No abstract

Let {Xn, n ≥ 1} be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call Xn a δ-record if Xn > max{X1, . . . , Xn−1} + δ, where δ is an integer constant. We use martingale arguments to show that the counting process of δ-records among the first n observations, suitably centered and scaled, is asymptotically normally distributed for δ = 0. In particular, taking δ = −1 we obtain a central limit theorem for the number of weak records. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2007, Vol. 13, No. 3, 754-781. This reprint differs from the original in pagination and typographic detail.

Let (X t ) and (Y t ) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X t ) and (Y t ) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

No abstract

Consider a sequence (X n ) of independent and identically distributed random variables taking nonnegative integer values, and call X n a record if X n > max{X 1 , . . . , X n−1 }. By means of martingale arguments it is shown that the counting process of records among the first n observations, suitably centered and scaled, is asymptotically normally distributed.