br Acknowledgment br This work was supported

Acknowledgment
This work was supported by the Ministry of Science and Education of the Russian Federation (Program 3.329.2014/K).
According to Kotelnikov\'s works , normal fluctuation noise consists of a large number of pulses following each other in random time intervals; it UM 171 is also possible for such pulses to overlap. Ref. and some other publications (see, for example, ), define the noise analytically as a trigonometric expansion. However, this model, characterized by a discrete spectrum, is not adequate for solving certain radiophysical problems. For example, in problems on radio-wave reflection in the lower (turbulent) ionosphere, it is more practical to use fluctuations with a continuous spectrum. Let us take as a basis a probabilistic description of the possible trajectories of stationary random processes using the probability density functional (()), first introduced in Ref. . In the case of a process that exhibits a flat power spectrum with a specific bandwidth, this functional is given by where () are the trajectories depending on the time is the quantity depending on the partition rank of the interval on which the trajectories are examined (it is the same for all trajectories if the partition rank tends to zero); is the height of the power spectrum; is the time interval under consideration. It is easy to see that under these conditions, when the functional is expressed by , the wavelet implementation of the process is more likely than the sine one. This follows from the fact that the integral in diverges with an increase in values for the sine function (), while for the wavelet it is finite. To further develop the results obtained in , our study has analytically defined the noise that is a superposition of elementary random processes as an expansion in terms of a wavelet basis. The proof of the theorem on the expansion of a stationary random process in terms of a wavelet basis is offered as a new result of our study. For the purposes of further discussion, it is practical to use instead of the dimensional quantities and (that have a time dimension) the dimensionless ones, by dividing them, for example, to =1s; previous notations, namely and , will be used throughout the text for these quantities
for the wavelets and examine wavelets () and () satisfying the following conditions when → ∞:
Let us write the coefficients () of the expansion () in terms of these wavelets :
The implementation of () is, generally speaking, an alternating function with an arbitrary number of zeroes (in its medium) that can be represented by tuples of positive and negative bursts, so that the expressions are true. Let us consider the positive bursts and apply the mean value theorem known from mathematical analysis to the first integral of :
Let us evaluate the modulus of the covariance of random quantities and using the Cauchy–Bunyakovsky–Schwarz inequality: where
Assuming that the quantity is finite, we have: when → ∞, → 0 which proves that the quantities and are uncorrelated. Let us now apply Lyapunov\'s central limit theorem to the random variables :
Let us find the mathematical expectation as a mathematical expectation of the product of uncorrelated random quantities:
Taking into account the property given by , the integral in the right-hand side of is equal to zero, consequently, . Let us now find the variance Taking into account the above-proved lack of correlation between and , we obtain:
Let us rewrite the last expression using the property given by formula ():
Taking into account and , we find from :
Similar transformations for the random variables yield:
Let us rewrite according to and :
determines the coefficients of expanding the process () in terms of the wavelet basis. Recovering () from the found coefficients () , we obtain .□