On first-passage problem for a non-singular Gaussian discrete-time series

For a non-singular Gaussian discrete-time series, the evaluation of the first passage time probability distribution through a bounded time-varying boundary is considered by using orthant probabilities. Sufficient conditions are given for the existence of finite mean and variance of the first passage time random variable. The computation of orthant probabilities is examined regard to the existing literature. As working examples, the first passage time problem for the discrete-time versions of the fractional white noise is considered.