MSC 37H05, 60K37

Valery Doobko111Research Center of the Institute of Cybernetics, Kiev,
Ukraine<EMAIL_ADDRESS>and Elena Karachanskaya222Far Eastern State
Transport University, Khabarovsk, Russia<EMAIL_ADDRESS>
Indicator Random Processes and its Application for Modeling Open Stochastic
Systems

###### Abstract

The authors present a method of indicator random processes, applicable to
constructing models of jump processes associated with diffusion process.
Indicator random processes are processes that take only two values: 1 and 0,
in accordance with some probabilistic laws. It is shown that the indicator
random process is invariant when reduced to an arbitrary positive degree.
Equations with random coefficients used in modeling dynamic systems, when
applying the method of indicator random processes, can take into account the
possibility of adaptation to external changes, including random ones, in order
to preserve indicators important for the existence of the system, which can be
continuous or discrete. In the case of indicator random processes, defined as
functions of the Poisson process, equations for dynamic processes in a media
with abruptly changing properties are constructed and studied. To study the
capabilities of the proposed method, dynamic models of the diffusion process
in media with delay centers and diffusion processes during transitions by
switching from one subspace to another were studied. For these models,
equations for characteristic functions are constructed. Using the method of
indicator random processes, a characteristic function for the Kac model was
constructed. It is shown that in the case of dependence of the indicator
random process on the Poisson process, the equation for the characteristic
function corresponds to the telegraph equation. This result coincides with the
result of Kac.

Keywords: Indicator random processes; stochastic dynamical models.

##  1 Introduction

An open system is a system that has external interactions. Therefore, a
process modeling in an open system is difficult due to external random
disturbances that greatly affect the process.The modeling of diffusion
processes with jumps is still a pressing problem, since many applied problems
are described by similar models. Sharp fluctuations in option prices have led
to the need to study jump diffusion models, for example, the Merton model, and
Kou’s model.

Stochastic processes that can instantly, due to a jump, change their
parameters, described by systems of stochastic differential equations with
switching, are also used in problems of financial mathematics. Switching jump-
diffusion models are also used for problems in control theory, and diffusion
in random media [1] , [2], [3], and others.

In contrast to the models mentioned above, we will consider the Itô stochastic
differential equations with regime switching based on the method, which we
propose below. These equations can be used in the simulation of dynamic
systems that can adapt to external changes, including random ones. These
changes in the structure of the system can be continuous, discrete or mixed.
If changes in the coefficients of the equations is jump-like, then one of the
ways to model transitions from one state of the system to another is to use
methods from the theory of queuing systems. Nevertheless, to describe the
dynamics of the implementation of these states, one needs to use a stochastic
equation.

The purpose of the article is to show the possibility of studying random
processes described by the stochastic differential Itô equations, the
properties of which can jump-like change at random times, using the method of
indicator random processes.

In our previous article [4], we proposed a method and an idea for its use in
modeling diffusion with delay centers. In this article, we consider several
models of random processes associated with diffusion processes. For the
presented models of random processes, characteristic functions have been
constructed that determine the probabilistic properties of these processes.
The application of the proposed method of indicator random processes for
finding solutions to equations for characteristic functions is demonstrated.

##  2 Indicator random process

Indicator functions are often used to identify subsets, points, and properties
from a certain set. An indicator function in the form of a "delta function" is
used, for example, to make the transition from the original, generally
nonlinear dynamic system to an equivalent description in terms of linear
partial differential equations.

Let us introduce the concept of an indicator random process. Here and below,
all random processes under consideration are defined on a common probability
space.

######  Definition 1.

A random process $\chi(t)$ is called an indicator random process if it takes
only two values: 1 or 0.

To study the properties of the process $\chi(t)$, we use following notations:
$Prob(A)$ is the probability of an event A, $E[\chi(t)]$ is the mathematical
expectation of a random process $\chi(t)$, $t\geqslant 0$ is a time.

######  Property 1.

For any time $t$ the following conditions are satisfied:

$(\chi(t))^{\alpha}=\chi(t),\ \ \ (1-\chi(t))^{\alpha}=1-\chi(t)\ \ \
\forall\,\alpha>0.$ (1)

######  Property 2.

The following equalities are satisfied:

$\begin{gathered}E[\chi(t)]=Prob(\chi(t)=1),\\\
Prob(\chi(t)=0)=1-E[\chi(t)].\end{gathered}$ (2)

######  Definition 2.

Two and more random processes $\xi_{1}(t),\xi_{2}(t),...$, are called
incompatible processes if for every $t\geqslant 0$ only one process from this
collection is nonzero:

$\begin{array}[]{c}\xi_{k_{1}}(t_{1})\neq 0,\ \ \ \ \xi_{j}(t_{1})=0,\ \ for\
any\ j\neq k_{1};\\\ \xi_{k_{2}}(t_{2})\neq 0,\ \ \ \ \xi_{j}(t_{2})=0,\ \
for\ any\ j\neq k_{2};\\\ \ldots\end{array}$ (3)

######  Lemma 1.

Let $\chi_{j}(t)$, $j=1,2,\ldots,n-1$, be independent indicator random
processes. Then the random processes

$\begin{gathered}z_{1}(t)=\chi_{1}(t);\ \ \
z_{k}(t)=\chi_{k}(t)\prod\limits_{j=1}^{k-1}(1-\chi_{j}(t)),\ \ \
k=2,3,\ldots,n-1;\\\
z_{n}(t)=\prod\limits_{j=1}^{n-1}(1-\chi_{j}(t)),\end{gathered}$ (4)

form a complete group of incompatible processes for every $t\geqslant 0$.

###### Proof.

1\. Each of the processes (4) can take only two values: 1 and 0. Let us
consider the product of random processes $z_{l}(t)z_{k}(t)$ at any time $t$
for any $1\leqslant k,l<n$. Taking into account the Definition 1 of an
indicator random process and Property 1, we obtain

$\displaystyle z_{l}(t)z_{k}(t)$
$\displaystyle=\chi_{k}(t)\prod\limits_{j=1}^{k-1}(1-\chi_{j}(t))\chi_{l}(t)\prod\limits_{j=1}^{l-1}(1-\chi_{j}(t))$
(5)
$\displaystyle=\chi_{k}(t)(1-\chi_{k}(t))\prod\limits_{j=1}^{k-1}(1-\chi_{j}(t))^{2}\chi_{l}(t)\prod\limits_{i=k+1}^{l-1}(1-\chi_{i}(t))=0.$

Similarly, for any $1\leqslant k<n$ the following holds: $z_{k}(t)z_{n}(t)=0$.
Therefore, random processes (4) are incompatible (see Definition 2).

2\. Consider the process $Z_{n}(t)=\sum\limits_{j=1}^{n}z_{n}(t)$:

$\displaystyle Z_{n}(t)$
$\displaystyle=\chi_{1}(t)+\sum\limits_{k=2}^{n-1}\chi_{k}(t)\prod\limits_{j=1}^{k-1}(1-\chi_{j}(t))+\prod\limits_{j=1}^{n-1}(1-\chi_{j}(t))$
(6)
$\displaystyle=\chi_{1}(t)+(1-\chi_{1}(t))[\chi_{2}(t)+\chi_{3}(t)(1-\chi_{2}(t))+\ldots$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}+\chi_{n-1}(t)(1-\chi_{2}(t))\cdots(1-\chi_{n-2}(t))+(1-\chi_{2}(t))\cdots(1-\chi_{n-1}(t))].$

At any time $t$, the process (6) will take the value 1 (due to the Definition
1 of processes $\chi_{j}(t)$). Thus, at any given time $t$, only one of the
processes (4) will take the value 1.

The results (5), (6) lead to the statement of the lemma. ∎

######  Remark 1.

Using a given set of incompatible processes that form the complete group, and
knowing their probabilities, one can proceed to constructively specifying the
implementations of random processes with mixture structure (see Section 4.5).

##  3 Characteristic function for a sum of incompatible processes

Let $g_{1}(t),\ldots,g_{n}(t)$ be random processes. Consider the random
process $Y(t)=\sum\limits_{j=1}^{n}y_{j}(t)$, where

$\begin{gathered}y_{1}(t)=\chi_{1}(t)g_{1}(t);\ \ \
y_{k}(t)=\chi_{k}(t)\prod\limits_{j=1}^{k-1}(1-\chi_{j}(t))g_{k}(t),\ \ \
k=2,3,\ldots,n-1;\\\
y_{n}(t)=\prod\limits_{j=1}^{n-1}(1-\chi_{j}(t))g_{n}(t),\end{gathered}$

and indicator random processes $\chi_{j}(t)$, $j=1,2,\ldots,n$, are
independent of each other and with the random processes $g_{l}(t)$,
$l=1,\ldots,n$. Then the random process $Y(t)$ has the form:

$Y(t)=\chi_{1}(t)g_{1}(t)+\sum\limits_{k=2}^{n-1}\chi_{k}(t)\prod\limits_{j=1}^{k-1}(1-\chi_{j}(t))g_{k}(t)+\prod\limits_{j=1}^{n-1}(1-\chi_{j}(t))g_{n}(t).$
(7)

In accordance with Lemma 1, at any time $t$ only one of the processes
$g_{k}(t),\ k=1,\ldots,n,$ will be accomplished.

Let us construct the characteristic function of the random process (7).

