Polya, inverse Polya, and circular Polya distributions of order k for l-overlapping success runs

The Polya-Eggenberger sampling scheme is considered, i.e., a ball is drawn at random from an urn containing w white (success) balls and b black (failure) balls, its color is observed, and then it is returned to the urn along with s additional balls of the same color of the ball drawn. This scheme is repeated n times, and N-n,N-k,N-l,N-s denotes the number of l-overlapping success runs of length k. If the balls drawn are arranged on a circle, the number of I-overlapping success runs of length k is denoted by N-n,k,l,s(c).. Finally, W-r,W-k,W-l,W-s denotes the number of drawings according to the Polya-Eggenberger sampling scheme until the rth occurrence of the l-overlapping success run of length k. Polya, inverse Polya, and circular Polya distributions of order k for l-overlapping success runs of length k are introduced as the distributions, respectively, of N-n,N-k,N-l,N-s, W-r,W-k,W-l,W-s, and N-n,k,l,s(c). These distributions include as special cases known and new distributions of order k. Exact formulae are derived for their probability distribution functions and means, which generalize several results on well-known distributions of order k. Asymptotic results of the new distributions are also given, relating them, respectively, to the binomial, negative binomial, and circular binomial distributions of order k for l-overlapping success runs of length k.