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Elements of the Kopula (eventological copula) theory

New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced. The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar's theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N-sets of events.


Introduction
Long time ago and little by little, the incentive for this work materialized in the theory of sets of events, eventology [1], where the need to locate the classes of event-probability distributions (e.p.d.) of the sets of events (s.e.), which were so arbitrary and spacious to be able without let or hindrance to deal with the relationships between pairs, triples, quadruplets, etc., of events, in other words, to understand the structure of statistical dependencies and generalities between events from some s.e. A similar need is perhaps the only one that has always fueled the development of the probability theory and statistics, which in one way or another are theories of studying and evaluating the structures of statistical dependencies and generalities in the distributions of sets of events.
The classical copula theory [2,3,4,5], existing since the 50s of the last century, allows us to construct classes of joint distribution functions that have given marginal distribution functions. In eventology, the theory of sets of events, proposed in the paper the theory of Kopula (eventological copula) allows us to solve a similar problem -to build classes of e.p.d's of sets of events whose events happen with given probabilities of marginal events. To specify a family of sets of 2 probabilities of the 1st kind (1.1), it is necessary and sufficient to specify a family of sets from the 2 − 1 parameters, since all the probabilities in each set must be nonnegative and give in the sum of one. And to specify a family of sets of 2 probability values of the 1st kind (1.1) with given probabilities of marginal events that form the X-set it is necessary and sufficient to specify a family of sets from the 2 − − 1 parameters, since for each collection there must be another constraints for events ∈ X: (1.3) Therefore, "to define the family of functions of e.p.d.'s of the 1st kind with given probabilities of marginal events" means "to define a family of sets from the 2 − −1 parameters" as sets of functions of these probabilities. Eventological theory should solve this problem with the help of a convenient tool, the Kopula, which allows us to define a family of sets from the 2 − − 1 parameters as sets arXiv:1802.08179v1 [stat.OT] 17 Feb 2018 of functions from marginal probabilities, which in turn can be made dependent on a number of auxiliary parameters.
In general, the -Kopula in the eventological theory is an instrument for defining the family of probability distributions of the 1st kind of thes.e., in the form of a family of 2 -set of functions of the probabilities of their marginal events.

Prolegomena of the Kopula theory
The main results of this paper are presented in a rather rigorous mathematical manner. And although the definitions, statements and proofs are provided with examples and illustrations, in order to visualize the ideas underlying the Kopula theory, in my opinion, a number of preliminary explanations in a less strict context, which are collected in several prolegomena, may be necessary.
If in some set of events X, some events from the subset ⊆ X are replaced by their complements, then we get a new set of events X ( | ) = + (X − ) ( ) , which is called the -phenomenon of s.e. X. The set of all such -phenomena for ⊆ X is called the 2 X -phenomenon-dom of s.e. X. In [6] a rather distinct theory of set-phenomena and the phenomenon-dom of some s.e.
Similarly, the theory of set-phenomena [6] defines the phenomena and phenomenon-dom of the set˘= { , ∈ X } of the probabilities of marginal events ∈ X: the -phenomenon˘( | ) = { , ∈ } + { 1 − , ∈ X − } of the set of marginal probabilities˘is obtained by replacing the marginal probabilities by complementary marginal probabilities 1 − when ∈ X − . Prolegomenon 1 (set-phenomenon of a set of events and a set of probabilities of events). The main conclusion of the above theory is obvious: probability distribution of the s.e. X characterizes the probability distribution and the set of marginal probabilities of each set-phenomenon from its 2 X -phenomenondom.
Prolegomenon 2 (set-phenomenal transformation). 1) For each pair X ( | ) , X ( | ) set-phenomena of the set of events X their probability distributions are related to each other set-phenomenal transformations. 2) In each pair˘( | ) ,˘( | ) set-phenomena of the set of marginal probabilities˘events from X are also interconnected by set-phenomenal transformations.
The event ∈ X is called half-rare [6] if the probability = P( ) with which it happens is not more than half: 1/2. If all events from the s.e. X are half-rare, we speak of a set of half-rare events, or a half-rare s.e.
Prolegomenon 3 (sets of half-rare events and its Kopulas). 1) It is not difficult to guess that for any s.e. X, in the 2 X phenomenon-dom of the sets of events X ( | ) , ⊆ X, and in the X-phenomenon-dom of the sets of its marginal probabilities˘( | ) , ⊆ X, there is always a half-rare set-phenomenon. If, in addition, there are no events in X happening with probability 1/2, then such a half-rare set-phenomenon is unique. 2) A Kopula of some family of half-rare -s.e.'s is generated by 2 functions from half-rare variables defined on the half-hypercube [0, 1/2] with values from [0, 1] that are continued by the set-phenomenal transformations of the half-rare variables to the corresponding half-hypercubes, all together completely filling the unit hypercube.
Prolegomenon 4 (an invariance of the copula with respect to the order of half-rare events). Our task is to construct a Kopula of a family of arbitrary (unordered) sets of half-rare events, i.e. a 1-function, the arguments of which form an unordered set of probabilities of their marginal events. Therefore, it is natural to require such a function to be invariant with respect to the order of its arguments; with respect to the order of events in these sets. In other words, it is natural to consider this 1-function as a function of a set of arguments, rather than a vector of arguments with ordered components, as it is usually assumed.
Prolegomenon 5 (insertable sets of half-rare events and a frame half-rare event).
Two insertable sets of half-rare events for a given set of half-rare events X = { 0 } + with the frame half-rare event 0 ∈ X, happening with the highest probability among all events from X, are two sets of half-rare events ′ = { 0 }(∩) and ′′ = { 0 }(∩) that partition the set of other events = X − { 0 } into two: = ′ (+) ′′ . The events of one of them, ′ , are contained in the frame half-rare event 0 , and the events of the other, ′′ , are contained in its complement 0 = Ω − 0 .
Prolegomenon 6 (the insertable sets of events and conditional e.p.d.'s of a set of events with respect to the frame event and its complement).
Conditional e.p.d.'s of the 1st kind of one s.e. X with respect to the other s.e. Y are defined in the traditional way [1]. However, until now attempts to define such a "conditional" s.e., which would have given a conditional e.p.d. of the 1st kind, turned out to be completely impractical [7]. The concept of two insertable s.e.'s in a frame event is a well-defined "ersatz" of such "conditional" s.e.'s. The e.p.d.'s of this "ersatz", although they do not coincide with two conditional e.p.d.'s of the 1st kind with respect to the frame event and its complement, but they are fully characterized by them. The converse is also true: the e.p.d.'s of two frame s.e.'s characterize the corresponding two conditioned e.p.d.'s of the 1st kind.
Prolegomenon 7 (a frame method of constructing a Kopula of an arbitrary set of half-rare events).
A frame method constructs a Kopula of a family of arbitrary sets of half-rare events on the basis of a conditional scheme by means of a recurrence formula via conditional e.p.d.'s concerning the frame event and its complement. A recurrent formula associates this Kopula with two Kopulas of families of their insertable sets of half-rare events of smaller dimension, which are characterized by the corresponding conditional e.p.d.'s of the 1st kind (see Prolegomenon 6).
Prolegomenon 8 (a set-phenomenal transformation of a half-rare Copula to an arbitrary one).
To construct the Kopula of a family of arbitrary s.e.'s it is enough to construct the Kopula of the family of their halfrare set-phenomena and apply a set-phenomenal transformation to this Kopula.

