Embedding of Urn Schemes into Continuous Time Markov Branching Processes and Related Limit Theorems

Limit theorems for triangular urn schemes

“…An important case, which includes many applications, is when the replacement Date: March 24, 2004; revised March 18, 2005. matrix is irreducible; in our setting with two colours this means that β, γ > 0. Limit theorems for the irreducible case have been given by many authors, for example [2,3,4]; see also [14] and the further references given there. In contrast, we will here study the case of a triangular replacement matrix, i.e.…”

“…We will use the embedding method of Athreya and Karlin [2,3]: Let (X (t), Y(t)), t ≥ 0 be a continuous time Markov branching process with particles of two types (black and white); each particle lives a random time with an exponential distribution Exp (1), and on its death a black [white] particle is replaced by 1 + α black and β white [γ black and 1 + δ white] new particles. It is then easy to see that (X (t), Y(t)) observed at the (a.s. distinct) times of deaths gives the urn process above.…”

“…In the case of an irreducible replacement matrix, it is wellknown that the type of the asymptotics depends on the relation between the eigenvalues of the replacement matrix, and in particular on whether the largest eigenvalue is at least twice the real part of any other eigenvalue (in which case we have asymptotic normality), see [2] and [14]. For two colours we have two eigenvalues λ 1 ≥ λ 2 , both real, and the three cases are λ 2 < 1 2 λ 1 , λ 2 = 1 2 λ 1 , and λ 2 > 1 2 λ 1 ; in the two first cases we have asymptotic normality (with a log factor in the asymptotic variance in the second case), but not in the third.…”

Generalized Pólya urn Designs with Null Balance

“…sequence of random variables, then H n = H for any n (i.e. the GPU model is said to be homogeneous) and Athreya and Karlin [1] and Smythe [25] derived the asymptotic behaviour of {W n } n∈N and {N n } n∈N under some conditions on the spectral structure of the generating matrix. In such a case, they assumed that the expected number of balls added at each step is a positive constant, namely that 2…”

“…Bernoulli random variables, A n ∼ Ber(t) and B n ∼ Ber(s), so that the adding rule does not depend on the number of balls already in the urn and the BN 1 -GPU becomes homogeneous. From Proposition 1, in order to obtain a sequential procedure which is asymptotically balanced we set s = t. In such a case the stationary distribution of the ergodic chain {W n,1 } n∈N is binomial with π(·) = Bin(2w, 1 2 ). Moreover, from the spectral representation given in [21] the transition matrix P has eigenvalues λ x = 1 − sx/w for x = 0, .…”