######  Theorem 1.

Let $\chi_{j}(t)$, $j=1,2,\ldots,n-1$, be independent indicator random
processes, and $Prob(\chi_{i}(t)=1)=p_{i}(t)$. Then the characteristic
function of the process (7) has the form:

$J(t)=p_{1}(t)E[e^{i\beta g_{1}(t)}]+\sum\limits_{k=2}^{n-1}E[e^{i\beta
g_{k}(t)}]p_{k}(t)\prod\limits_{j=1}^{k-1}(1-p_{j}(t))+E[e^{i\beta
g_{n}(t)}]\prod\limits_{j=1}^{n-1}(1-p_{j}(t)).$

###### Proof.

Construct the characteristic function for the process (7):

$\displaystyle J(t)$ $\displaystyle=E[e^{i\beta Y(t)}]$
$\displaystyle=E\biggl{[}\exp\biggl{\\{}i\beta\left(\chi_{1}(t)g_{1}(t)+\sum\limits_{k=2}^{n-1}\chi_{k}(t)\prod\limits_{j=1}^{k-1}(1-\chi_{j}(t))g_{k}(t)+\prod\limits_{j=1}^{n-1}(1-\chi_{j}(t))g_{n}(t)\right)\biggl{\\}}\biggr{]}$
$\displaystyle=E[e^{i\beta
g_{1}(t)}]\prod\limits_{k=2}^{n-1}\exp\biggl{\\{}i\beta\chi_{1}(t)\prod\limits_{j=1}^{k-1}(1-\chi_{j}(t))g_{k}(t)\biggr{\\}}\exp\biggl{\\{}i\beta\prod\limits_{j=1}^{n-1}(1-\chi_{j}(t))g_{n}(t)\biggr{\\}}.$

Further, we apply the exponential series expansion, taking into account Lemma
1, Property 2, and the mutual independence of $g_{k}(t)$ and $\chi_{l}(t)$ for
any indices $k,l$. Next, we calculate the mathematical expectation, and taking
into account (2), we obtain the statement of the theorem. ∎

##  4 Application of indicator random processes

Let us examine several interesting examples of using indicator random
processes to construct mathematical models of physical processes which
associated with diffusion processes.

###  4.1 Processes in an environment with jump-like changes in properties

Using the properties of indicator random processes $\chi_{1}(t)$ and
$\chi_{2}(t)$, we can construct an equation for dynamic processes with jump-
like changes in properties. Let us consider the system of Itô stochastic
differential equations:

$\displaystyle dx(t)$
$\displaystyle=\chi_{1}(t)a_{1}(x(t),t)dt+\chi_{2}(t)B_{1}(x(t),t)d{\rm w}(t)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}+(1-\chi_{1}(t))a_{2}(x(t),t)dt+(1-\chi_{2}(t))B_{2}(x(t),t)d{\rm w}(t),$

where, in the general case, $x(t),\,a_{j}(x(t),t)\in\mathbf{R}^{n}$, ${\rm
w}(t)$ is the $m$-dimensional Wiener process with independent components,
$B_{j}(x(t),t)$ is the matrix of size $n\times m$, $j=1,2$.

Using the indicator random process $\chi(t)$, one can also construct a model
of the diffusion process with transitions from one subspace to another:

$\displaystyle dx(t)$
$\displaystyle=\chi(t)[a_{1}(x(t),y(t),t)dt+B_{1}(x(t),y(t),t)d{\rm w}(t)],$
(8) $\displaystyle dy(t)$
$\displaystyle=(1-\chi(t))[a_{2}(x(t),y(t),t)dt+B_{2}(x(t),y(t),t)d{\rm
w}(t)],$

etc. Such problems arise in the course of simulating a diffusion process with
a non-random modulus of speed, when the magnitude of the velocity modulus can
change abruptly under the influence of external random disturbances, and
remain constant between these jumps [5].

If the coefficients of (8) satisfy to conditions

$B_{j}(x(t),y(t),t)=0,\ \ \ a_{1}(x(t),y(t),t)=-a_{2}(x(t),y(t),t)=-c,\ \ \
c=const>0,$

then the process $x(t)+y(t)$ corresponds to the Kac model of particle motion
with random changes in the direction of velocity [6].

###  4.2 Diffusion process with random time delay centers

We propose a new version of the model of dynamic process with delay centers
employs indicator random processes.

A delay center, or a time absorption center, arises, for example, for queuing
processes: a device becomes a delay center when serving a customer. For the
diffusion process, such centers can be considered points in space where a
particle temporarily stuck and leaves them at a random moment in time.

We will correlate the randomness of the moment of stopping the particle’s
movement and the moment of resumption its movement with a random function of a
non-random integer value $N(t)$ with independent increments (indicator random
process) [4]:

$\chi(t)=\bar{\chi}(N(t)).$

######  Definition 3.

An indicator random function $\chi(t)$ is called conditionally periodic if it
satisfies the condition

$\chi(t)=\bar{\chi}(N(t)+2k),\ \ \ \ \ k\in\mathbf{N}.$ (9)

The conditions (1), (9) are satisfied by the function

$\chi(t)=\bar{\chi}(N(t))=0.5(1+\cos[\pi N(t)]).$ (10)

This function has the following properties:

$\chi(t)=\begin{cases}0.5(1+\cos[\pi N(t)])=1,&\text{for}\ \ N(t)=2s,\ \
s\in\mathbf{N}\cup\\{0\\},\\\ 0.5(1+\cos[\pi N(t)])=0,&\text{for}\ \
N(t)=2s+1,\ \ s\in\mathbf{N}\cup\\{0\\}.\end{cases}$

For example, as $N(t)$ we can take a homogeneous Poisson process: $E[N(t)]=\mu
t$. Then $E[\bar{\chi}(N(t))]=0.5(1+\exp\\{-2\mu t\\})$.

To model diffusion processes that have centers of random time delay (diffusion
interruption) and subsequent restoration of the movement process, let us use
the following Itô stochastic differential equation:

$dx(t)=\chi(t)[a(x(t),t)dt+B(x(t),t)d{\rm w}(t)],$ (11)

where, in the general case, $x(t),\,a(x(t),t)\in\mathbf{R}^{n}$, ${\rm w}(t)$
is the $m$-dimensional Wiener process with independent components, $B(x(t),t)$
is the matrix of size $n\times m$.

As is known, diffusion occurs by several mechanisms. Surface diffusion is a
general process involving the motion of molecules, and atomic clusters at
solid material surfaces, and the corresponding model is the system of
equations (11) in $\mathbf{R}^{2}$. Bulk diffusion, i. e. diffusion in the
bulk of the material, can be modelled by the system of equations (11) in
$\mathbf{R}^{3}$.

Let us consider the equation

$dx(t)=a(t)dt+0.5(1+\cos[\pi N(t)])b(t)d{\rm w}(t),\ \ \ x(0)=0.$ (12)

This model corresponds to the case when the state of the system changes
according to a deterministic law, and then, over a random period of time, it
is affected by random disturbances. Let us make a change of variables:

$y(t)=x(t)-\displaystyle\int_{0}^{t}a(\tau)d\tau.$

Then the equation (12) takes the form

$dy(t)=0.5(1+\cos[\pi N(t)])b(t)d{\rm w}(t),$ (13)

where $y(t),\,a(t),\,b(t),\,\in\mathbf{R}$, ${\rm w}(t)$ is one-dimensional
Wiener process, $N(t)$ is Poisson process, and $N(t)$ and ${\rm w}(t)$ are
mutually independent ones. Suppose, that $y(0)=0$.

The characteristic function for the random process (13) is

$J_{1}(t)=E[e^{i\beta
y(t)}]=E\biggl{[}\exp\biggl{\\{}0.5i\beta\displaystyle\int_{0}^{t}(1+\cos[\pi
N(\tau)])b(\tau)d{\rm w}(\tau)\biggr{\\}}\biggl{]}.$

If, during the simulation, we take into account that the average time interval
between a particle being at rest and in motion are different, then as $N(t)$
we can choose, for example, the random process $N(t)=N_{1}(t)N_{2}(t)$, where
$N_{1}(t)$, $N_{2}(t)$ are independent Poisson processes for which
$E[N_{1}(t)]=\mu_{1}t$, $E[N_{2}(t)]=\mu_{2}t$. Since the relation holds

$\displaystyle\sum_{s=0}^{\infty}Prob(N(t)=2s)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=1-\sum_{s_{1}=0}^{\infty}Prob(N_{1}(t)=2s_{1}+1)\sum_{s_{2}=0}^{\infty}Prob(N_{2}(t)=2s_{2}+1),$

i.e., the probability of a particle in motion will be greater than the
probability of being at rest.

If the situation is the opposite, then we proceed to using the process

$\hat{\chi}(t)=\chi(N(t)+1).$

Applying Itô formula, taking into account (1), and using the explicit form of
the Poisson distribution [7], we obtain

$\displaystyle D(t)=\frac{dE\left[y^{2}(t)\right]}{dt}$
$\displaystyle=0.5b^{2}\langle 1+\cos[\pi N(t)]\rangle$ (14)
$\displaystyle=\frac{1}{4}b^{2}[4-(1-e^{-2\mu_{1}t})(1-e^{-2\mu_{2}t})],$

where $b^{2}$ is the diffusion coefficient without time delay centers. It
follows from equality (14) that

$D(0)=b^{2},\ \ \ D(\infty)=\displaystyle\frac{3}{4}b^{2}.$

Note that the difference between the probabilities for even and odd $N(t)$ is
due to the fact that $N(t)\in\mathbf{N}\cup\\{0\\}$ and $N(0)=0$, i.e., $N(t)$
does not start with an odd number. This leads to a discrepancy among the
analytical expressions for the probabilities:

$\displaystyle\sum_{s=0}^{\infty}Prob(N(t)=2s)$ $\displaystyle=e^{-\mu
t}\sum_{s=0}^{\infty}\frac{(\mu t)^{2s}}{(2s)!}=e^{-\mu t}\cosh\mu
t=0.5(1+e^{-2\mu t}),$ $\displaystyle\sum_{s=0}^{\infty}Prob(N(t)=2s+1)$
$\displaystyle=e^{-\mu t}\sum_{s=0}^{\infty}\frac{(\mu
t)^{2s+1}}{(2s+1)!}=e^{-\mu t}\sinh\mu t=0.5(1-e^{-2\mu t})$
$\displaystyle=1-\sum_{s=0}^{\infty}Prob(N(t)=2s).$

The equality of probabilities is only asymptotic:

$\lim\limits_{t\to\infty}\sum_{s=0}^{\infty}Prob(N(t)=2s)=\lim\limits_{t\to\infty}\sum_{s=0}^{\infty}Prob(N(t)=2s+1)=0.5.$

Thus, if in a homogeneous media there is a nonlinear time dependence of the
average square displacement of a particle and an asymptotic decrease in the
diffusion coefficient, we can conclude that there exist time delay centers.