The Kopula: definition, theorem and the simplest Kopulas
We consider the general probability space of Kolmogorov events (Ω, , P), some particular probability space of events(Ω, , P) and the -set of events ( -s.e.) X ⊂ with the event-probability distribution (e.p.d. 2 ) of the 1st kind (X) = { ( / /X) : ⊆ X}, 2 The abbreviations: e.p.d. and e.c.d. are used for the eventprobability distribution and for the event-covariance distribution. and of the second kind which, recall, are related to each other by the Mobius inversion formulas: Definition 1 (set-phenomena of a s.e. and its phenomenon-dom). Every -s.e. X ⊂ generates its own 2 X -phenomenon-dom, defined as a 2 -family composed of -s.e. in the form which for each ⊆ X is called its set-phenomen [6], more precisely, -phenomen, where is an -complement of the s.e. ⊆ X.
We also recall that probabilities of the second kind Definition 2 (set-phenomena of the set of probabilities of events from a s.e. and its phenomen-dom). The -set of probabilities of events from X = { : ∈ X} also generates its 2 X -phenomen-dom, the 2 -totality composed of -sets in the form and defined for ⊆ X as the -set of probabilities of events from -phenomenon X ( | ) of the s.e. X where for In particular, for = X We denote by a nonnegative bounded numerical function defined on the set-product [8], X-hypercube Arguments of form the -set which generates its own 2 X -phenomenon-dom, the 2 -totality of -sets of arguments: be the family of all the nonnegative bounded numerical functions on the X-hypercube.
i.e., the sum of its values on all the -sets of arguments from 2 X -phenomenon-dom 2 ( |˘) is one.
A function ∈ Ψ X is called a 1-function on the X-hypercube if for all˘∈ [0, 1] ⊗X -marginal equalities are satisfied for all ∈ X: i.e., the sum of its values on -halves of -sets of arguments from the 2 X -phenomenon-dom 2 ( |˘) is .
Denote by the family of functions, normalized on the Xhypercube; and by the family of 1-functions on the X-hypercube.
1 (properties of 1-functions on the { , }square ). A strict inclusion is fair: Proof. In other words, the lemma states: { , } is a normalized function on the { , }square; 2) among the normalized functions from Ψ 0 { , } there is one which is not a 1-function. But this is obvious, as it is confirmed by the following simple examples.
First, indeed, for the doublet of events X = { , } by the definition of a 1-function, we have The sums (2.9) and (2.12) as well as the sums (2.10) and (2.11) as a result give i.e., ∈ Ψ 0 { , } is a normalized function on the { , }-square.
Of course, in the general case, for an arbitrary s.e. X the same lemma is fulfilled.
Lemma 2 (properties of 1-functions on the Xhypercube ). A strict inclusion is fair: Proof is similar.
Note 1 (a representation of a 1-function on the X-hypercube in the form of 2 |X| -set of functions).

Characterization of a set of events by Kopula
The eventological analogue and the preimage of the well-known Sklar theorem on copulas [2] is the following theorem.
Proof is a direct consequence of the properties of e.p.d. of the 1st kind of the s.e. X and the |X|-Kopula. First, if the e.p.d. of the 1st kind = { ( / /X) : ⊆ X} of some s.e. X with the X-set of marginal probabilities˘= { : ∈ X} is defined, then from properties of probabilities of the 1st kind it follows that for ∈ X i.e., the function K, defined by the e.p.d. of the 1st kind and formulas (2.3), satisfies -marginal equalities for ∈ X: (required for being a 1-function:: K ∈ Ψ 1 X ) and serves as the |X|-Kopula.
Second, if the function K is the |X|-Kopula, then by Lemma 1: K ∈ Ψ 1 X ⊂ Ψ 0 X , i.e., it is normalized and, consequently, by (2.20) the function is normalized too: In addition, from (2.20) and from the fact that the |X|-Kopula is a 1-function, (2.21) follows for all ∈ X. Therefore, the function is a e.p.d. of the 1st kind of the s.e. X with the X-set of marginal proba-bilities˘. The theorem is proved.

Convex combination of Kopulas
Lemma 3 (convex combination of Kopulas ). A convex combination of an arbitrary set of Kopulas of one and the same s.e. is its Kopula too.
Proof without tricks. Let X be a s.e., and be some set of its Kopulas. Let us prove that their convex combination (where, of course, 1 + . . . + = 1, 0, = 1, . . . , ) is also a Kopula. For this it suffices to prove that K is a 1-function. In other words, that for ∈ X ∑︁ ∈ ⊆X K(˘( | / /X) ) = . (2.26) Since each Kopula from the set (2.24) has a property of a 1-function, then for ∈ X we get what is required: (2.27)

Corollary 1 (convex combination of Kopulas ).
For every set of events X the space of 1-functions Ψ 1 X , as well as the space of its Kopulas, is a convex manifold.

Kopula of free variables: A computational aspect
Without set-phenomenon transformations and variable transformations, analytic work on sets of half-rare events (s.h-r.e.'s) (see [6]) and sets of their marginal probabilities, is unlikely to be effective. However, in specific calculations at first, because of their unaccustomedness, these compulsory wisdoms can cause misunderstandings, leading to errors. Therefore, it is useful, in order to avoid unnecessary stumbling during calculations, to introduce separate notation for half-rare marginal probabilities events from s.h-r.e's. X and its set-phenomena, that is, probabilities that are not greater than half, in order to distinguish them from free marginal probabilities, to the values of which there are no restrictions.
So, we will talk about half-rare variables (h-r variables) and free variables, assigning special notation to them 5 : (2.28) 5 Just remember [6], that the formula of -renumbering any X-set of probabilities of events has the form for ⊆ X: and always interpreting them as probabilities of events. In particular, for the half-rare doublet X = { , } we have: −half-rare function of half-rare variables, −free function of free variables.

