We consider closure properties in the class of positively decreasing distributions. Our results stem from different types of dependence, but each type belongs in the family of asymptotically independent dependence structures. Namely, we examine the closure property with respect to minimum, maximum, convolution, convolution roots, and convolution product. Furthermore, we take into account some closure properties of the class of generalized subexponential positively decreasing distributions, as also we introduce and study the class of the generalized long-tailed positively decreasing distributions. Further, we consider the convolution closure problem of subexponentiality in the case of a subexponential positively decreasing class. In some classes we discuss the closure property of randomly stopped sums. Finally, we revisit some problems of infinity divisibility distributions in a subexponential positively decreasing class of distributions, and we study the asymptotic relation between jump measure and Lévy measure of superpositions of the Ornstein--Uhlenbeck process in the case where jump measure has positive and finite Matuszewska indexes.

It is well known that the convergence properties for weighted sums of random variables are important topics in probability limit theory. In this paper, under some suitable conditions, we study the complete convergence and complete moment convergence for randomly weighted sums of arrays of rowwise widely negative dependent random variables in sublinear expectation space, which extend the corresponding ones in classical probability space to the case of sublinear expectation space. As an application, we obtain strong law of large numbers for the randomly weighted sums of arrays of rowwise widely negative dependent random variables.

In this work, by using the von Bahr--Esseen type inequality and Rosenthal type inequality for extended negatively dependent random variables under sublinear expectations, some results on complete convergence for weighted sums of general normalizing extended negatively dependent random variables are established. The results extend some corresponding results in the classical probability space to sublinear expectation space.

In the white Gaussian noise (WGN) model on the half-line, $dV(x)=f(x)\,dx+\varepsilon \, dW(x)$, $x>0$, minimax linear estimators of the functional $x^{1/2}_0f(x_0)$ are sought for a given $x_0>0$. Here, $f$ is an unknown signal satisfying a certain restriction $f\in\mathfrak{F}$, $\varepsilon>0$ is a given parameter, and $W(\,{\cdot}\,)$ is the standard Wiener process. With the use of the Mellin transform, it is shown that the above problem is equivalent to a similar (but better known) WGN model on the whole line, $dY(u)=g(u)\,du+\varepsilon \, dW(u)$, $-\infty<u<\infty$, where minimax linear estimators of the functional $g(u_0)$ are sought under a matching restriction $g\in \mathfrak{F}_0$, for $u_0=\log x_0.$ This demonstrates a close connection between the two models, which permits one to translate results on minimax linear estimation easily from one model to the other. In particular, it leads to minimax linear estimators on the half-line for a variety of ellipsoidal and cuboidal function classes $\mathfrak{F}$.

We find the limit distribution of the statistic describing the mutual deviation of projection estimators of the distribution densities in $p\geq 2$ independent samples. A goodness-of-fit test is also constructed, and examples are given.

In this paper, we propose a condition on a probability distribution that enables us to determine the distribution uniquely using all of its moments. This condition applies to both the Hamburger case (distributions on the entire real line) and the Stieltjes case (distributions on the positive half-line). The proposed condition is easily verifiable and is connected to both the Lin condition and the converse Krein condition. In addition, we provide another determinacy condition for both Hamburger and Stieltjes moment problems, which is similar to the Lin condition but does not necessitate the converse Krein condition.

In this paper, we introduce a new algorithm based on the one-step improvement to solve the optimal multiple stopping problem under Knightian uncertainty in the discrete time case. This algorithm induces a monotonically increasing sequence to approximate the value functions, which coincides with the value functions after finitely many iteration steps independent of the times of exercise rights. We also present the stability of this algorithm and some numerical simulations.

Van Moerbeke [Arch. Rational Mech. Anal., 60 (1976), pp. 101--148] proved that there is a correspondence between optimal stopping problems on finite and infinite time intervals. In the present paper, such a correspondence is constructed via different ideas from statistical sequential analysis.

In this paper, we obtain a new generalization of a Chebyshev--Markov type inequality. Then we apply it to the symmetric distribution and the skewed distribution respectively, showing that it is sharper than the original Chebyshev inequality.