###  4.3 Diffusion with random change in direction of movement. The Kac model

Let us consider one-dimensional particle movement with speed $v$, when the
direction of movement changes at random times [6]:

$\left\\{\begin{aligned} dx^{+}(t)&=\chi(t)vdt,\\\
dx^{-}(t)&=-(1-\chi(t))vdt.\end{aligned}\right.$

Since we are interested in the total displacement $x(t)=x^{+}(t)+x^{-}(t)$, we
obtain the equation

$dx(t)=(2\chi(t)-1)vdt.$ (15)

For $\chi(t)$ we take the representation (10):
$\chi(t)=\bar{\chi}(N(t))=0.5(1+\cos[\pi N(t)]).$ Let $v=const=c>0$. Then the
characteristic function for the process $x(t)$ takes the form:

$I(t)=E\biggl{[}\exp\biggl{\\{}i\beta\int_{0}^{t}\cos[\pi
N(\tau)]cd\tau\biggr{\\}}\biggr{]}.$ (16)

######  Theorem 2.

If $N(t)$ is the stationary Poisson process with parameter $\lambda$, the
characteristic function for the process $x(t)$ subordinate to the system (15)
is a solution to the Cauchy problem:

$\displaystyle\frac{d^{2}I(t)}{dt^{2}}+2\lambda\frac{dI(t)}{dt}+c^{2}\beta^{2}I(t)=0,\
\ \ I(0)=0,\ \ \ \frac{dI(0)}{dt}=ic\beta.$ (17)

###### Proof.

Let us differentiate (16):

$\frac{dI(t)}{dt}=E\biggl{[}i\beta\cos[\pi
N(t)]c\exp\biggl{\\{}i\beta\displaystyle\int_{0}^{t}\cos[\pi
N(\tau)]cd\tau\biggr{\\}}\biggr{]}.$ (18)

Let $f(x)\in\mathcal{C}^{\infty}$. Then

$df(N(t))=[f(N(t)+1)-f(N(t))]dN(t),$ (19)

where $dN(t)$ is an advanced increment, i.e., it is independent of the
previous values of $N(t)$. Since $E[dN(t)]=\lambda dt,\,\lambda>0$, then
taking into account (18), we obtain

$\displaystyle d\frac{dI(t)}{dt}$
$\displaystyle=-c^{2}\beta^{2}E\biggl{[}\cos^{2}[\pi
N(t)]\exp\biggl{\\{}i\beta\int_{0}^{t}\cos[\pi
N(\tau)]cd\tau\biggr{\\}}\biggr{]}dt$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {}+i\beta
cE\biggl{[}(\cos[\pi(N(t)+1)]-\cos[\pi
N(t)])\exp\biggl{\\{}i\beta\int_{0}^{t}\cos[\pi
N(\tau)]cd\tau\biggr{\\}}\biggr{]}\lambda dt$
$\displaystyle=-c^{2}\beta^{2}E\biggl{[}\exp\biggl{\\{}i\beta\int_{0}^{t}\cos[\pi
N(\tau)]cd\tau\biggr{\\}}\biggr{]}dt$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {}+2i\beta c\lambda
E\biggl{[}\cos[\pi N(t)]\exp\biggl{\\{}i\beta\int_{0}^{t}\cos[\pi
N(\tau)]cd\tau\biggr{\\}}\biggr{]}dt$
$\displaystyle=-c^{2}\beta^{2}I(t)dt-2\lambda\frac{dI(t)}{dt}dt.$

From the last equality we obtain the statement of the theorem. ∎

######  Remark 2.

As is known, the characteristic function allows one to find the distribution
density function. Applying the inverse Fourier transform to the equation (17)
from the Theorem 2, we obtain the telegraph equation for the distribution
density function $\rho(x,t)$:

$\frac{\partial^{2}\rho(x,t)}{\partial
t^{2}}+2\lambda\frac{\partial\rho(x,t)}{\partial
t}-c^{2}\frac{\partial^{2}\rho(x,t)}{\partial x^{2}}=0,$

which coincides with the results obtained [6].

Note that the Kac model is finding new applications. In particular, it is used
to study the model of random colliding particles interacting with the infinite
reservoir at a fixed temperature and chemical potential [8]. This is the so-
called thermostat problem, in which particles can leave the system towards the
reservoir or enter the system from the reservoir at random times. Accordingly,
the proposed random indicator process method can also be used to solve the
thermostat problem.

###  4.4 Diffusion model with random transitions from one subspace to another.
2-dimensional case

Let us consider the following diffusion model:

$\left\\{\begin{aligned} dx(t)&=\tilde{\chi}(t)bd{\rm w}(t),\\\
dy(t)&=(1-\tilde{\chi}(t))bd{\rm w}(t),\end{aligned}\right.$ (20)

where $b$ is the diffusion coefficient, i.e., at random moments of time the
process occurs either in the space $x(t)$ or $y(t)$. Such models can describe
the diffusion process in random porous media. As a random process
$\tilde{\chi}(t)$ we choose the representation $\bar{\chi}(t)=0.5(1+\cos
N(t)).$ Then the characteristic function for the process $\\{x(t),y(t)\\}$, in
accordance with (20), takes the form:

$\displaystyle J(t)$
$\displaystyle=E\biggl{[}\exp\biggl{\\{}i\alpha\int_{0}^{t}0.5(1+\cos
N(\tau))bd{\rm w}(\tau)+i\beta\int_{0}^{t}0.5(1-\cos N(\tau))bd{\rm
w}(\tau)\biggr{\\}}\biggr{]}.$

######  Theorem 3.

If $(N(t)/\pi)$ is the stationary Poisson process with parameter $\lambda$,
$J(t)$ is a solution to the equation:

$\frac{d^{2}J(t)}{dt^{2}}+0.5(4\lambda+[\alpha^{2}+\beta^{2}]b^{2}\frac{dJ(t)}{dt}+0.5(\lambda\alpha^{2}+0.5\alpha^{2}\beta^{2}b^{2}+\lambda\beta^{2})b^{2}\beta^{2}J(t)=0.$
(21)

###### Proof.

Since the processes $N(t)$ and ${\rm w}(t)$ are independent, we obtain

$\displaystyle J(t)$
$\displaystyle=E\biggl{[}\exp\biggl{\\{}i\int_{0}^{t}0.5[(\alpha+\beta)+(\alpha-\beta)\cos
N(t)]bd{\rm w}(\tau)\biggr{\\}}\biggr{]}$ (22)
$\displaystyle=E\biggl{[}\exp\biggl{\\{}-\int_{0}^{t}2^{-3}[(\alpha+\beta)+(\alpha-\beta)\cos
N(t)]^{2}b^{2}d\tau\biggr{\\}}\biggr{]}$
$\displaystyle=E\biggl{[}\exp\biggl{\\{}-\int_{0}^{t}2^{-3}[(\alpha+\beta)^{2}+2(\alpha^{2}-\beta^{2})\cos
N(t)+(\alpha-\beta)^{2}]b^{2}d\tau\biggr{\\}}\biggr{]}$
$\displaystyle=E\biggl{[}\exp\biggl{\\{}-\alpha^{2}\int_{0}^{t}2^{-2}(1+\cos
N(t))b^{2}d\tau-\beta^{2}\int_{0}^{t}2^{-2}(1-\cos
N(t))b^{2}d\tau\biggr{\\}}\biggr{]}.$

For compactness, let us denote by $f(t)$ the last expression under the
mathematical expectation sign in (22):

$f(t)=\exp\biggl{\\{}-\alpha^{2}\displaystyle\int_{0}^{t}2^{-2}(1+\cos
N(t))b^{2}d\tau-\beta^{2}\int_{0}^{t}2^{-2}(1-\cos
N(t))b^{2}d\tau\biggr{\\}}.$ (23)

Next we get

$\displaystyle\frac{dJ(t)}{dt}=-E[f(t)\\{\alpha^{2}2^{-2}(1+\cos
N(t))b^{2}+\beta^{2}2^{-2}(1-\cos N(t))b^{2}\\}].$ (24)

Taking into account that $dN(t)$ is an advanced increment, i.e., it does not
depend on the previous values of $N(t)$, for which, due to the properties of
the Poisson distribution,

$E[d(N(t)/\pi)]=\lambda dt,\ \ \ \lambda>0.$

Let us calculate the differential (compare with (19)):

$d\cos N(t)=[\cos(N(t)+\pi)-\cos N(t)]d(N(t)/\pi)=-2\cos N(t)d(N(t)/\pi).$

The process $\cos N(t)$ have the following properties:

$\begin{gathered}\cos N(t)\cos N(t)\equiv 1,\\\ (1-\cos N(t))(1+\cos
N(t))=1-\cos^{2}N(t)\equiv 0,\\\ (1-\cos N(t))^{2}=1-\cos N(t),\\\ (1+\cos
N(t))^{2}=1+\cos N(t).\end{gathered}$

