Statistical Properties and Applications of Correlated Length Intervals of Some Stationary Random Point Processes

The point process, a sequence of random univariate random variables derived from correlated 
bivariate random variables as modeled by Arnold and Strauss, has been examined. Statistical 
properties of the time intervals between the points as well as the probability distributions of the 
number of points registered in a finite interval have been analyzed specifically in function of the 
coefficient of correlation. The results have been applied to binary detection and to the transmission 
of information. Both the probability of error and the cut-off rate have been bounded. Several 
simulations have been generated to illustrate the theoretical results.


Introduction
It is known that the detection of an optical field at a low level of power is a sequence of events which is a set of distinct time instants {ϑ j }, 0 ≤ j < ∞, such that ϑ j+1 ≥ ϑ j , for all j. These {ϑ j }s, which are the time instants of interaction between the photons and the detector device, for example, a photomultiplier, constitute a random point process (RPP) for which a positive instantaneous density λ(ϑ j ) can be defined. We here consider only simple and homogeneous stationary processes.
Reciprocally, the existence of a physical optical field, given the knowledge of all of the properties of such an RPP, is not an easy problem to solve. For example, given an RPP of nonclassical properties [1] does not necessarily correspond to a nonclassical physical optical field, although its feasibility can be demonstrated [2]. This problem, mainly due to the quantum nature of the interaction between radiation and matter is not examined here.
Nevertheless, using a parameterized RPP, whose special values of such a parameter (denoted here c) correspond to a physical field, for example, a coherent or a thermal state, then it is reasonable to admit that intermediate values of c corresponds to realistic optical fields. Despite these limitations, the properties of the RPPs studied here are important in statistical optics.
As already pointed out, there are two types of processing that can be utilized to characterize such RPP: the time interval distributions (TIDs) and the probability of number distributions (PNDs) [3]. If we choose to characterize the RPP by the TID, we can define the time interval between points θ j = ϑ j+1 − ϑ j , called residual time (or lifetime sometimes). Its probability distribution function (PDF) w(θ) will be called here a triggered PDF. When ϑ j = t 0 is arbitrary, that is to say not a point of the (RPP), the v(θ), PDF of the corresponding θ j will be called a relaxed PDF.
If we now choose to characterize the RPP by the PND, we can as above, define the relaxed PND P(n, t) of the random variable (RV) N(t 0 , t 0 + t) which is the number of instants occurring within the time interval [t 0 , t 0 + t], t 0 and t / = ϑ j , for all j. As previously we may as well define Q(n, t) a triggered PND of the RV N(ϑ j , ϑ j + t), t / = ϑ j , the counting process being started by a point of the RPP.
The purpose of this paper is firstly to calculate the statistical properties of the RPPs, both in terms of the TIDs and the PNDs. Secondly we apply the results to the calculation of the performances, namely, the probability of error in binary detection and the cut-off rate in the binary transmission of the information, and their variations with respect to the coefficient of correlation c. The one-dimensional processes, such as the Poisson and the geometric PNDs, are finally utilized for establishing bounds to the performances.

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In Section 2, the notations are defined and the basic equations are briefly recalled. Section 3 establishes the main results of the PDFs both those dealing with TIDs and PNDs depending on extreme values of c: low values and high values. Finally, Section 4 presents theoretical results and curves that explain and illustrate the results of the numerical simulation.

Basic Equations
We need to define the Laplace transform of f (t) (1) and the functions , The PDF of the number of events registered between 0 and t, denoted by PND, reads where I is the indicator function I(t ∈ T) = 1(0), in case the event occurs or not within t ∈ T. Therefore P(n; t) = L −1 P(n; s) where the symbol L −1 denotes the inverse Laplace transform. From (4), it is easily seen that (see e.g., [4,5]) The symbol E denotes the mathematical expectation. In the following P(n; t) will be called the relaxed PDF. Another interesting PDF that can be derived from (4), will be called the triggered PDF. In fact, it is known [3,6] that these PDFs can be expressed using the moments of the time-integrated density J(t) = λt for the Poisson RPP. Therefore For all RPPs that we deal with here, all the PDFs depend only on t (not on t 0 ) due to the stationary property. When t is a parameter, P(n; t) and Q(n; t) are simply denoted P(n) and Q(n).
In terms of TIDs, we just recall the basic formulas of w(t) and v(t) (see [6]) Notice that g(0) w(t = 0)/v(t = 0) the ratio of the values at t = 0 is related to σ 2 J , the variance of J, such that g(0) −

