ABSTRACT
Further generalization of random walks are the so called continu ous time random walks (CTRW).
Definition 5. Suppose that {(Tn, Xn), n = 1 , 2 , . . . } is a sequence of i.i.d. random vectors with state space [0, oo) x R d for d > 1. The random variables Tn and Xn can be dependent or independent. Now a CTRW process is defined as a sum of random variables Xn
The d-dimensional random vector R(t),t > 0 describes the posi tion of a particle which starts form the origin and moves in the d - dimensional Euclidian space. The jumps occur at the renewal epochs of the process 5n, n = 0 , 1 , 2 , . . . and the magnitude of the jumps is given by the r.v. Xn. The limiting behavior of R(t) is of interest for many physical processes and it has been investigated widely in the literature. Kotulski (1995) has given a survey of the results in this direction and has proved new and known results using probabilis tic methods. He worked out the limiting distributions of R(t)/t as t —► oo depending on different assumptions about the distribution of (Tn,X n). In the case when Xi — Ti (in other words the walk is on the real line and the jumps coincide with the waiting times) and ET{ = oo, Pr{T< > t ) = Ct 0 < 0 < 1 the following limit is valid (Case 3.2, Kotulski (1995)):
Hence, In Example 5.6, a d-dimensional CTRW R(t), t > 0 is constructed
Stoyanov and Pirinsky (2000) investigated a class of discrete time Markov chains with continuous state space-the interval (0,1). It is constructed in such way that different beta distributions axe obtained as limiting distributions. The construction is as follows:
Suppose the particle D is located at some starting point xo in the interval (0,1) and its motion, always along a line segment is determined by the following 2-stage rule:
Theorem 18. Prom the initial position xq the particle D moves al ternatively in the direction of 0, of 1, of 0, of 1, etc (Stage 1 is deter ministic). Every time D changes the direction at a random point cho sen uniformly from the corresponding interval (Stage 2 is random). Then the random sequence Xn does not converge. However, it is split into two subsequences A"2n+i, n = 0 , 1 , 2 , . . . and X 2n, n = 0 , 1 , 2 , . . . each being convergent in distribution:
Hence,
it is clear that this sequence is a Markov chain with the state space - the interval (0,1).