Then

$\displaystyle\frac{d^{2}J(t)}{dt^{2}}$
$\displaystyle=E[f(t)\\{0.5\alpha^{2}b^{2}\cos N(t)-\beta^{2}2^{-2}2b^{2}\cos
N(t)\\}]\lambda$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {}+E[f(t)\\{\alpha^{2}2^{-2}(1+\cos
N(t))b^{2}+\beta^{2}2^{-2}(1-\cos N(t))^{2}b^{4}\\}^{2}]$
$\displaystyle=E[f(t)\\{0.5\alpha^{2}b^{2}\cos N(t)-0.5\beta^{2}b^{2}\cos
N(t)\\}]\lambda$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {}+E[f(t)\\{\alpha^{4}2^{-4}(1+\cos
N(t))^{2}b^{4}+\beta^{4}2^{-4}(1-\cos N(t))^{2}b^{4}\\}]$
$\displaystyle=\lambda E[f(t)\\{0.5\alpha^{2}(1+\cos
N(t))b^{2}+0.5\beta^{2}(1-(\cos N(t)))b^{2}\\}]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}-0.5\lambda(\alpha^{2}+\beta^{2})b^{2}J(t)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}+E[f(t)\\{\alpha^{4}2^{-3}(1+\cos N(t)b^{4}+\beta^{4}2^{-3}(1-\cos
N(t))b^{4}\\}].$

This imply that

$\displaystyle\frac{\partial^{2}J(t)}{\partial t^{2}}$
$\displaystyle=-2\lambda\frac{\partial J(t)}{\partial
t}-0.5\lambda(\alpha^{2}+\beta^{2})b^{2}J(t)$ (25)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}+E[f(t)\alpha^{4}2^{-3}(1+\cos N(t))b^{4}]+E[f(t)\beta^{4}2^{-3}(1-\cos
N(t))b^{4}].$

Let us continue the transformation for the last terms in (25). For simplicity,
let us examine each of them separately.

Considering the first term:

$\displaystyle E[f(t)\alpha^{4}2^{-3}(1+\cos N(t))b^{4}]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=0.5\alpha^{2}b^{2}E[f(t)\\{\alpha^{2}2^{-2}(1+\cos
N(t))b^{2}+\beta^{2}2^{-2}(1-\cos N(t))b^{2}\\}]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}-\alpha^{2}\beta^{2}b^{4}E[f(t)2^{-3}(1-\cos N(t)\\}],$

and taking into account (24), we have

$\displaystyle E[f(t)\alpha^{4}2^{-3}(1+\cos N(t))b^{4}]$ (26)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=-0.5\alpha^{2}b^{2}\frac{dJ(t)}{dt}-\alpha^{2}\beta^{2}b^{4}E[f(t)2^{-3}(1-\cos
N(t)\\}].$

Let us transform the second term:

$\displaystyle E[f(t)\beta^{4}2^{-3}(1-\cos N(t))b^{4}]$ (27)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=0.5\beta^{2}b^{2}E[f(t)\\{\alpha^{2}2^{-2}(1+\cos
N(t))b^{2}+\beta^{2}2^{-2}(1-\cos N(t))b^{2}\\}]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}-\alpha^{2}\beta^{2}b^{4}E[f(t)2^{-3}(1+\cos N(t))]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=-0.5\beta^{2}b^{2}\frac{dJ(t)}{dt}-\alpha^{2}\beta^{2}b^{4}E[f(t)2^{-3}(1+\cos
N(t))]$

Substituting the resulting representations (26) and (27) into (25), we obtain:

$\alpha^{2}\beta^{2}b^{4}E[f(t)2^{-3}(1-\cos
N(t)]+\alpha^{2}\beta^{2}b^{4}E[f(t)2^{-3}(1+\cos
N(t))]=\alpha^{2}\beta^{2}b^{4}2^{-2}J(t).$

As a result, taking into account the notation $E[f(t)]=J(t)$, we obtain

$\displaystyle\frac{d^{2}J(t)}{dt^{2}}$
$\displaystyle=-2\lambda\frac{dJ(t)}{dt}-0.5\lambda(\alpha^{2}+\beta^{2})b^{2}J(t)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}-0.5\alpha^{2}\beta^{2}b^{2}\frac{dJ(t)}{dt}-\alpha^{2}\beta^{2}b^{4}2^{-2}E[f(t)]$
$\displaystyle=-(2\lambda+0.5\alpha^{2}\beta^{2}b^{2})\frac{dJ(t)}{dt}-[\alpha^{2}\beta^{2}b^{4}2^{-2}+0.5\lambda(\alpha^{2}+\beta^{2})b^{2}]J(t)$

The statement of Theorem 3 is obtained. ∎

The equation (21) can be solved. However, its construction will be cumbersome.
Using the result obtained in Theorem 3, we find an explicit view of the
characteristic function for a simpler process. We will solve the equation for
a simpler model, and then we will show how to move to the explicit form of
solving the equation (21).

As such a simple model, consider the characteristic function (13) for the
diffusion model with delay centers:

$\frac{d^{2}I(t)}{dt^{2}}+2\lambda\frac{dI(t)}{dt}+c^{2}\beta^{2}I(t)=0,\ \ \
I(0)=0,\ \ \ \frac{dI(0)}{dt}=ic\beta.$

The equation for the characteristic function $J_{1}(t)$ should coincide with
(21) if we set $\alpha=0$. Formally, this corresponds to the following
substitutions of coefficients in (21):

$\displaystyle 0.5(4\lambda+[\alpha^{2}+\beta^{2}]b^{2})$
$\displaystyle\Rightarrow$ $\displaystyle 0.5(4\lambda+\beta^{2}b^{2}),$ (28)
$\displaystyle
0.5(\lambda\alpha^{2}+0.5\alpha^{2}\beta^{2}+\lambda\beta^{2})b^{2}$
$\displaystyle\Rightarrow$ $\displaystyle 0.5\lambda\beta^{2}b^{2}.$

Accordingly, the characteristic function $J_{1}(t)$ for the diffusion model
with delay centers is a solution to the equation:

$\frac{d^{2}J_{1}(t)}{dt^{2}}+0.5(\beta^{2}b^{2}+4\lambda)\frac{dJ_{1}(t)}{dt}+0.5\lambda\beta^{2}b^{2}J_{1}(t)=0.$
(29)

Since the conditions are satisfied

$\displaystyle 2^{-2}(\beta^{2}b^{2}+4\lambda)^{2}-2\lambda\beta^{2}b^{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=2^{-2}\beta^{4}b^{4}+2\beta^{2}b^{2}\lambda+4\lambda^{2}-2\beta^{2}b^{2}\lambda$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
=2^{-2}\beta^{4}b^{4}+4\lambda^{2}>0\ \ \ \forall\,\lambda>0,$

then the solution to the equation (29) ([9], p. 375, formula 235(a)) will be
as follows:

$\displaystyle J_{1}(t)$
$\displaystyle=C_{1}\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)+t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{\\}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}+C_{2}\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)-t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{\\}}.$

Since
$\displaystyle\frac{1}{4}(\beta^{2}b^{2}+4\lambda)^{2}>\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}$,
then the first and second terms decrease with increasing $t$, and therefore
$\lim\limits_{t\to\infty}J_{1}(t)=0$. Taking into account the initial
conditions

$\frac{dJ_{1}(t)}{dt}\biggl{|}_{t=0}=-0.5\beta^{2}b^{2},\ \ \
J_{1}(t)\bigl{|}_{t=0}\bigr{.}=1,$

we obtain an equation for determining the constants:

$C_{1}+C_{2}=1,$ $\begin{gathered}\begin{aligned}
0.5\beta^{2}b^{2}&=C_{1}\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)+t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{\\}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}+C_{2}\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)-t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{\\}},\\\
0.5\beta^{2}b^{2}&=(1-C_{2})\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)+t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{\\}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}+C_{2}\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)-t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{\\}},\end{aligned}\\\
\end{gathered}$ $2\lambda=C_{2}\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}.$

Having solved this equations system, we establish that

$\begin{gathered}C_{2}=2\lambda\biggl{(}\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{)}^{-1}=\biggl{(}\frac{1}{16}\lambda^{-2}\beta^{4}b^{4}+1\biggr{)}^{-0.5},\\\
C_{1}=1-\biggl{(}\frac{1}{16}\lambda^{-2}\beta^{4}b^{4}+1\biggr{)}^{-0.5}.\end{gathered}$

Thus, the solution to the equation (29) takes the form

$\displaystyle J_{1}(t)$
$\displaystyle=\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)+t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{\\}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}+\biggl{(}\frac{1}{16}\lambda^{-2}\beta^{4}b^{4}+1\biggr{)}^{-0.5}\biggl{[}\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)-t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\bigl{.}\biggr{\\}}\bigr{.}\bigr{.}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{}-\exp\biggl{\\{}-0.5t(\beta^{2}b^{2}+4\lambda)+t\sqrt{\frac{1}{4}\beta^{4}b^{4}+4\lambda^{2}}\biggr{\\}}\bigl{.}\biggr{]}.$

Since the condition is satisfied

$\biggl{(}\frac{1}{16}\lambda^{-2}\beta^{4}b^{4}+1\biggr{)}^{-0.5}<1,$

then $J_{1}(t)>0$ for all $t\geqslant 0.$

Using inverse substitutions of coefficients based on relations (28), we obtain
the solution to the equation (21). Taking into account the relationship
between characteristic functions and moments, we can find random moments for
the processes under consideration.