Model for Correlated Variables
Among the several models proposed to deal with correlated variables, we consider the Arnold-Strauss model [7,8] where a, b, c > 0. The time intervals t, u are of course positive RVs in [0, ∞[. The constant of normalization is where Ei(m, x) are the exponential integral functions [9] for m = 1, 2, . . ., and x ∈ R + . They verify the approximate expressions given in (A.1a)-(A.2a) in Appendix A.1. The constant γ = 0.5772157 is the Euler constant. The marginal distribution of t is deduced from (10) and is given by In the following, we report some calculations that can be obtained under closed forms depending on the value of the parameter c. Let us first consider the case where c 1.
In what follows, most of the calculations are done up to O(c 3 ).

TID.
It can be shown that (see, e.g., [6]) Given E[t], the first moment of t, the average value of the density of the process λ = 1/E[t] is given by Equation (13) yields ν(0) = λ and dν/dt = −λw(t) which prove that ν(t) is a monotonic continuously decreasing function of t. More precisely, we can show that From (15), we deduce On the other hand, we can see that which positivity is a characteristic of classical processes [10].
When only a few values of P(n, t) or only the moments are needed, it is however, better, to use (3) and (5). Based on their expansion as series in s, we obtain, at the second order of c, where we denote ω = c/b. Seeking the exact expressions of the PDFs of the number seems difficult to obtain. However, as just seen, approximations of closed expressions are simple. Thus, using (4), it is easy to calculate approximate expressions of the PDFs of the number for n = 0, 1, 2 and t = 1, as given in (A.3) in Appendix A.2. Therefore, we can prove that which is negative for all c 1. We will numerically illustrate this property by simulations. When we choose the triggered processing, we obtain which is here again negative for all c. The case c 1 (but finite) is perhaps more interesting although closed forms of the moments are difficult to obtain.

c
1. The calculations are now done up to O(1/c 3 ).

TID.
We can show that the normalized w(t) is given by and the unnormalized PDF, evaluated up to O(1/c 2 ) and leading to where we denote χ = γ + log(ab) and assume that λ a(log c − χ). It is seen that h t (0, c 1) given by (25) is now positive and increases more slowly than h t (0, c 1) ∼ c 2 given by (18). From (23), we can also deduce that Both moments tend to 0 when c → ∞.

PND. The approximate expressions given in (A.4) in Appendix A.2 lead to
We can calculate the first two moments of the number for the specific case a = 1 and b = 1, As for the previous case, it can be shown that k n (c) ≤ 0, for all c.

