In many experimental studies, the presence of some phenomenon is established by observing a "signal", i.e. by establishing that some, as we will say, elementary event A is realized. However, this elementary event can also be caused by secondary causes not related to the phenomenon under study. In such a case, only repeated occurrences of event A will allow us to judge with sufficient certainty the presence of the phenomenon under study. Let us conduct a series of n experiments, where event A can appear with some probability p. The phenomenon of interest to us (we will call the implementation of this phenomenon event M) can be connected to varying degrees with elementary event A. It is natural to assume that there is a rather weak dependence between event M and elementary event A if, in order to obtain certainty of the realization of M, it is necessary to observe A sufficiently often in n trials (for example, at least once t times in a row). In this paper, we limit ourselves to considering only three types of dependence of events M and A, the most important for applications (see the article).
The aim of this paper is to study the properties of integral transformations of the form ϕ (χ) = ʃ r(y) K(x, y) dy, performed on the function r (y) by means of the stochastic kernel K(x, y). According to the generally accepted terminology, the kernel is called stochastic if it satisfies the conditions (see the article). The study of the properties of the transformed function ϕ (χ) naturally splits into two cases: symmetric and asymmetric F(x, y). The article considers questions about the functional moments of symmetric correlation and about the functional moments of asymmetric correlation.
