Joint distributions of numbers of occurrences of a discrete pattern and weak convergence of an empirical process for the pattern

For a {0, 1}-pattern of finite length, an empirical process is introduced in order to describe the number of overlapping occurrences of the pattern at each level t∈[0,1]t∈[0,1] in a sequence of the corresponding indicators of i.i.d. [0, 1]-valued observations of length nn. A method for obtaining the exact finite-dimensional distributions of the empirical process is given. The weak convergence of the process to a Gaussian process in D[0,1]D[0,1] as nn tends to infinity is also established. The limiting process depends on the given pattern. The exact covariance function is compared with the asymptotic covariance function in a numerical example.