Simulation and Results
We have used the algorithm recently described [11] for several values of c.
In Figure 1, results of the simulated data of the TIDs are plotted. The w(t), the triggered PDF follows very well (23). The v(t), the relaxed one has been well fitted by the expression for a = 1, b = 1, c = 30 and where ν 0 = 0.07 and ν 1 = 0.2. By the way, it is interesting to remark that the PDF of S = j=0 θ j , 1, which is the addition of several correlated identically distributed positive random time interval, deviates from the Gaussian profile. We may conclude that the application of the central limit theorem requires the addition of a very high number of correlated random variables.
On the other hand, for high values of c, the theoretical approximations of the PNDs are given by where leading to The values of the parameters that correctly fit the simulated results of the PDFs, as seen in Figure 2, are b 1 = 0.2, a 1 = 1, b 2 = 0.62, a 2 = 5.2, and E[N] = 3.64. The variations of the moments with respect to c are derived from the expressions recalled in the Section 2. We obtained Regarding the reduced moments h n and k n , an excellent fit of the simulated results of Figure 3 is carried out with α 1 = 4, β 1 = 0.17 and α 2 = 3, β 1 = 0.24 As an application of these results to communications, we consider a system of communication processed with a direct threshold detector. The decision device operates such that n n s (c) −→ H 0 −→ P c=0 (n) = e −a a n n! , (35a) based on the binary hypotheses: H 0 (no correlation) and H 1 (correlation with the parameter c). The threshold is obtained from the likelihood ratio Λ(n), We can also utilize the TIDs as a useful tool of processing [12]. The threshold would be based on the likelihood ratio Θ(t) = w c (t)/λe −λt > 1 → t s (c) and the decision would operate as follows where t s (c) = (1/c)(K/a − b), K being given by (11).
Here, we focus on the method based on PNDs because it is generally more efficient. To simplify the calculations, the decision is not randomized [13]. Now, the probability of error in detection when processing with the relaxed PND is given by Similarly, we have for the processing with the triggered PND It is first seen that Q err ≤ P err , the inequality which is demonstrated for a special case in the Appendix B. Furthermore, both probabilities of error in detection decrease with E[N], then with c because E[N] ∝ log c as it is calculated in (28a), (33a) and is shown in Figure 3. This is within the range of c : 0 ≤ c 100. For instance, with the help of (30)-(31b) and (38)-(39) calculated for n s = 1 and c = 100, we obtained P err 0.28 and Q err 0.16, the values which are in excellent agreement with the simulated results of Figure 4.
On the other hand, the bounds to Q err and P err can easily be calculated. In fact, denoting where E[N] = a + ξ. Now, taking into account (35a), we have up to O(ξ 2 ), for 0 ≤ ν / = 1. Therefore P (1) err is the approximation of the exact P err for n s = 1 plotted as the curve quoted "1" in Figure 4 where a = 1. Similarly, taking into account (35a), we have

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Again, P (2) err is the approximation of the exact P err for n s = 1 plotted as the the curve quoted "2" in Figure 4 for a = 1. Finally, we have P (2) err ≤ Q err ≤ P err ≤ P (1) err .
Let us conclude this analysis with a brief comment on information. We will concentrate on the cut-off rate which is known as a useful criterion for evaluating the performances of a channel. Thus, for a binary noiseless channel, when the transmission of messages "0" and "1" is done via the probabilities P c=0 and P c / = 0 respectively (the probabilities a priori are taken equal to 1/2) the cut-off rate expresses as which is generally interpreted as a lower bound to the channel capacity [14]. The P c=0 (n) is the Poisson PND of parameter a = 1 and P c / = 0 (n) will be, as above, either the relaxed or the triggered PND yielding the cut-off rates R p and R q , respectively. Because exact closed expressions seem difficult to attain, the bounds are very useful and can easily be established. The first bound is obtained using geometric PNDs where ν is given by (40) and ν 0 = a/(1 + a). This is a lower bound to R p . The second bound is obtained using the Poisson PNDs which is an upperbound to R p . In Figure 5, the results of the simulation R p and R q and the bounds (curves quoted "1" and "2") given by (45b) and (46) are plotted versus E[N]. Here again, it is seen that the processing with the triggered PND performs much better that the processing with the relaxed PND in the limit of 0 ≤ c ≤ 100. However, because the cut-off rate is a monotonic increasing function with respect to E[N], the inequalities (47) may be extrapolated to for all c.
In conclusion, the binary performances, as summarized by the inequalities (43) and (47), show that the processing with the triggered PND is the preferable mode of operation. Both performances, in detection and information transmission, are improved with the coefficient of correlation.

B. Inequality between Probabilities of Error in Binary Detection
To demonstrate that the triggered processing yields better performance than the relaxed processing, Δ e = Q err − P err ≤ 0, we begin with (