We consider the tail behavior of the product of two dependent random variables X and $\Theta $. Motivated by Denisov and Zwart (J Appl Probab 44:1031–1046, 2007), we relax the condition of the existing $\alpha \,+\,\epsilon $ th moment of $\Theta $ in Breiman’s theorem to the existing $\alpha $th moment and obtain the similar result as Breiman’s theorem of the dependent product $X \Theta $, while X and $\Theta $ follow a copula function. As applications, we consider a discrete-time insurance risk model with dependent insurance and financial risks and derive the asymptotic tail behaviors for the (in)finite-time ruin probabilities.

Let G be a finite group and $\mathscr {F}$ a saturated formation of finite groups. Then G is a quasi-$\mathscr {F}$-group if for every $\mathscr {F}$-eccentric chief factor H / K of G and every $x\in G$, x induces an inner automorphism on H / K. In this article, we obtain some results about the quasi-$\mathscr {F}$-groups and use them to give the conditions under which a group is quasisupersoluble.

This paper proposes a new customer lifetime model: the Gamma/Weibull distribution (G/W). Similar to the Pareto/NBD model, we propose a G/W/NBD model by combining the G/W distribution with a negative binomial distribution (NBD) and study its properties such as (i) the probability that a customer to be alive at a time point; (ii) the expectation and variance of the number of transactions for a customer during a fixed time period; (iii) the conditional expectation and conditional variance of the number of future transactions for a customer during a fixed time period. Several simulation studies are conducted to investigate the forecasting accuracy and flexibility of the proposed model. A CDNOW data set is analyzed by the proposed model.

Given a binary quadratic polynomial $f(x_1,x_2)=\alpha x_1^2+\beta x_1x_2+\gamma x_2^2\in \mathbb {Z}[x_1,x_2]$, for every $c\in \mathbb Z$ and $n\ge 2$, we study the number of solutions $\mathrm {N}_J(f;c,n)$ of the congruence equation $f(x_1,x_2)\equiv c\bmod {n}$ in $(\mathbb {Z}/n\mathbb {Z})^2$ such that $x_i\in (\mathbb {Z}/n\mathbb {Z})^\times $ for $i\in J\subseteq \{1,2\}$.

Let $\varGamma $ be a distance-regular graph of diameter 3 with strong regular graph $\varGamma _3$. The determination of the parameters $\varGamma _3$ over the intersection array of the graph $\varGamma $ is a direct problem. Finding an intersection array of the graph $\varGamma $ with respect to the parameters $\varGamma _3$ is an inverse problem. Previously, inverse problems were solved for $\varGamma _3$ by Makhnev and Nirova. In this paper, we study the intersection arrays of distance-regular graph $\varGamma $ of diameter 3, for which the graph ${\bar{\varGamma }}_3$ is a pseudo-geometric graph of the net $PG_{m}(n, m)$. New infinite series of admissible intersection arrays for these graphs are found. We also investigate the automorphisms of distance-regular graph with the intersection array $\{20,16,5; 1,1,16 \}$.

We focus on the complexity of a special flow built over an irrational rotation of the unit circle and under a roof function on the unit circle. We construct a weak mixing minimal special flow with bounded topological complexity. We also prove that if the roof function is $C^\infty $, then the special flow has sub-polynomial topological complexity and the time one map meets the condition of Sarnak’s conjecture.