For every
⊂ X phenomenon replacement of half-rare variables˘∈ [0, 1/2] ⊗X by free variables˘∈ [0, 1] ⊗X and vise-versa is defined for ⊆ X by mutually inverse formulas of the set-phenomenon transformation of the form: where for ∈ X the following agreement is always accepted (see, for example, paragraph 11.7): (2.31)

Kopula for a monoplet of events
Let's construct the 1-Kopula K ∈ Ψ 1 X of a family of e.p.d.'s of monoplet of events X = { } with the e.p.d. of the 1st kind In other words, let's construct a 1-function on the unit X-segment, i.e., a such nonnegative bounded numerical function Since for ⊆ X = { } then a marginal and global normalization of the function K are written as: and the global normalization obviously follows from the marginal one, which agrees with Lemma 2; and from the marginal normalization it follows that the 1-copula K of an arbitrary monoplet of events X = { } is defined for free variables˘= { } ∈ [0, 1] by a one formula: which provides two values on each 2 ( |˘)penomenon-dom by "free" formulas:

Kopulas for a doublet of events
Let's construct an example of 2-Kopulas K ∈ Ψ 1 X of families of a doublet of events X = { , }, in other words, let's construct on the unit { , }-square the such nonnegative bounded numerical functions Since composed from forth its set-phenomenȃ and normalizations for every˘∈ [0, 1] ⊗{ , } are written as: The e.p.d. of the 1st kind of doublet of events is defined by the 2-Kopula for ⊆ { , } in half-rare variables by general formulas: where (˘) is functional parameter that has a sense of probability of double intersection. This e.p.d. of the 1st kind of doublet of events in the free functional parameters and variables (after replacement (2.31)) has the form:

Kopula for a doublet of independent events
The simplest example of a 1-function on a { , }square is the so-called independent 2-Kopula, which for free variables˘∈ [0, 1] ⊗{ , } is defined by the formula: (2.40) This provides it on each 2 ( |˘) -phenomenon the following four values: (2.41) Indeed, the so-defined independent 2-Kopula is a 1function because The e.p.d. of the 1st kind of doublet of independent events with the { , }-set of probabilities of events is defined by four values of the independent 2-Kopula (2.40) on its 2 ( |˘) -penomenon-dom by general formulas in half-rare variables (see Fig. 2), i.e., for ⊆ { , }: (2.42)

2-Kopula of free variables: A computational aspect
With the phenomenal substitution (2.30) half-rare 2-Kopula as a function of the free variables takes the equivalent form:  We rewrite this formula a pair of times, in order to understand the properties of the phenomenon substitution of variables and not get confused in the calculations: (2.44) and again with restrictions in the form of the familiar "human" inequalities 6 : are the Fréchet inequalities for as the halfrare probability of double intersection of half-rare events: either half-rare events or , or their halfrare complements, when events or are not halfrare.
Note that the conditional formulas (2.43), (2.44) and (2.45) can not be rewritten as four unconditional formulas, because these conditions are in the right, and not in the left. This is explained exclusively by the properties of the phenomenon replacement of half-rare variables by free ones (2.30), which, for this reason, leads to formulas that are convenient for calculations. (2.47)

Upper 2-Kopula of Fréchet
An example of a 1-function on a { , }-square is the so-called upper 2-Kopula of Fréchet, which suggests the probabilities of a double intersection to be its upper Fréchet boundary. In other words, the only functional free parameter in (2.39) is: The upper 2-Kopula of Fréchet from free variables ∈ [0, 1] ⊗{ , } is defined by the formulas: (2.49) After simple transformations, these formulas provide the upper 2-Kopula of Fréchet on each 2 ( |˘)phenomenon-dom the following four values of free variables: The same four values from the half-rare variables have the form: (2.51) If , this formula takes the form: (2.52) And if < , then this formula takes the form: (2.55)

Lower 2-Kopula of Fréchet
A once more example of a 1-function on a { , }square is the so-called lower 2-Kopula of Fréchet, which suggests the probabilities of a double intersection to be its upper Fréchet boundary. In other words, the only functional free parameter in (2.39) is:  The lower 2-Kopula of Fréchet from free variables ∈ [0, 1] ⊗{ , } is defined by the formulas: After simple transformations, these formulas provide the lower 2-Kopula of Fréchet on each 2 ( |˘)phenomenon-dom the following four values of free variables: The same four values from the half-rare variables have the more simple form: So the definite lower 2-Kopula of Fréchet is indeed a 1-function, due to the fact that for all half-rare variables (2.60)

Convex combinations of the lower, upper and independent 2-Kopulas of Fréchet
A rather general example of a 1-function on a { , }square is the convex combinations of the upper, lower, and independent 2-Kopulas ofwise Fréchet, which propose the probabilities of a pair intersection to become a convex combination (this is allowed by the lemma 3) of its upper and lower Fréchet boundaries, as well as the probability of double intersection of independent events with some functional weighting parameter.

Convex combination of the lower and upper 2-Kopulas of Fréchet
A convex combination of the lower and upper 2-Kopula of Fréchet can be ensured by the unique functional free parameter (˘) in (2.39), in which the probability of double intersection is computed by the following formula: are the lower and upper Fréchet-boundaries of probability of double intersection.

For
= −1, the probability (˘) = − (˘) coincides with the lower Fréchet boundary of halfrare marginal probabilities; for = 1, the probability (˘) = + (˘) coincides with the upper Fréchet boundary of marginal probabilities. Unfortunately, these are the properties of a convex combination such that for = 0 the probability of double intersection is equal to half of the sum of its lower and upper Fréchet boundaries: Figure6), and not an independent 2-Kopula, no matter how much we want it. This "blunder" of the convex combination can easily be corrected by conjugation of two convex combinations, as done below. In

