Approximations of the tail probability of the product of dependent extremal random variables and applications

•The tail probability of products of dependent random variables.•Extend the bivariate product to the multivariate case.•Results for different max-domain of attractions.•Ruin probabilities for a discrete-time insurance risk model as applications.

In this paper, we investigate the tail probability of the product X∏i=1nYi, where (X,Y1,…,Yn) follows a multivariate Sarmanov distribution. An explicit asymptotic formula is established for the tail probability of the product when X belongs to the Fréchet, Gumbel, or Weibull max-domain of attraction. As applications, we consider a discrete-time risk model with dependent insurance and financial risks, and obtain the asymptotic behavior for the (in)finite-time ruin probabilities.