diverse phenomena from the real-world can be modeled using random matrices, allowing matrix-variate distributions to be considered. The normal distribution is often employed in this modeling, but usually the mentioned random matrices do not follow such a distribution. An asymmetric non-normal model that is receiving considerable attention due to its good properties is the Birnbaum–Saunders (BS) distribution. We propose a statistical methodology based on matrix-variate BS distributions. This methodology is implemented in the statistical software R. A simulation study is conducted to evaluate its performance. Finally, an application with real-world matrix-variate data is carried out to illustrate its potentiality and suitability.

We investigate some properties of the Harris-G class of distributions (Sankhya B 73 (2011) 70–80). We demonstrate that the density function of the Harris-G class can be expressed as a linear combination of density functions of the exponentiated baseline distribution. We provide general formulas for moments (raw, centered, incomplete and factorial), quantile function, generating functions and entropies. Two numerical examples are presented to demonstrate the potentiality of the models in this class. The first one applies the Harris–Burr XII distribution to model bimodal data. The second example uses the Harris-exponential distribution to model SAR image data. The results of the fitted models look promising.

Cosmic microwave background (CMB) Anisotropies is a subject of intensive research in recent years, and therefore it is necessary to develop suitable theory and methods for the analysis of isotropic fields on spheres. The main object of our paper is to show that the polyspectra can be given as the coefficients of the orthogonal expansion of cumulants of the field in terms of irreducible tensor products of spherical harmonics. We obtain necessary and sufficient conditions for isotropy of a non-Gaussian field and the conditions are stated in terms of higher order spectra (polyspectra). The relation between cumulants and spectra gives a new method of estimating spectra.

The residual entropy function introduced by Ebrahimi [Sankhyā A 58 (1996) 48–56], is viewed as a dynamic measure of uncertainty. This measure finds applications in modeling and analysis of life time data. In the present work, we propose nonparametric estimators for the residual entropy function based on censored data. Asymptotic properties of the estimator are established under suitable regularity conditions. Monte Carlo simulation studies are carried out to compare the performance of the estimators using the mean-squared error. The methods are illustrated using two real data sets.

This paper has three specific aims. First, some probability inequalities, including Hölder’s inequality, Lyapunov’s inequality, Minkowski’s inequality, concentration inequalities and Fatou’s lemma for Choquet-like expectation based on a monotone measure are shown, extending previous work of many researchers. Second, we generalize some theorems about the convergence of sequences of random variables on monotone measure spaces for Choquet-like expectation. Third, we extend the concept of uniform integrability for Choquet-like expectation. These results are useful for the solution of various problems in machine learning and made it possible to derive new efficient algorithms in any monotone system. Corresponding results are valid for capacities, the usefulness of which has been demonstrated by the rapidly expanding literature on generalized probability theory.

This paper studies the asymptotic behavior of the number of leaves $L_{n}$ in a random recursive tree $T_{n}$ with $n$ nodes. By utilizing the size-bias method, we derive an upper bound on the Wasserstein distance between the distribution of $L_{n}$ and a standard normal distribution. Furthermore, we obtain a weak version of an Erdös–Rényi type law and a large deviation principle for $L_{n}$.