The conjugation of two convex combinations of the independent one with the lower and upper 2-Kopula of Fréchet
We construct two convex combinations of the the independent 2-copula and the lower and upper 2-Kopulas of Fréchet. The conjugation of these two convex combinations can be ensured by the unique functional free parameter (˘) in (2.39) by the following conjugation formula for two convex combinations: for every ⊆ the -inserted s.e.'s, generated by , in the frame s.e. are s.e.'s, which are denoted by (∩ / / ) , and defined as the following Mintersection 7 : and have the e.p.d., which coincides with the projection of the joint e.p.d. (3.1) for fixed ⊆ and every ⊆ : Definition 7 (event-probabilistic pseudodistribution of an inserted s.e.).
For each ⊆ the -inserted s.e.
with the e.p.d. (3.3) has the event-probabilisticpseudo-distribution, which is defined as a set of probabilitieso of terraced events that coincide with probabilities from the e.p.d. (3.3) for all ⊆ excepting = ∅: The sum of all probabilities from every -pseudodistribution (3.5) is ( / / ) = P(ter( / / )), the probability of a terraced event, generated by the frame s.e. , in which the given s.
Thus, the only difference of e.p.d.'s of -inserted s.e.'s from their event-probabilistic -pseudodistributions, lies in the fact that the sums of the probabilities of the terraced events, from which the -pseudo-distributions are composed, are normalized not by unity, but by the probabilities of the corresponding frame terraces events ( / / ). And the sum of the normalizing constants by ⊆ is obviously equal to one.
In Definition 6 the s.e. and can always be swapped, i.e., to take the s.e. on a role of the frame one, and the s.e.
to take on a role of s.e., that generates -inserted s.e.'s for every ⊆ : The Msum 8 of the all -inserted s.e.'s (∩ / / ) for ⊆ forms the given s.e. : Note 7 (charcterization of -inserted s.e.'s by conditional e.p.d.'s of the 1st kind).
(∩ / / ) with every ⊆ has a form for ⊆ : where for every ⊆ i.e., the probabilities of the 1st kind, forming for ⊆ the -conditional e.p.d. of the 1st kind of the s.e.
with respect to terraced event ter( / / ) generated by the s.e. .
(∩ / / ) for ⊆ are characterized by formulas (3.8) andconditional e.p.d.'s of the 1st kind of the s.e. with respect to the terraced event ter( / / ), generated by the s.e. . The connection between eachpseudo-distribution of the -inserted s.e. with the corresponding -conditional e.p.d. looks simpler. It is sufficient for each fixed ⊆ to normalize all its probabilities of "inserted" terraced events by the probability of a terraced event ( / / ) in order to obtain corresponding to the -conditional probabilities regarding the fact that the corresponding frame terraces event ter( / / ) happened. As a result, we have the following obvious inversion formulas:

(3.10)
Note 9 (about the appropriateness of the concept of inserted s.e.'s). It would seem, why develop a theory of inserted s.e.'s, pseudo-distributions of which are simply characterized by conditional e.p.d.'s. Is not it better to instead practice the theory of conditional e.p.d., especially since this theory has long had excellent recommendations in many areas. However, in eventology, as the theory of events, which prefers to work directly with sets of events, there is one rather serious objection. The fact is that conditional e.p.d., as any e.p.d. in eventology, there must be a set of some events, in this case, a set of well-defined "conditional events". But until now it has not been possible to give a satisfactory definition of the "conditional event", except for my impractical definition in [7]. So, the inserted s.e.'s are a completely satisfactory "surrogate" definition of the sets of "conditional events". Such that e.p.d.'s of inserted s.e.'s, although they do not coincide with the desired conditional e.p.d.'s, but are associated with them by well-defined mutual-inverse transformations. As a result, inserted s.e.'s play the role of a convenient eventological tool for working with conditional e.p.d.'s of a one set of events regarding terrace events generated by another set of events.
Example 1 (two inserted s.e.'s in a frame monoplet). Let in formulas (3.1) the s.e. is an arbitrary set, and the s.e. = { } is a frame monoplet of events, which have the joint e.p.d. in a form: Then there is the { }-inserted s.e. and the ∅inserted s.e.: which by formulas (3.5) define two -pseudodistributions for ⊆ : (3.14) First of all, note that the sum of the probabilities of terraced events from the { }-pseudo-distribution (3.14) is , and the probabilities of the ∅-pseudodistribution (3.15) is 1 − ; and secondly, that these two pseudo-distributions define a joint e.p.d. of the s.e. and the monoplet { }, i.e., e.p.d. of the s.e. + { }, which is related to them by fairly obvious formulas for ⊆ + { }: (3.16) The formulas (3.16) are recurrent, connecting the e.p.d. of s.e. + { } with two pseudo-distributions of the inserted s.e. ′ = (∩{ }/ / ) and ′′ = (∩∅/ / ) whose power is less by one. The inversion formulas (3.10) allow recurrence formulas (3.16) to express the e.p.d. of + { } via the conditional e.p.d. with respect to one of its events ∈ + { } and its complements = Ω − : where ⊆ + { }. Note that these formulas, like (3.16), can be used recursively to express the e.p.d. of s.e. + { } through two conditional e.p.d.'s of the s.e. whose power is less by one.

Inserted and conditional Kopulas of a family of sets of events with respect to the set of events
wherȇ is the set of probabilities of "inserted" marginal events from the (∩ / / ) , and are -phenomena of the set of "inserted" marginal probabilities˘( ) .
We also need to define an inserted -pseudo-Kopula with respect to the s.e. , which characterizes the -pseudo-distribution of the -inserted s.e.
(∩ / / ) , inserted into the terraces event ter( / / ) generated by the s.e. . Although the -pseudo-Kopula is not a Kopula, i.e., is not a 1-function, it has properties very reminiscent of the Kopula properties.
with respect to the s.e. is a such function K ( ) on -hypercube with sides [0, ( / / )] that 1) is non-negative: 2) satisfies the -marginal equalities for ∈ X: is a -phenomenon of the -set of marginal probabilities of the -pseudo-distribution of -inserted s.e. (∩ ) , i.e., And from (3.23) and (3.27) it follows the terraceby-terrace probabilistic -normalization of pseudo-Kopula: for ⊆ X.
Definition 10 (conditional Kopulas). The -Kopulas, characterizing -inserted e.p.d.'s of the 1st kind of the -s.e. with respect to the terraced event ter( / / ), generated by the s.e. , i.e., e.p.d.'s of the 1st kind, defined by joint e.p.d. and by formulas with fixed ⊆ for ⊆ : are called the -conditional -Kopulas of the -s.e. with respect to the terraced event ter( / / ), generated by the s.e. .
Such -cvonditional -Kopulas characterize theconditional e.p.d. of the 1st kind (3.32) by formulas for ⊆ : is a set of conditional marginal probabilities of events ∈ with respect to the terraced event ter( / / ), and are -phenomenon of the set of conditional marginal probabilities˘| . Conditional marginal probabilities are connected with "inserted" marginal probabilities for ∈ by the formula of conditional probability: since "inserted" marginal probabilities (3.20) are probabilities of intersections of events ∈ with the terraced event ter( / / ). The connection between the corresponding set of conditional "inserted" marginal probabilities we shall write in the similar way: (3.37) Note 11 (connection between conditional Kopulas and inserted pseudo-Kopulas). From Definition 10 of conditional Kopula and Definition 9 of inserted pseudo-Kopula with respect to the s.e. , and also from the formula (3.37) it follows the simple inversion formulas that connect conditional Kopulas and inserted Pseudo-Kopulas of the family of sets of events for ⊆ : (3.38) Note 12 (two formulas of full probability for a Kopula). The Kopula K of s.e. is expressed through -conditional Kpulas K | for ⊆ by the usual formula of full probability: From (3.39) and (3.38) we obtain an analogue of the formula of total probability -the representation of the Kopula of s.e. in the form of sum ofpseudo-Kopulas by ⊆ : Note 13 (Kopula of a sum of sets). A Kopula of sum + of two s.e.'s and characterizes their joint e.p.d. of the 1st kind and by definition has the form is the ( + )-phenomenon of the set of marginal probabilities for the sum + .
From previous formulas (3.32), (3.33), and (3.29) for a inserted pseudo-Kopula and conditional Kopula we obtain formulas that for each ⊆ connect the Kopula of sum + with the product of -conditional Kopula with respect to and the value of Kopula atphenomenon; and also with the -inserted pseudo-Kopula of which is inserted in the terraced event ter( / / ), generated by .

