Thinning of point processes—covariance analyses

Abstract

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

Another characterization of homogeneous Poisson processes

“…Theorem 1, whose proof is given at the end of Section 2, is most closely related to the findings of [8,7,15] give that, when either T 1 has the distribution of the integrated tail of T 2 with (implicitly) ET 2 P p0, 8q, or else when T 1 has the same distribution as T 2 , PpT 2 ă 8q " 1 and T 2 is non-arithmetic (respectively, when Pp0 ă T 1 q " 1 and (as implicitly assumed in the proof; not all the assumptions appear to be given explicitly) PpT 1 " 8q ă 1; when Pp0 ă T 1 , 0 ă T 2 q " 1 and PpT 1 ă ǫq ą 0 for all ǫ ą 0), then covpN 0 t , N 1 t q " 0 for all t P p0, 8q implies N " HPPpθq for some θ P p0, 8q. (Strictly speaking the quoted result of [8] is false.…”

“…Theorem 1, whose proof is given at the end of Section 2, is most closely related to the findings of [8,7,15]. Let us see how it compares.…”

“…‚ N has jumps of size 1 a.s. and its inter-arrival times are independent, identically, exponentially with mean c ´1 (notation: Exppcq), distributed [9,Theorem 6.5.5(d) 6.15] in the context of Palm theory of random measures; [18,14] for the order statistics property; [10,16,13] concerning age (a.k.a. spent or current life) and residual life; finally [15,8,7] that deal with marking (thinning) of renewal processes.…”

“…For still further characterizations see [11,Sections 2.2,2.3 and 4.3] dealing with the property of 'complete randomness', distribution form and various operations on stationary renewal processes, respectively; Slivnyak-Mecke's characterization of Poisson processes [12, Proposition 13.1.VII] [17, Lemma 6.15] in the context of Palm theory of random measures; [18,14] for the order statistics property; [10,16,13] concerning age (a.k.a. spent or current life) and residual life; finally [15,8,7] that deal with marking (thinning) of renewal processes.…”

On a study of the exponential and Poisson characteristics of the Poisson process

“…for certain pairs of r and s, where a and b are independent of t, then A has to be a Poisson process. When A is a Poisson process, the fact that (1) and (2) hold are based on the so called order statistics property. More precisely, for the Poisson process A, given At nY S 1 S 2 Á Á Á S n are distributed as the order statistics of n i.i.d.…”