###  4.5 Application of a random indicator process to specify implementations
of random processes with variable structure

######  Lemma 2.

Let the following be given: a collection of independent set-events ${\rm
A}_{j}$, $j=1,2,\ldots,n-1$ and a complete group of incompatible events ${\rm
B}_{j}$, $j=1,2,\ldots,n$:

$B_{1}={\rm A}_{1};\ \ \ \ \ \ {\rm B}_{j}={\rm
A}_{j}\bigcap\biggl{(}\,\bigcap_{k=1}^{j-1}\bar{{\rm A}}_{k}\biggr{)},\ \ \
j=2,3,\ldots,n-1;\ \ \ \ \ \ {\rm B}_{n}=\bigcap_{k=1}^{n-1}\bar{{\rm
A}}_{k}.$

Let us assume that a set of probabilities is given:

$Prob({\rm B}_{j}),\ \ \ j=1,2,\ldots,n-1;\ \ \ \ \ \ Prob({\rm
B}_{n})=1-\sum\limits_{j=1}^{n-1}Prob({\rm B}_{j}).$

Then it is possible to establish a one-to-one correspondence between the sets
$Prob({\rm A}_{j})$ and $Prob({\rm B}_{r})$, $r,j=1,2,\ldots,n-1$.

###### Proof.

Due to the independence of ${\rm A}_{j},\ \ j=1,2,\ldots,n-1$, we get the
equalities:

$\displaystyle Prob({\rm B}_{1})$ $\displaystyle=Prob({\rm A}_{1}),$
$\displaystyle Prob({\rm B}_{2})$ $\displaystyle=Prob({\rm
A}_{2})Prob({\rm\bar{A}}_{1}),$ $\displaystyle Prob({\rm B}_{3})$
$\displaystyle=Prob({\rm A}_{3})Prob({\rm\bar{A}}_{2})Prob({\rm\bar{A}}_{1}),$
$\displaystyle\ldots,$ $\displaystyle Prob({\rm B}_{n-1})$
$\displaystyle=Prob({\rm
A}_{n-1})\prod\limits_{k=1}^{n-2}Prob({\rm\bar{A}}_{k})$

or

$\displaystyle Prob({\rm B}_{1})$ $\displaystyle=Prob({\rm A}_{1}),$
$\displaystyle Prob({\rm B}_{2})$ $\displaystyle=Prob({\rm A}_{2})(1-Prob({\rm
A}_{1})),$ $\displaystyle Prob({\rm B}_{3})$ $\displaystyle=Prob({\rm
A}_{3})(1-Prob({\rm A}_{2}))(1-Prob({\rm{A}}_{1})),$ $\displaystyle\ldots,$
$\displaystyle Prob({\rm B}_{n-1})$ $\displaystyle=Prob({\rm
A}_{n-1})\prod\limits_{k=1}^{n-2}(1-Prob({\rm{A}}_{k})).$

Switching from one equality to another, we obtain

$\displaystyle Prob({\rm B}_{1})$ $\displaystyle=Prob({\rm A}_{1}),$
$\displaystyle Prob({\rm B}_{2})$ $\displaystyle=Prob({\rm A}_{2})(1-Prob({\rm
A}_{1}))=Prob({\rm A}_{2})(1-Prob({\rm B}_{1}))$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Rightarrow\ \ \ Prob({\rm A}_{2})=\frac{Prob({\rm B}_{2})}{1-Prob({\rm
B}_{1})},$ $\displaystyle Prob({\rm B}_{3})$ $\displaystyle=Prob({\rm
A}_{3})(1-Prob({\rm A}_{2}))(1-Prob({\rm{A}}_{1}))$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Rightarrow\ \ \ Prob({\rm A}_{3})=\frac{Prob({\rm B}_{3})}{1-Prob({\rm
B}_{1})-Prob({\rm B}_{2})},$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \ldots,$ $\displaystyle Prob({\rm
B}_{n-1})$ $\displaystyle=Prob({\rm
A}_{n-1})\prod\limits_{k=1}^{n-2}(1-Prob({\rm{A}}_{k}))$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Rightarrow\ \ \ Prob({\rm A}_{n-1})=\frac{Prob({\rm
B}_{n-1})}{1-\sum\limits_{k=1}^{n-2}Prob({\rm B}_{k})}.$

Thus, a one-to-one correspondence established. ∎

Consider an example. Let $\chi_{j}(t)$ be independent indicator random
processes. For example, $\chi_{j}(t)$ can be as follows:

$\chi_{j}(t)=0.5(1-\cos(\pi N_{j}(t)),$ (30)

where $N_{j}(t)$ are independent Poisson processes with variable intensity
$\lambda_{j}(t)$, and

$Prob(N_{j}(t)=m)=\frac{a_{j}^{m}(t)}{m!}e^{-a_{j}(t)},\ \ \ m=0,1,2,\ldots,\
\ \ a_{j}(t)=\int_{0}^{t}\lambda_{j}(\tau)d\tau.$

Let the events ${\rm B}_{j}$, $j-1,\ldots,n$, be incompatible events that
constitute a complete group. Let us assume that events ${\rm A}_{j}$ are
associated with process $\chi_{j}(t)$, and events ${\rm B}_{j}$ are associated
with process $N_{j}(t)$.

Let the events ${\rm B}_{j}$, $j-1,\ldots,n$, be incompatible events that form
a complete group. Let us assume that the event process ${\rm A}_{j}$ is
associated with the process $\chi_{j}(t)$, and the event process ${\rm B}_{j}$
is associated with the probabilities $Prob_{t}({\rm B}_{j})=p_{j}(t)$.

Taking into account (30), events ${\rm A}_{j}$ will correspond only to odd
values of the process $N_{j}(t)$. Therefore,

$\displaystyle Prob_{t}(\bar{{\rm A}}_{j})$
$\displaystyle=\sum_{k=0}^{\infty}Prob(N_{j}(t)|N_{j}(t)=2k)$
$\displaystyle=e^{-a_{j}(t)}\sum\limits_{m=0}^{\infty}\frac{(a_{j}(t))^{2m}}{(2m)!}=e^{-a_{j}(t)}\cosh
a_{j}(t)=0.5(1+e^{-2a_{j}(t)}).$

Let us proceed to comparing the distributions $Prob_{t}({\rm A}_{j})$ and
$Prob_{t}({\rm B}_{j})$:

$\begin{gathered}0<Prob_{t}({\rm
A}_{j})=0.5(1-e^{-2a_{j}(t)})=\frac{Prob_{t}({\rm
B}_{j})}{1-\sum\limits_{k=1}^{j-1}Prob_{t}({\rm
B}_{k})}=\frac{p_{j}(t)}{1-\sum\limits_{k=1}^{j-1}p_{k}(t)}=p_{j}(\chi(t))<1,\\\
1-2p_{j}(\chi(t))=e^{-2a_{j}(t)}=1-\frac{2p_{j}(t)}{1-\sum\limits_{k=1}^{j-1}p_{k}(t)}=\frac{-\sum\limits_{k=1}^{j}p_{k}(t)-p_{j}(t)}{1-\sum\limits_{k=1}^{j-1}p_{k}(t)}\geqslant
0.\end{gathered}$

From this equality it follows:

$a_{j}(t)=0.5\ln\left[\frac{1-\sum\limits_{k=1}^{j-1}p_{k}(t)}{1-\sum\limits_{k=1}^{j}p_{k}(t)]-p_{j}(t)}\right].$
(31)

As the numerator in the equality (31) is positive,

$1-\sum\limits_{k=1}^{j-1}p_{k}(t)=\sum\limits_{k=j}^{n}p_{k}(t)\geqslant 0,$

it is necessary to establish the conditions when the denominator is positive:

$1-\sum\limits_{k=1}^{j}p_{k}(t)\geqslant p_{j}(t)\ \ \ \Rightarrow\ \ \
\sum\limits_{k=j+1}^{n}p_{k}(t)\geqslant p_{j}(t)\ \ \
\forall\,j=1,\ldots,n-1.$ (32)

The requirement follows from (32):

$p_{j+1}(t)\geqslant p_{j}(t).$

Such ranking is always possible, and therefore we consider it as the initial
one.

Since $a_{j}(t)\geqslant 0$, the following inequality must hold:

$1-\sum\limits_{k=1}^{j-1}p_{k}(t)\geqslant
1-\sum\limits_{k=1}^{j}p_{k}(t)-p_{j}(t),$

which is always held:

$\biggl{[}1-\displaystyle\sum\limits_{k=1}^{j}p_{k}(t)\biggr{]}+p_{j}(t)\geqslant\biggl{[}1-\sum\limits_{k=1}^{j}p_{k}(t)\biggr{]}-p_{j}(t)\
\ \ \ \Rightarrow\ \ \ \ 0\geqslant-2p_{j}(t).$

## Conclusion

The proposed method of indicator random processes using the method of
characteristic functions allows us to consider both previously known models
and new ones that have physical interpretations, such as diffusion processes.
Theorems have been proven in which equations for the characteristic functions
of the random processes under consideration are obtained. In the future, these
characteristic functions can be used to find the probabilistic properties of
random processes.

Also we note that if a complete group of incompatible random processes is
given, then it is established that there is a set of independent indicator
random processes. Based on the latter, it is possible to construct a complete
group of events whose distribution will coincide with the distribution of a
given group of incompatible events.