Theory of the frame method for constructing Kopula
The basis of the frame method of constructing Kopula is a rather simple idea of composing an arbitrary -s.e. X using the recurrence frame formula: are composed from two ( − 1)-s.e.'s ′ and ′′ by set-theoretic operation of -union 9 and defined as the inserted s.e.'s in the frame monoplet { 0 } by the following formulas: This simple idea allows us to find the recurrent frame formulas for the -Kopula of s.e. X as functions of the set of marginal probabilities˘= The frame method relies on formulas (3.16) and (3.17) and also correspondingly on (3.44) and (3.45), and constructs two recurrent formulas: for the -Kopula of -s.e. X through known probability 0 of the event 0 and together with it through two known inserted pseudo-( −1)-Kopulas (see Definition 9), i.e., through pseudo-( −1)-Kopulas of inserted ( − 1)-s.e.'s ′ and ′′ in the frame monoplet { 0 }, either through two known conditional ( − 1)-Kopulas (see Definition 10) with respect to the frame monoplet { 0 } of the same ′ and ′′ .
Note, that for the sake of brevity in the formulas (3.49) and (3.50) we use the following abbreviations, of course, given that X = + { 0 }:   [6]. This, however, does not detract from the generality of its application, since the set-phenomenon transformations are any s.e. can be obtained from its half-rare projection [6].
We will make the following useful half-rare, i.e., its marginal probabilities from = { 0 , 1 , ..., −1 } are not more than half, for example: Lemma 4 (about independent half-rare s.e.'s, constructed by the frame method from two inserted half-rare s.e.'s). That in the family of half-rare s.e.'s X with sets of marginal probabilitiues˘, constructed by the frame method from two inserted half-rare s.e.'s ′ and ′′ , there was an independent half-rare s.e., it is necessary so that the sets of marginal probabilities are related to the marginal probabilities of the frame s.e. X = { 0 } + = { 0 } + ( ′ (+) ′′ ) by the following way: (3.56) and sufficient so that the e.p.d. of the 1st kind of inserted s.e.'s ′ and ′′ to be calculated from the formulas for ⊆ : Proof. The necessity is obvious, since the inserted marginal probabilities of the independent s.e. X are probabilities of double intersections of independent events which have the required form for = 1, ..., − 1: The sfficiency follows from (3.57) 10 and formulas that connect the e.p.d. of the 1st kind of frame s.e. X with the e.p.d. of the 1st kind of inserted s.e.'s ′ and ′′ , which have the form for ⊆ : (3.60) Demanding (3.60) to perform sufficient conditions (3.57), we get As a result, for the s.e. X we have the e.p.d. of the 1st kind of independent events: The lemma is proved.

The Kopula theory for monoplets of events
which which provides 2 values on each 2 ( |˘)phenomenon-dom by general formulas for ⊆ { }: However, the formula (4.2) can be generalized in the following simple way: where K { } is any function such that in half-rare variables: and in free variables:

The frame method for constructing a half-rare doublet of events
In order to construct by the frame method the˘-:::::::: ordered frame half-rare doublet of events let's suppose that we have at our disposal two halfrare inserted monoplets of events with known 1-Kopulas: By Definition of inserted monoplets (5.3) (see Fig.  7) and also because of the˘-ordering assumption (5.2), we get that Consequently, the 1-Kopulas of inserted monoplanes (5.4) and (5.5) are bound by the sum of their marginal probabilities: Consequently, the 1-Kopulas of inserted monoplets (5.4) and (5.5) are bound by the restriction on the sum of their marginal probabilities: and depend on only one parameter: We get the following formulas: Rewrite this in a form of an explicit recurrent formula: Considering (2.34) and (5.9), we will continue: We note, by the way, that the restriction (5.9) by the assumption of˘-ordered (5.2) is a special case of Fréchet-inequalities: (5.14) Note 17 (frame mathod for otherwise˘-ordered halfrare doublet of events).
For otherwise˘-ordered half-rare doublet of events and also because of the assumption and also because of the assumption of another˘-ordering, we get that Consequently, 1-copulas of inserted monoplanes are connected by a restriction on the sum of their marginal probabilities: We note, as above, that the restriction (5.21) by the assumption of another˘-ordering is a special case of Fréchet inequalities: (5.26)

The frame method: recurrent formulas for a half-rare doublet of events
The formulas (5.11), as well as formulas ( (5.28) In the formulas (5.27) the pseudo-Kopulas K ({ }) and K (∅) of inserted monoplets ′ and ′′ , correspondingly, are defined by the first and the second pairs of probabilities from (5.13) correspondingly, i.e., by formulas:  In the formulas (5.28) the conditional Kopulas K |{ } and K |∅ are defined by the first and the second pairs of probabilities from (5.13), normalized by and by 1 − correspondingly, i.e. by the formulas:   that provides it on each 2 ( |˘) -phenomenon-dom the following 2 3 values: for ⊆ X. Indeed, as in the case of the doublet of events this function is a 1-function, since for all ∈ X The e.p.d. of the 1st kind of independent triplet of events X with the X-set of probabilities of eventsȋ s defined by 2 3 values of the independent 3-Kopula (6.1) on 2 ( |˘) -phenomenon-dom by the general formulas of half-rare variables, i.e., for ⊆ { , }:

Three-dimensional maps of the independent 3-Kopula
In Fig. 9 it is shown the results of visualization of the three-dimensional graph of independent 3-Kopula (8.1) of the triplet X = { , , }, defined on the cube [0, 1] 3 , in projections on planes, which are orthogonal to the axis .

