On the comparison of point processes

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4]. In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

On the comparison of point processes

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4].In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

Sticky Brownian motion as the limit of storage processes

The paper considers a modified storage processwith state space [0,∞). Away from the origin,Wbehaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls toIt is shown thatWcan be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift. When these same two transformations are applied to Brownian motion, one obtains sticky (or slowly reflected) Brownian motionW∗on [0,∞). ThusW∗is the natural diffusion approximation forW, and it is shown thatWconverges in distribution toW∗under appropriate conditions. The boundary behavior ofW∗is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.

Sticky Brownian motion as the limit of storage processes

The paper considers a modified storage process with state space [0,∞). Away from the origin, W behaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls to It is shown that W can be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift. When these same two transformations are applied to Brownian motion, one obtains sticky (or slowly reflected) Brownian motion W∗ on [0,∞). Thus W∗ is the natural diffusion approximation for W, and it is shown that W converges in distribution to W∗ under appropriate conditions. The boundary behavior of W∗ is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.