## References

  * [1] Hanson, F.B. Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation; SIAM: Philadelphia, USA, 2007; 472 p.
  * [2] Averina, T.A.; Rybakov, K.A. Solving approximately a prediction problem for stochastic jump-diffusion systems. Numer. Anal. Appl. 2017, 10, 1–10.
  * [3] Averina, T.A.; Karachanskaya, E.V.; Rybakov, K.A. Statistical analysis of diffusion systems with invariants. Russ. J. Numer. Anal. Math. Model. 2018, 33, 1–13.
  * [4] Doobko, V.A. Simulation of distribution of impurities in the environment with delay centers using indicator functions. In Proceedings: Agrarian science and education in the 21st century: Problems, prospects and innovations, Ukraine, 17th–18th, May, 2018. [in Ukrainian].
  * [5] Karachanskaya, E.V. Simulation Processes with Invariants, Pacific National Univ. Publ.: Khabarovsk, Russia, 2014; 144 p. [in Russian].
  * [6] Kac, M. Probability and Related Topics in Physical Sciences, Interscience Publ.: London, UK, 1959; 266 p.
  * [7] Gihman, I.; Skorohod, A. Stochastic Differential Equations; Springer-Verlag: Berlin, New York, USA, 1972; 354 p.
  * [8] Beck, J.; Bonetto, F. Grand canonical evolution for the Kac model. J. Stat. Phys. 2022, 188, 7.
  * [9] Kamke, E. Differential Equations. Solution Methods and Solutions. Vol. 1: Ordinary differential equations; Akademische Verlagsgesellschaft Becker & Erler: Leipzig, Germany, 1951; 665 p.


# Most Cantor sets in $\mathbb{R}^{N}$ are in general position with respect to
all projections

Olga Frolkina
Chair of General Topology and Geometry
Faculty of Mechanics and Mathematics
M.V. Lomonosov Moscow State University
and
Moscow Center for Fundamental and Applied Mathematics
Leninskie Gory 1, GSP-1
119991 Moscow, Russia
E-mail<EMAIL_ADDRESS>
###### Abstract

We prove the theorem stated in the title. This answers a question of John Cobb
(1994). We also consider the case of the Hilbert space $\ell_{2}$.

††2020 _Mathematics Subject Classification_ : Primary 54B20; Secondary 54E52,
54C25, 46C05.†† _Key words and phrases_ : Euclidean space, Hilbert space,
projection, Cantor set, dimension, general position, Baire Category Theorem.

##  1 Introduction

Already at the end of the 19th century, mathematicians were interested in
behaviour of projections of totally disconnected subsets of Euclidean space.
In 1884, G. Cantor described a continuous surjection of what is now called the
standard middle-thirds Cantor set $\mathcal{C}$ onto the segment $I=[0,1]$; it
takes a point $\frac{x_{1}}{3}+\frac{x_{2}}{3^{2}}+\ldots\in\mathcal{C}$
(where each $x_{i}\in\\{0;2\\}$) into the point
$\frac{1}{2}\left(\frac{x_{1}}{2}+\frac{x_{2}}{2^{2}}+\ldots\right)\in I$. In
1906, L. Zoretti recalls as a known fact (“ce resultat bien connu etant
acquis…”) that the graph of any such surjection is a Cantor set in plane whose
projection to $y$-axis coincides with $I$. (See [8] or [9] for historic
details.)

In 1994, John Cobb asked [5, p. 126]: “given integers $m>n>k>0$, is there a
Cantor set in $\mathbb{R}^{m}$ each of whose projections into $n$-hyperplanes
will be exactly $k$-dimensional?” Examples of such sets are known for the
following triples $(m,n,k)$: $(2,1,1)$ [2, 9, p.272; fig.2, p.273], [5, p.
124, Example], [6, Prop. 1]; $(m,n,n)$ [4]; $(3,2,1)$ [5]; $(m,n,n-1)$ [7];
$(m,m-1,k)$ [3]. Other constructions for the cases $(m,{m-1},{m-2})$ and
$(3,2,1)$ are described in the papers [8] and [10] which also contain further
references. For each integer $n$, there is a Cantor set in the Hilbert space
$\ell_{2}$ all of whose projections into $n$-planes are $(n-1)$-dimensional
[3].

In the same paper, Cobb posed another question [5, p. 128]: “Cantor sets that
raise dimension under all projections and those in general position with
respect to all projections are both dense in the Cantor sets in
$\mathbb{R}^{m}$ — which (if either) is more common, in the sense of category
or dimension or anything?” In [9], I answered a weaker version of this
question showing that all projections of a typical Cantor set in Euclidean
space are Cantor sets; thus the examples listed above are “a rarity”. In this
article, we fully answer the category part of this question: a typical Cantor
set in $\mathbb{R}^{N}$ is in general position with respect to all
projections. We also show that a typical Cantor set in $\ell_{2}$ is in
general position with respect to all projections with finite-dimensional
kernels. (By a projection of $\mathbb{R}^{N}$ or $\ell_{2}$, we always mean
its orthogonal projection onto a non-zero linear subspace.)

##  2 Preliminaries

###  2.1 Baire Category Theorem

Let $\mathcal{X}$ be a non-empty topological space. We say that a set
$\mathcal{A}\subset\mathcal{X}$ is _meager in $\mathcal{X}$_, or _of first
category in $\mathcal{X}$_, if it is the union of countably many nowhere dense
subsets of $\mathcal{X}$. A set $\mathcal{A}$ that is not of first category in
$\mathcal{X}$ is said to be _non-meager in $\mathcal{X}$_, or _of second
category in $\mathcal{X}$_. A set $\mathcal{A}\subset\mathcal{X}$ is called
_comeager in $\mathcal{X}$_, or _residual in $\mathcal{X}$_, if it is the
complement of a meager set; equivalently, if it contains the intersection of a
countable family of dense open sets. The two last definitions make sense only
if $\mathcal{X}$ is of second category (in itself).

A topological space $\mathcal{X}$ is called a _Baire space_ if any non-empty
open set in $\mathcal{X}$ is non-meager in $\mathcal{X}$; equivalently, every
comeager set is dense in $\mathcal{X}$; equivalently, for each countable
family $\\{\mathcal{U}_{n},n\in\mathbb{N}\\}$ of open dense subsets of
$\mathcal{X}$, the intersection $\bigcap\limits_{n=1}^{\infty}\mathcal{U}_{n}$
is dense in $\mathcal{X}$. Every Baire space is non-meager in itself. The
_Baire Category Theorem_ states that _a completely metrizable space is Baire_
; see e.g. [12, 8.4] for a proof. A $G_{\delta}$ subset of a completely
metrizable space is itself completely metrizable [12, 3.11].

Let $\mathcal{A}$ be a subset of a non-empty Baire space $\mathcal{X}$. _A
generic (or typical) element of $\mathcal{X}$ is in $\mathcal{A}$_ or _most
elements of $\mathcal{X}$ are in $\mathcal{A}$_ if $\mathcal{A}$ is comeager
in $\mathcal{X}$. This notion is well-defined: in a Baire space, the
statements “most elements are in $\mathcal{A}$” and “most elements are in
$\mathcal{X}-\mathcal{A}$” exclude each other. In a Baire space $\mathcal{X}$,
a subset $\mathcal{A}$ is comeager iff it contains a dense $G_{\delta}$ subset
of $\mathcal{X}$.

###  2.2 Hyperspaces of compact (Cantor) subsets

_A Cantor set_ is a space homeomorphic to the standard middle-thirds Cantor
set $\mathcal{C}$. By the Brouwer theorem, these spaces can be characterized
as non-empty metric zero-dimensional perfect compacta [12, 7.4].

Let $X$ be a complete metric space. The space $\mathcal{K}(X)$ of all non-
empty compact subsets of $X$ is endowed with the Hausdorff metric; this space
is complete [12, 4.25]. The corresponding topology on $\mathcal{K}(X)$
coincides with the Vietoris topology [12, 4.F, 4.21]. If, moreover, $X$ has no
isolated points, then the set $\mathcal{C}(X)$ of all Cantor sets in $X$ is a
dense $G_{\delta}$ subset of $\mathcal{K}(X)$ [14, Prop. 2] (see also [13,
Lemma 1] for the case of $\mathbb{R}^{N}$, or [11, Lemma 2.1, Remark 2.2]).
Hence $\mathcal{C}(X)$ is itself a Baire space. Note that the Hausdorff metric
on $\mathcal{C}(X)$ may be not complete.

##  3 Main result

###  3.1 Statements

The next definition is introduced in [5, p. 127] for $\mathbb{R}^{N}$. We
extend it to the case of $\ell_{2}$.

######  Definition 3.1.

Let $H$ be a non-zero proper linear subspace of $\mathbb{R}^{N}$ (or of
$\ell_{2}$). Let $p_{H}:\mathbb{R}^{N}\to H^{\bot}$ (or $p_{H}:\ell_{2}\to
H^{\bot}$) be the orthogonal projection onto $H^{\bot}$. For a non-empty
subset $M$ of $\mathbb{R}^{N}$ (or of $\ell_{2}$), denote the fibers of
$p_{H}|_{M}$ by $F(A)=\left(p_{H}|_{M}\right)^{-1}\left(p_{H}(A)\right)$,
where $A\in M$. Call a fiber $F(A)$ _non-degenerate_ if $|F(A)|>1$. A set
$M\subset\mathbb{R}^{N}$ (or $M\subset\ell_{2}$) is said to be _in general
position with respect to $p_{H}$ or $H$_ if, denoting $k=\dim H$, we have:
$p_{H}|_{M}$ has only finitely many non-degenerate fibers; each non-degenerate
fiber $F(A)$ consists of $\leqslant k+1$ points which form the vertices of a
simplex of dimension $|F(A)|-1$; and $\sum\limits_{A\in
M}\left(|F(A)|-1\right)\leqslant k$. We say that $M$ is _in general position
with respect to all projections_ if, for _each_ $H$, $M$ is in general
position with respect to $H$. Finally, $M\subset\ell_{2}$ is _in general
position with respect to all projections with finite-dimensional kernels_ if
$M$ is in general position with respect to $H$ for _each_ finite-dimensional
$H$.