The frame method for constructing a half-rare triplet of events
In order to construct by the frame method thȇ -::::::: ordered frame half-rare triplet of events X = { , , } with the X-set of marginal probabilities˘= { , , }, where let's suppose that and in our disposal we have two inserted half-rare doublets of events with the known 2-Kopulas (see Fig. 10) which by the definition satisfy the following inclusions: In view of the assumptions made (see Fig. 10) these 6 terraced events are defined. All of them are generated by the frame half-rare triplet X, with the  that express the e.p.d. of the 1st kind of frame halfrare triplet of events X through the e.p.d. of the 1st kind of inserted half-rare doublets ′ and ′′ , and the probability of frame event .
In the language of e.p.d. of the 1st kind assumptions (6.6) mean that 12) or the same in the language of probabilities events: In addition, the third pair of inclusions under the assumptions (6.6) means that Taking into account the Fréchet inequalities the restrictions (6.13) and (6.14) are equivalent to the following inequalities for 4 parameters ′ , ′ , ′ ′ ′′ ′′ of inserted doublets ′ and ′′ : Let's write the formulas (6.11), using only these 4 parameters and remembering the restrictions (6.16): (6.18)

The frame method: recurrent formulas for a half-rare triplet of events
The formulas (6.18) as well as the formulas (6.11) can be written in the form of special cases of recurrence formulas (3.52) and (3.53) from Note 15 for the triplet  In the formulas (6.19) the inserted pseudo-Kopulas K ({ }) and K (∅) are defined by the first and the second four probabilities from (6.18) correspondingly, i.e., by the formulas: where, for example, In the formulas (6.20) the conditional Kopulas K |{ } and K |∅ are defined by the first and the second four probabilities from (6.18), normalized by and by 1 − correspondingly, i.e., by the formulas: Remind, that the four functional parameters ′ , ′ , ′ ′ and ′′ ′′ in the recurrent formulas (6.19), (6.20), and also in the formulas for pseudo-Kopulas (6.21), (6.22), and the conditional Kopulas (6.25), (6.26), obey the Frechet-constraints (6.16).

The frame method for constructing a half-rare quadruplet of events
In order by the frame method to construct the˘- let's suppose that and we have two inserted half-rare triplets of events with the known 3-Kopulas, which by definition satisfy the following inclusions (see Fig. 11): The recurrent formulas, which express the e.p.d. of the 1st kind of˘-ordered half-rare quadruplet of events X through the e.p.d. of the 1st kind of two inserted triplets ′ and ′′ , follow from the general recurrent formulas (3.52) and (3.53) in Note 15 as well as in cases of a doublet and a triplet of events. And therefore, and because of the cumbersomeness, these formulas are not represented here, but are only illustrated by Venn diagrams (see Fig.  11).

Recurrent Fréchet-restrictions in the frame method
Let us dwell in more detail on Fréchet-restrictions for 11 = 2 4 − 4 − 1 functional parameters of a Kopula of quadruplet of events, to derive the recurrent sequence of such Fréchet-restrictions, which begins with Fréchet-restrictions for a doublet of events (5.36), continues with Fréchet-restrictions for a triplet of events (6.16), and should be supported by Fréchet-restrictions for parameters of a Kopula of quadruplet of events X = { , , , } and so on.
To this end, we first recall Fréchet-restrictions for parameters of Kopulas of a doublet and a triplet of events.

Fréchet-restrictions for a doublet of events
For a Kopula of doublet of events, the Fréchetrestrictions of a 1 = 2 2 − 2 − 1 parameter of inserted monoplets ′ and ′′ have the form:

Fréchet-restrictions for a triplet events
For a Kopula of doublet of events, the Fréchetrestrictions of 4 = 2 3 − 3 − 1 parameters of inserted doublets ′ and ′′ have the form: The case of triplet of events gives a new level of Fréchet-restrictions (the two last Fréchetboundaries in (7.6)), when probabilities of double intersections of events from inserted doublets have Fréchet-boundaries that depend not only on marginal probabilities of the triplet, but and on inserted marginal probabilities on which, in turn, the usual Fréchet-restrictions mentioned above are imposed.

Fréchet-restrictions for a quadruplet of events
For a Kopula of quadruplet of events the Fréchetrestrictions of 11 = 2 4 − 4 − 1 parameters of the in-serted triplet ′ and ′′ have the form: The such Fréchet-restrictions and Fréchetboundaries for a doublet (7.4), a triplet (7.5,7.6), a quadruplet (7.7,7.8) of events and so on, will call the recurrent Fréchet-restrictions and recurrent Fréchet-boundaries.

The frame method: recurrent formulas for a half-rare quadruplet of events
The recurrent formulas for Kopula of a quadruplet of events immediately can be written in the form of special cases of recurrence formulas (3.52) and (3.53) from Note 15 for the quadruplet X = { , , , In the formulas (7.9) the inserted pseudo-Kopulas K ({ }) and K (∅) are defined by the octuples of probabilities, i.e., by the formulas: (7.13) and the corresponding sets of marginal probabilities of inserted triplets ′ and ′′ have the form (7.14) In the formulas (7.10) the condirional Kopulas K |{ } and K |∅ are defined by the octuples of probabilities that are normalized by and by 1 − correspondingly, i.e., by the formulas: The corresponding sets of marginal conditional probabilities of events , ∈ with respect to the frame terraced events ter({ }/ /{ }) = and ter(∅/ /{ }) = correspondingly have the form: in the recurrent formulas (7.11, 7.12) and (7.15, 7.16) obey the Fréchet-restrictions (7.7) and Fréchet-boundaries (7.8).

