Strong Gaussian approximations of product-limit and quantile processes for truncated data under strong mixing

In this paper, we consider the product-limit quantile estimator of an unknown quantile function under a truncated dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and normed product-limit quantile process are constructed with rate O((logn)−λ) for some λ>0. The strong Gaussian approximation of the product-limit process is then applied to derive the law of the iterated logarithm for the product-limit process.