In $\mathbb{R}^{N}$, $N\geqslant 2$, there exist Cantor sets in general
position with respect to all projections [5, Theorem 5] (where a sketch proof
is given). Our main result says that in the sense of category, a typical
Cantor set in $\mathbb{R}^{N}$ is in general position with respect to all
projections; this answers the category part of Cobb’s question [5, p. 128].

######  Main Theorem 3.2.

For each integer $N\geqslant 2$, Cantor sets in $\mathbb{R}^{N}$ that are in
general position with respect to all projections form a dense $G_{\delta}$
subset in the space $\mathcal{C}(\mathbb{R}^{N})$. Cantor sets in $\ell_{2}$
that are in general position with respect to all projections with finite-
dimensional kernels form a dense $G_{\delta}$ subset in
$\mathcal{C}(\ell_{2})$.

Recall that $\mathcal{C}(X)$ is a dense $G_{\delta}$ subset in
$\mathcal{K}(X)$ if $X$ is a complete metric space without isolated points.
Hence our main result immediately follows from

######  Theorem 3.3.

For each integer $N\geqslant 2$, non-empty compact sets in $\mathbb{R}^{N}$
that are in general position with respect to all projections form a dense
$G_{\delta}$ subset in the space $\mathcal{K}(\mathbb{R}^{N})$. Non-empty
compacta in $\ell_{2}$ that are in general position with respect to all
projections with finite-dimensional kernels form a dense $G_{\delta}$ subset
in $\mathcal{K}(\ell_{2})$.

Concerning the case of $\ell_{2}$, it is appropriate to remind the reader of
Theorem 3 from [3] which says that for any integer $k$ and any compact subset
$M\subset\ell_{2}$, the restriction $p_{H}|_{M}$ is an embedding for all $H$
in some dense subset of the Grassmann space $\mathcal{G}_{k}(\ell_{2})$.

As a consequence, we get the following result. Its first part was proved in
[9] by a straightforward geometric reasoning.

######  Corollary 3.4.

For each integer $N\geqslant 2$, all projections of a typical Cantor set in
$\mathbb{R}^{N}$ are Cantor sets. For a typical Cantor set in $\ell_{2}$, any
of its projections with finite-dimensional kernel is a Cantor set.

This corollary follows from Theorem 3.2 together with Brouwer’s
characterization of Cantor sets [12, 7.4].

###  3.2 Partial confirmation

Before we proceed to the proof of Theorem 3.3, let us recall the following
standard

######  Definition 3.5.

A subset $M$ of $\mathbb{R}^{N}$ (or of $\ell_{2}$) is said to be _in general
position_ if each of its subsets of $d+1$ pairwise distinct points
$A_{0},A_{1},\ldots,A_{d}$, where $d\leqslant N$ (respectively, where $d$ is
any integer), is affinely independent; that is, the affine subspace generated
by $\\{A_{0},A_{1},\ldots,A_{d}\\}$ has dimension $d$.

A Cantor set which satisfies Definition 3.1 is clearly in general position.
After minor modification, direct geometric reasoning from [13, proof of Lemma
2] give us

######  Proposition 3.6.

For any $N\geqslant 2$, the set of all non-empty compacta in $\mathbb{R}^{N}$
which are in general position form a dense $G_{\delta}$ subset in
$\mathcal{K}(\mathbb{R}^{N})$. The set of all non-empty compacta in $\ell_{2}$
which are in general position form a dense $G_{\delta}$ subset in
$\mathcal{K}(\ell_{2})$.

###### Proof.

Let $X$ be $\mathbb{R}^{N}$ (where $N\geqslant 2$) or $\ell_{2}$. For
arbitrary positive integers $d$ and $t$ let $\mathcal{H}_{d,t}$ be the subset
of $\mathcal{K}(X)$ consisting of all non-empty compacta $K\subset X$ with the
property: there exists an affinely dependent finite subset
$\\{A_{0},A_{1},\ldots,A_{d}\\}\subset K$ such that $d\leqslant\dim X$ and

$\frac{1}{t}\leqslant\min\\{\rho(A_{i},A_{j})\ |\ i=0,\ldots,d;\
j=0,\ldots,d;\ i\neq j\\}.$

As in [13, proof of Lemma 2], each $\mathcal{H}_{d,t}$ is closed and nowhere
dense in $\mathcal{K}(X)$. A non-empty compactum $M\subset X$ is in general
position iff
$M\in\bigcap\limits_{d,t\in\mathbb{N}}\left(\mathcal{K}(X)\setminus\mathcal{H}_{d,t}\right)$.
∎

But to fulfill all conditions of Definition 3.1, we need additional work. The
main ingredient of our proof is the Mycielski-Kuratowski Theorem. (Proposition
3.6 will not be used in our proof, although it could have been.)

###  3.3 Proof of Theorem 3.3

###### Proof of Theorem 3.3.

Let $X$ be $\mathbb{R}^{N}$ (where $N\geqslant 2$) or $\ell_{2}$. Call an
array of integers $(k;s;i_{1},\ldots,i_{s})$ _admissible_ if $0<k<\dim X$,
$s\geqslant 1$, $i_{1}\geqslant 2$, …, $i_{s}\geqslant 2$, and
$\sum\limits_{j=1}^{s}(i_{j}-1)\geqslant k+1$. For each admissible array,
define the relation $R_{k;s;i_{1},\ldots,i_{s}}\subset X^{i_{1}+\ldots+i_{s}}$
as the set of all

$(A_{1}^{1},\ldots,A_{i_{1}}^{1};A_{1}^{2},\ldots,A_{i_{2}}^{2};\ldots;A_{1}^{s},\ldots,A_{i_{s}}^{s})\in
X^{i_{1}+\ldots+i_{s}}$

such that the linear span of the set

$\Bigl{\\{}\overrightarrow{A_{1}^{1}A_{2}^{1}},\overrightarrow{A_{1}^{1}A_{3}^{1}},\overrightarrow{A_{1}^{1}A_{4}^{1}},\ldots,\overrightarrow{A_{1}^{1}A_{i_{1}}^{1}};\\\
\overrightarrow{A_{1}^{2}A_{2}^{2}},\overrightarrow{A_{1}^{2}A_{3}^{2}},\overrightarrow{A_{1}^{2}A_{4}^{2}},\ldots,\overrightarrow{A_{1}^{2}A_{i_{2}}^{2}};\\\
\ldots\\\
\overrightarrow{A_{1}^{s}A_{2}^{s}},\overrightarrow{A_{1}^{s}A_{3}^{s}},\overrightarrow{A_{1}^{s}A_{4}^{s}},\ldots,\overrightarrow{A_{1}^{s}A_{i_{s}}^{s}}\Bigr{\\}}$
(3.1)

has dimension $\leqslant k$. (Note that the set (3.1) consists of
$\sum\limits_{j=1}^{s}(i_{j}-1)\geqslant k+1$ vectors.) For an integer $q$ and
an ordered $q$-tuple of points $(A_{1},\ldots,A_{q})\in X^{q}$, denote

$\mathcal{V}(A_{1},\ldots,A_{q}):=\\{\overrightarrow{A_{1}A_{2}},\overrightarrow{A_{1}A_{3}},\overrightarrow{A_{1}A_{4}},\ldots,\overrightarrow{A_{1}A_{q}}\\}.$

In this notation, the set $(\ref{vectsyst})$ can we written briefly as
$\bigcup\limits_{j=1}^{s}\mathcal{V}(A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{i_{j}})$.

For each admissible array $(k;s;i_{1},\ldots,i_{s})$, the set
$R_{k;s;i_{1},\ldots,i_{s}}$ is closed and nowhere dense in
$X^{i_{1}+\ldots+i_{s}}$. Indeed, it is clear that the interior of
$R_{k;s;i_{1},\ldots,i_{s}}$ is empty; and it remains to show that
$X^{i_{1}+\ldots+i_{s}}\setminus R_{k;s;i_{1},\ldots,i_{s}}$ is open in
$X^{i_{1}+\ldots+i_{s}}$. For this, just observe that the linear subspace
spanned by the system $(\ref{vectsyst})$ has dimension $\geqslant k+1$ if and
only if the set $(\ref{vectsyst})$ contains a subset
$\\{\vec{v}_{1},\ldots,\vec{v}_{k+1}\\}$ with a non-zero Gram determinant

$\begin{vmatrix}(\vec{v}_{1},\vec{v}_{1})&(\vec{v}_{1},\vec{v}_{2})&(\vec{v}_{1},\vec{v}_{3})&\ldots&(\vec{v}_{1},\vec{v}_{k+1})\\\
(\vec{v}_{2},\vec{v}_{1})&(\vec{v}_{2},\vec{v}_{2})&(\vec{v}_{2},\vec{v}_{3})&\ldots&(\vec{v}_{2},\vec{v}_{k+1})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
(\vec{v}_{k+1},\vec{v}_{1})&(\vec{v}_{k+1},\vec{v}_{2})&(\vec{v}_{k+1},\vec{v}_{3})&\ldots&(\vec{v}_{k+1},\vec{v}_{k+1})\end{vmatrix}\neq
0.$

The Mycielski-Kuratowski Theorem [15, Thm. 1], [14, Cor. 3] (see [12, 19.1]
for a short proof) now implies that for each admissible array
$(k;s;i_{1},\ldots,i_{s})$ the set

$\mathcal{M}_{k;s;i_{1},\ldots,i_{s}}:=\\{K\in\mathcal{K}(X)\ |\
(K)^{i_{1}+\ldots+i_{s}}\subset X^{i_{1}+\ldots+i_{s}}\setminus
R_{k;s;i_{1},\ldots,i_{s}}\\}$

is a dense $G_{\delta}$ subset in $\mathcal{K}(X)$. Here $(K)^{q}$ denotes the
set of all sequences $(A_{1},\ldots,A_{q})\in X^{q}$ whose elements are
pairwise distinct: $A_{i}\neq A_{j}$ for each $i\neq j$. Consider

$\mathcal{P}:=\bigcap\mathcal{M}_{k;s;i_{1},\ldots,i_{s}},$

where the intersection is taken over all admissible arrays
$(k;s;i_{1},\ldots,i_{s})$. The set $\mathcal{P}$ is a dense $G_{\delta}$
subset in $\mathcal{K}(X)$.