Independent -Kopula
First, without the frame method, which is not required here, let's consider the simplest example of the -Kopula K ∈ Ψ 1 X of an -set of events X, i.e., a 1-function on the unit X-hypercube. In other words, we construct a nonnegative bounded numerical function A such simplest example of a 1-function on the unit X-hypercube is the so-called independent -Kopula which for all free variables˘∈ [0, 1] ⊗X is defined by the formula: that provides it on each 2 ( |˘) -phenomenon-dom the following 2 values: for ⊆ X. Indeed 11 as in the case of doublet of events this function is a 1-function, since for all ∈ X The e.p.d. of the 1st kind of ondependent -s.e. X with the X-set of probabilities of events˘is defined by 2 values of the independent -Kopula (8.1) on the 2 ( |˘) -phenomenon-dom by the general formulas of half-rare variables, i.e., for ⊆ { , }:

The frame method for constructing a half-rare set of events
The general recurrent formulas (3.52, 3.53) of the frame method for constructing a Kopula of a set of events are derived in Note 15. Recall these formulas: We write out more detailed formulas for corresponding pseudo-( −1)-Kopulas: and for conditional ( −1)-Kopulas: where

Recurrent formulas for Fréchet-boundaries and Fréchet-restrictions
Now let us consider recurrent formulas for Fréchet-boundaries Fréchet-restrictions and for the 2 − −1 functional parameters of an -Kopula of -set of events where is the X-set of probabilities of marginal events from X, i.e., = P( ), = 0, 1, . . . , −1.

Note 18 (denotations for subsets of fixed power).
To more conveniently represent the recurrent Fréchet-restrictions, agree to denote (8.14) the subsets consisting of events. In this notation, for example, the X-set of marginal probabil-ities˘( |X/ /X) is written as the X-set probabilities of the second kind that are numbered by monoplets of events 1 = { }, ∈ X: The set of probabilities of double intersection of events ∈ X, i.e., the set of probabilities of the second kind that are numbered by doublets, has the form: And the set of probabilities of triple intersections of events ∈ X, i.e., the set of probabilities of the second kind that are numbered by triplets, has the form: and so on.
Note 19 (recurrent formulas for Fréchetboundaries and Fréchet-restrictions). Probabilities of -intersections ( = 2, ..., −1) of events from the inserted s.e.'s ′ and ′′ have the recurrent Fréchetrestrictions (see paragraph 7.2) that are written by denotations from Note 18 by the following way with respect to ′ : are recurrent the lower and upper Fréchetboundaries.
And the lower Fréchet-boundary we can write somewhat differently after simple transformations:

Inserted triple covariances and Fréchet-correlations
We recall first that an absolute triple Fréchetcorrelation [1] of three events , and is defined similarly to the double one: is the triple covariance of events , and , and The definition of the inserted triple Fréchetcorrelation differs of the definition of absolute one (9.8) in that its the lower and upper Fréchetboundaries must depend on the e.p.d. of the inserted doublets ′ and ′′ . So they differ from absolute Fréchet-boundaries (9.10) and have the form For this reason, it is necessary to modify the definitions of two inserted triple covariances and, respectively, -inserted the lower and upper Fréchetboundaries of these covariances. , 14) We introduce some notation for brevity of the formulas: Two inserted triple covariances are defined by the formulas: (9.16) and inserted the lower and upper Fréchetboundaries of these covariances -by the formulas:

For any modification definitions of two inserted
Fréchet-correlations look in the usual way: (9.18)

Parametrization of functional parameters ′ ′ and ′′ ′′
The Fréchet-restriction of two functional parame- that in the frame method have a sense of probabilities of triple intersections of events: is based on the notion of the inserted triple Fréchetcorrelation.
From definitions (9.18) and (9.11) we get the parametrization of functional parameter ′ ′ of the inserted triple Fréchet-correlation Kor ({ }) in the following form: The parametrization of functional parameter ′′ ′′ of the inserted triple Fréchet-correlation Kor (∅) follows from the same definitions (9.18) and (9.11):       2-Kopula of Frank, ∈ R − {0}:  Proof follows immediately from the formulas of the set-phenomenon renumbering the terraced events of the 1st kind of the s.e. and of its setphenomena proved in [6].

Useful denotations for a doublet of events that are invariant relative to the˘-order
The following special denotations for a doublet of events X = { , } and the X-set of marginal proba-bilities˘= { , } that are invariant relative to thȇ -order, are useful. In such invariant denotations, it is not difficult to write down the general recurrence formula for the half-rare 2-Kopula of the doublet X, united combining both orders: where by Definition (11.3) The mutual set-phenomenon inversion of 2-Kopulas of half-rare˘and free˘marginal probabilities has the form: (11.8) (11.9) For example, for ⊆ X = { , } )︁ , )︁ , (11.10)

Recurrent properties of the˘-ordering a
half-rare s.e.

Recurrent properties of the˘-ordering a half-rare doublets of events
Let us explain the role of˘-ordering in the frame method using the example of constructing a 2-Kopula of the˘-ordered half-rare events X = { , } with X-set of marginal probabilities of events˘= { , }, that is, 1/2 : where, when selected as a function parameter of the 1-Kopulas of inserted half-rare monoplates Let us explain the role of˘-ordering in the frame method using the example of constructing a 3-Kopula of the˘-ordered half-rare events X = { , , } with X-set of marginal probabilities of events˘= { , , }, that is, 1/2 :  Above we outlined the frame method for constructing -Kopulas of˘-::::::: ordered half-rare -s.e.'s. It remains to extend it to construct -Kopulas of˘-: disordered half-rare -s.e.'s using the following technique, based on the obvious invariance property of permutations of events in s.e.: "as events from some s.e. do not order, the s.e. will not change "; and very useful in practical calculations.
Note 21 (properties of functions of an unordered set of arguments). Equations (11.32) and (11.33) should not be regarded as a unique property of the Kopula invariance with respect to permutations of its arguments. This property is possessed by any Kopula, since it is a function of an unordered set of arguments. Therefore it is quite natural that the Kopula is invariant under permutations of the arguments, like any other such function. This property must be remembered only in practical calculations, when we volence-nolens must introduce an arbitrary order on a disordered set in order to be able to perform calculations.

Geometric interpretation of set-phenomenon renumberings
For a subset of events ⊆ X -phenomenon renumbering of the terrace events, generated by ( − 1)-s.h-r.e. X, is based on the replacement of events from the subset = X − by their complements: / /X ( | ) )︁ (11.40) for ⊆ X and ⊆ X.
Therefore, the -phenomenon renumbering of the terrace events, generated by ( −1)-s.h-r.e. X, by the formulas (11.40) is geometrically interpreted on the ( − 1)-dimensional Venn diagram of this s.e. as a reflection of the X-hypercube relative to those hyperplanes that are orthogonal to the -axes numbered by the events ∈ ⊆ X (see Fig. 21 for the doublet of events).

