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The variance of the correlation function of the probability distribution function estimate F*(x) of the stochastic process is defined based on (14.40) by substituting x1 = x2 = x:

If samples are uncorrelated, then the correlation function given by (14.40) is simplified since at k ≠ 0

In doing so, the variance of the probability distribution function estimate F*(x) of the stochastic process takes the following form:

As we can see from (14.44), the variance of the probability distribution function estimate F*(x) of the stochastic process depends essentially on the level x and reaches the maximum

at the level x corresponding to the condition F(x) = 0.5.

In addition to the finite observation time period of the realization x(t) of continuous ergodic process ξ(t), the characteristics of the probability distribution function estimate F*(x) of the stochastic process depends also on the stability of the threshold level that can possess a random component characterized by the value δ, i.e., instead of the level x we use the level x + δ in practice. Then, the obtained characteristics of the probability distribution function estimate F*(x) of stochastic process are conditional, and to obtain the unconditional characteristics of the probability distribution function estimate F*(x) of stochastic process, there is a need to employ an averaging by all possible values of the random variable δ. We assume that the correlation interval of the random variable δ is much more in comparison with the observation time interval [0, T]. Consequently, we can think that the random variable δ does not change during measurement time of the probability distribution function estimate F*(x) of stochastic process and has the same statistical characteristics for all possible values x. Additionally, it is natural to suppose that random variations of the threshold are negligible and a difference between F(x + δ) and F(x) is very small in average sense. In this case, the probability distribution function F(x + δ) can be expanded in Taylor series about a point x and can be limited by the first three terms of expansion:

where Eδ and Dδ are the mathematical expectation and dispersion of random variations of the threshold value x. Based on (14.27) and (14.34), we obtain that the random variations of the threshold level lead to the bias of the probability distribution function estimate F*(x) of stochastic process:

Naturally, the presence of random variable δ leads to an increase in the variance of the probability distribution function estimate F*(x) of stochastic process.

14.3  VARIANCE OF PROBABILITY DISTRIBUTION FUNCTION ESTIMATE

14.3.1  Gaussian Stochastic Process

The two-dimensional pdf of a Gaussian stochastic process with zero mathematical expectation is defined in the following form:

where

σ2 is the variance

(τ) is the normalized correlation function of the investigated stochastic process

However, the written form of the two-dimensional pdf of Gaussian stochastic process with zero mathematical expectation in (14.49) does not allow us to obtain the formulas for the variance of the probability distribution function estimate F*(x) that are convenient for further analysis. Therefore, we present the two-dimensional pdf of Gaussian stochastic process with zero mathematical expectation in the form of a series with respect to derivatives from the Gaussian Q-function, i.e., Q(x) given by (12.157), where we assume that the mathematical expectation is equal to zero (E = 0):

As we can see from Table 14.1, the av defining the variance of the probability distribution function estimate F*(x) of Gaussian stochastic process decreases with an increase in the number v. Taking into consideration that in the case of stochastic processes the factor

decreases also with an increase in the number v, in practice, we can be limited by the first 5–7 terms of series expansion.

When the observation time interval [0, T] is much more than the correlation interval of stochastic process, as earlier, the coefficients cv can be approximated by

If, in addition to the condition T τcor we assume that T → ∞, then, according to (14.59) and (14.54) the variance of the probability distribution function estimate F*(x) of the Gaussian stochastic process tends to approach zero, which should be expected; i.e., the probability distribution function estimate F*(x) of the Gaussian stochastic process is consistent.

In the case when the observation time is lesser than the correlation interval of the stochastic process, i.e., T < τcor, we have

v=1

Formula (14.60) characterizes the maximal variance of the probability distribution function estimate F*(x) of Gaussian stochastic process.

As an example, consider the Gaussian stochastic process with the exponential normalized correlation function given by (12.13). Substituting (12.13) into (14.58), we obtain

where p is given by (12.140). In doing so, if p 1, then

The root-mean-square deviation σ{F*(x)} of measurements of the probability distribution function of Gaussian stochastic process as a function of the normalized level z for various values of p is presented in Figure 14.3. As we can see from Figure 14.3, the maximal values of σ{F*(x)} are obtained at the zero level.

The maximal value of σ{F*(x)} as a function of the parameter p given by (12.40) is shown in Figure 14.4. As we can see from Figure 14.4 and based on (14.54) and (14.61), the variance of the probability distribution function estimate F*(x) decreases linearly as a function of p starting from p ≥ 10. This graph allows us to define the minimal required time to observe a realization of the

Rayleigh stochastic process tends to approach zero and is consistent for the considered case. The variance of the probability distribution function estimation F*(x) of Rayleigh stochastic process also tends to approach zero at z = 0 (x = 0) and z → ∞ (x → ∞), as in both cases the coefficients

tend to approach zero for any values of the ratio T/τcor.

When T τcor, the variance of the probability distribution function estimations F*(x) of Rayleigh stochastic process can be presented in the following form:

v=1

Table 14.3 represents the coefficients bv as a function of the number v and the normalized level z. As we can see from Table 14.3, the coefficients bv decrease slowly and nonmonotonically with an increase in the number v. In practice, we can be limited by 4–5 terms of the sum given by (14.84).

As an example, we consider the Rayleigh stochastic process with the normalized correlation function (τ) given by (12.13). In this case, we can write

TABLE 14.3

Coefficients bv as a Function of the Number v

and Normalized Level z