_A non-empty compact set $M\subset X$ belongs to $\mathcal{P}$ if and only if
$M$ is in general position with respect to: all projections if
$X=\mathbb{R}^{N}$; and all projections with finite-dimensional kernels if
$X=\ell_{2}$._ Let us prove this.

Necessity. Let $M\in\mathcal{P}$. Take any non-zero proper linear subspace $H$
of $X$ (in addition, assume that $H$ is finite-dimensional if $X=\ell_{2}$).
Denote $\kappa:=\dim H$.

(i) Suppose that $p_{H}|_{M}:M\to H^{\bot}$ has infinitely many non-degenerate
pairwise distinct fibers $F_{1},F_{2},\ldots$. Let $\varsigma:=\kappa+1$.
There exist integers $\varphi_{1}\geqslant 2$, $\varphi_{2}\geqslant 2$,…,
$\varphi_{\varsigma}\geqslant 2$ and pairwise distinct points

$A_{1}^{1},A_{2}^{1},\ldots,A_{\varphi_{1}}^{1}\in F_{1},$
$A_{1}^{2},A_{2}^{2},\ldots,A_{\varphi_{2}}^{2}\in F_{2},$ $\ldots$
$A_{1}^{\varsigma},A_{2}^{\varsigma},\ldots,A_{\varphi_{\varsigma}}^{\varsigma}\in
F_{\varsigma}.$

Note that the array
$(\kappa;\varsigma;\varphi_{1},\varphi_{2},\ldots,\varphi_{\varsigma})$ is
admissible. By construction,

$(M)^{\varphi_{1}+\varphi_{2}+\ldots+\varphi_{\varsigma}}\subset
X^{\varphi_{1}+\varphi_{2}+\ldots+\varphi_{\varsigma}}\setminus
R_{\kappa;\varsigma;\varphi_{1},\varphi_{2},\ldots,\varphi_{\varsigma}},$

hence

$\dim\left(\operatorname{Span}\left(\bigcup\limits_{j=1}^{\varsigma}\mathcal{V}(A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{\varphi_{j}})\right)\right)\geqslant\kappa+1.$

On the other hand, for each $j=1,\ldots,\varsigma$ the fiber $F_{j}$ is
contained in the affine subspace $A_{1}^{j}+H:=\\{A_{1}^{j}+\vec{v}\ |\
\vec{v}\in H\\}$, therefore
$\mathcal{V}(A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{\varphi_{j}})\subset H$. We get

$\bigcup\limits_{j=1}^{\varsigma}\mathcal{V}(A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{\varphi_{j}})\subset
H,$

and

$\dim\left(\operatorname{Span}\left(\bigcup\limits_{j=1}^{\varsigma}\mathcal{V}(A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{\varphi_{j}})\right)\right)\leqslant\dim
H=\kappa.$

This contradiction shows that $p_{H}|_{M}:M\to H^{\bot}$ has only finitely
many non-degenerate fibers.

(ii) Let $F$ be any non-degenerate fiber of $p_{H}|_{M}:M\to H^{\bot}$.
Suppose that $|F|\geqslant\kappa+2$; take pairwise distinct points
$A_{1},\ldots,A_{\kappa+2}\in F$. The array $(k;s;i_{1})=(\kappa;1;\kappa+2)$
is admissible. Thus $(M)^{\kappa+2}\subset X^{\kappa+2}\setminus
R_{\kappa;1;\kappa+2}$. Therefore

$\dim\left(\operatorname{Span}(\overrightarrow{A_{1}A_{2}},\overrightarrow{A_{1}A_{3}},\overrightarrow{A_{1}A_{4}},\ldots,\overrightarrow{A_{1}A_{\kappa+2}})\right)\geqslant\kappa+1.$

This leads to a contradiction since
$\\{\overrightarrow{A_{1}A_{2}},\overrightarrow{A_{1}A_{3}},\overrightarrow{A_{1}A_{4}},\ldots,\overrightarrow{A_{1}A_{\kappa+2}}\\}\subset
H$. Consequently each non-degenerate fiber of $p_{H}|_{M}:M\to H^{\bot}$
contains $\leqslant\kappa+1$ points.

(iii) Let $F=\\{A_{1},\ldots,A_{\mu}\\}$ be any non-degenerate fiber of
$p_{H}|_{M}:M\to H^{\bot}$. Let us show that its points form the vertices of a
$(\mu-1)$-dimensional simplex. Since $\mu\leqslant\kappa+1$ by (ii), it
suffices to show that the system $\\{A_{1},\ldots,A_{\mu}\\}$ is in general
position. Assume the contrary; then there exists a linear subspace
$\tilde{H}\subset H$ and an index $j_{0}\in\\{1,\ldots,\mu\\}$ such that

$|(A_{j_{0}}+\tilde{H})\cap F|\geqslant\dim\tilde{H}+2.$

On the other hand, the set $(A_{j_{0}}+\tilde{H})\cap F$ is a non-degenerate
fiber of the projection $p_{\tilde{H}}|_{M}:M\to\tilde{H}^{\bot}$. Applying
(ii) to $p_{\tilde{H}}|_{M}$, we get the inequality

$|(A_{j_{0}}+\tilde{H})\cap F|\leqslant\dim\tilde{H}+1,$

which is a contradiction. Therefore the system $\\{A_{1},\ldots,A_{\mu}\\}$ is
in general position.

(iv) Let $F_{1},F_{2},\ldots,F_{\varsigma}$ be the list of all non-degenerate
fibers of $p_{H}|_{M}:M\to H^{\bot}$. They are finite sets by (ii). Enumerate
their points:

$F_{1}=\\{A_{1}^{1},A_{2}^{1},A_{3}^{1},\ldots,A_{\varphi_{1}}^{1}\\},$
$F_{2}=\\{A_{1}^{2},A_{2}^{2},A_{3}^{2},\ldots,A_{\varphi_{2}}^{2}\\},$
$\ldots$
$F_{\varsigma}=\\{A_{1}^{\varsigma},A_{2}^{\varsigma},A_{3}^{\varsigma},\ldots,A_{\varphi_{\varsigma}}^{\varsigma}\\}.$

Our aim is to prove the inequality

$(\varphi_{1}-1)+(\varphi_{2}-1)+\ldots+(\varphi_{\varsigma}-1)\leqslant\kappa.$

Suppose the contrary: let

$(\varphi_{1}-1)+(\varphi_{2}-1)+\ldots+(\varphi_{\varsigma}-1)\geqslant\kappa+1.$

In this case, the array
$(\kappa;\varsigma;\varphi_{1},\ldots,\varphi_{\varsigma})$ is admissible. We
get

$(M)^{\varphi_{1}+\varphi_{2}+\ldots+\varphi_{\varsigma}}\subset
X^{\varphi_{1}+\varphi_{2}+\ldots+\varphi_{\varsigma}}\setminus
R_{\kappa;\varsigma;\varphi_{1},\ldots,\varphi_{\varsigma}},$

hence

$\dim\left(\operatorname{Span}\left(\bigcup\limits_{j=1}^{\varsigma}\mathcal{V}(A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{\varphi_{j}})\right)\right)\geqslant\kappa+1.$

This leads to a contradiction since
$\bigcup\limits_{j=1}^{\varsigma}\mathcal{V}(P^{j}_{1},P^{j}_{2},\ldots,P^{j}_{\varphi_{j}})\subset
H$, as before.

Sufficiency. Let us prove that for $N\geqslant 2$ any non-empty compactum
$M\subset\mathbb{R}^{N}$ (or $M\subset\ell_{2}$) which is in general position
with respect to all projections (in case of $\ell_{2}$, to all projections
with finite-dimensional kernels) lies in $\mathcal{P}$. Suppose, towards a
contradiction, that $M\notin\mathcal{M}_{k;s;i_{1},\ldots,i_{s}}$ for some
admissible array $(k;s;i_{1},\ldots,i_{s})$. Then there exists a sequence

$(A_{1}^{1},\ldots,A_{i_{1}}^{1};A_{1}^{2},\ldots,A_{i_{2}}^{2};\ldots;A_{1}^{s},\ldots,A_{i_{s}}^{s})$

of pairwise distinct points of $M$ such that

$\dim\left(\operatorname{Span}\left(\bigcup\limits_{j=1}^{s}\mathcal{V}(A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{i_{j}})\right)\right)\leqslant
k.$

Put
$H:=\operatorname{Span}\left(\bigcup\limits_{j=1}^{s}\mathcal{V}(A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{i_{j}})\right)$.
Consider the fibers of $p_{H}|_{M}:M\to H^{\bot}$. For each $j=1,\ldots,s$ the
set $\\{A^{j}_{1},A^{j}_{2},\ldots,A^{j}_{i_{j}}\\}$ is contained in one of
the fibers (these fibers may coincide for different $j$’s). Consequently

$\sum\limits_{A\in
M}(|F(A)|-1)\geqslant\sum\limits_{j=1}^{s}(i_{j}-1)\geqslant k+1.$

Hence $M$ is not in general position with respect to $p_{H}$, a contradiction.
This finishes the proof. ∎

### Acknowledgements

Supported by Russian Foundation of Basic Research (Grant No. 19–01–00169).

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