Projection of the 2 X -simplex on the X-hypercube
Take an arbitrary ( −1)-s.e. X ⊆ with e.p.d. of the 1st kind, which, as is known [1], is defined as the 2 X -set of probabilities of terrace events of the 1st kind { ( / /X), ⊆ X}. (11.41) Look at the X-set (11.41) as a 2 X -hyper-point from a half-rare 2 -vertex simplex to each vertex of which the degenerate e.p.d. corresponds. In this e.p.d., as is known, only one of the 1st kind of probability, equal to one, is different from zero. Number the vertex 2 X of the simplex S 2 X by the subset ⊆ X. The degenerate e.p.d. of the 1st kind with ( / /X) = 1 corresponds to this vertex. And associate the vertex with the hypercube [0, 1] , numbered by the -set: , e.p.d., , , of the 1st kind, , .
⊆ X 2 X -S 2 X , e.p.d. of the 1st kind ( / /X) = 1. And associate with it the vertex of X-hypercube [0, 1] ⊗X , numbering by the following X-set: is a convex combination of hypercube vertices, which, as known [1], is intepreted as the probability of event ∈ X.
With projection (11.45) vertices of the 2 X -simplex maps to vertices of the X-hypercube, and edges map to its edges or diagonals (see [9], [10] and Fig.  23).
Example 4 (projections of vertices of a 2 X -simplex). For example, the vertex of the 2 X -simplex enumerated by the subset 0 ⊆ X corresponds to the degenerate e.p.d. of the 1st kind with probabilities From (11.46) you obtain that Therefore, by (11.43) { , ∈ X} = {Υ 0/ /X ( ), ∈ X} is a vertex of the X-hypercube.
In particular, the ∅-vertex of the 2 X -simplex, i.e., the vertex numbered by the subset 0 = ∅ is projected into the X-set {0, ..., 0}, consisting of the zero probabilities of marginal events, in other words, projected into the ∅-vertex of the X-hypercube, i.e., to the vertex located at the beginning coordinates: (11.47) and the X-vertex of the 2 X -simplex, i.e., the vertex enumerated by the subset 0 = X is projected into the X-set {1, ..., 1}, consisting of the unit probabilities of marginal events, in other words, projected into the X-vertex of the X-hypercube, i.e., to the vertex opposite to the origin:  In general, due to the linearity of the projection (11.45), the set of such points of the 2 X -simplex that project into the same point of the X-hypercube is convex and forms a sub-simplex of smaller dimension.

Half-rare events on Venn ( −1)-diagram
We will figure out how a Venn ( − 1)-diagram of an arbitrary ( − 1)-s.e. is constructed on the basis of the projection (11.45), in which the role of the space of universal elementary events Ω is played by the unit ( − 1)-dimensional hypercube. Such a Venn ( −1)-diagram puts terraced hypercubes generated by dividing a unit hypercube in half orthogonal to each of the axes into a one-to-one correspondence with the terraced events generated by the given ( −1)-s.e.
If the correspondence between the terraced hypercubes and the terraced events looks natural, then for the sets of half-rare events X the correspondence between the terraced hypercubes and the numbering of the vertices of the 2 X -simplex projected into the corresponding vertices of the Xhypercube under the projection (11.45) is defined by the operation of the complement and requires a special {1, ∈ }+{0, ∈ X − } ∈ [0, 1] ⊗ter( Δ / /X) , (11.56) in particular, for the ∅-phenomenon X ( |∅) = X ( ) and for ∅-vertex and X-vertex we have: (11.57)

Set-phenomenon spectrum of functions on the X-hypercube
Definition 11 (set-phenomenon spectrum of functions on the X-hypercube ). With each function ∈ Ψ X , defined on the X-hypercube, the 2 functions are connected. These functions are defined on the X-hypercube by formulas:

Remaining behind the scenes
In the text and, in particular, in the Appendix, the value of the˘-ordering condition of the set of -vertices simplex on a cube, but two conditional schemes of these projections, which illustrate a connection of six permutations of events in a half-rare triplet of events with its 2 3 set-phenomena. The conditional scheme of the projection on the upper half of the cube is shown at the top, on the lower half -at the bottom. In the Venn diagram of half-rare events: is the left, is the right, and is the lower half of the cube. events is specified, which complicates the computational implementation of the above algorithms in the frame method of constructing Kopulas as set functions of the set of marginal probabilities. The reason for this complication lies in the properties of the set-functions, i.e., functions of a set that differ from the properties of arbitrary functions of several variables. The point is that the set-function of the set of marginal probabilities is necessarily a symmetric function of the marginal probability vector (Cartesian representation of Kopula, see Prolegomenon 9), to determine which it is sufficient to specify its values only on those vectors whose components are ordered, for example, in descending order, so that the remaining values can be determined by the appropriate permutations of the arguments. For example, the Cartesian representation of an -Kopula in R is sufficient to define on the 1/ ! part of the unit -hypercube so that this representation becomes definite on the whole hypercube by continuing permutations of arguments.
Although this task is purely technical, but its solution opens the way for the application of the proposed Kopula (eventological copula) theory to the construction of the eventological theory of ordinary copulas that determine the joint distribution of a given set of marginal distributions. The author encountered this when developing the program code, which calculated all the illustrations for the Kopula examples. The problem is solved programmatically, but requires a detailed description of this solution (see Fig. 24 and 25), which, of course, together with the eventological theory of copula deserves a separate publication.
In conclusion, I can not resist the temptation to quote the formulation of the tenth Prolegomenon of the Kopula theory, which reveals the content of these my next publications.

On the inevitable development of language
This first work on the theory of the eventological copula is over at the end of July 2015. It sums up the work on the eventological theory of probabilities, raising the theory of Kopula to its apex. The work is written in a mathematical language, in which the state of the eventological theory was reflected precisely at the time when the author unexpectedly, but by the way, got a brilliant example of two statisticians from sociology and ecology, who immediately forced him to postpone polishing of the Kopula theory for almost a year in order to immediately immerse themselves in the destructive creation of a new unifying eventological theory of experience and chance by the agonizing fusion of two dual theories: the eventological theory of believabilities and the eventological theory of probabilities.
Because of this, the mathematical language of this work is just a pretension to the eventological probability theory, which does not yet know that there is a very close twin that exists -the eventological theory of believabilities. Therefore, in the terminology of this work, those crucial changes in the basic concepts and notations that were invented to construct a unifying eventological theory did not find any worthy reflection. Of course, the new unifying theory suggests the development of the original mathematical language of dual Kopulas, one of which hosts the eventological probability theory, and the other -in the eventological believability theory.
The English version of this article was published on November 12, 2017. Therefore, my later works [11,12,13,14], which expand the themes of this work, are added to the list of references. Due to the arXiv.org limitation on the volume of publication, the work is reduced by removing some illustrations. In full, the work is available at: http: //www.academia.edu/35218637/.