А conditional linear random process (CLRP) has been defined as the stochastic integral of a random function with respect to a process with independent increments. When the process with independent increments is Poisson then CLRP represents the signal as a sum of a large amount of stochastically dependent impulses whose times of occurrence are the times of a Poisson process. For example, the electricity loads of the electrical power systems, also the processes of gas and water consumption, electrophysiological signals et al. can be modelled using CLRP. Moreover, the stochastic periodicity of the signals can be taken into account. A random coefficient autoregressive model has been shown to be a member of the class of discrete-time CLRP and suitable for estimation purposes. The main goal of the paper is to 1 / 4 N2 s7develop the procedure for the parameter estimation of random coefficient periodic autoregressive (RCPAR) model. The model has periodic parameters and consequently periodic probability distribution. The estimations have been obtained as a result of applying the least squares method to the set of L (where L is a period) stationary and jointly stationary subsequences of RCPAR model. The simulation results have been presented which confirm the consistency of the developed estimations, that is, the precision of the estimates increases with the increase in the sample size. The results of short-term electricity consumption forecasting of the enterprise (which belongs to the class of small and medium-sized) have been presented and analyzed using RCPAR model. References 16, figures 4, tables 2.

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The object of research is the process of mathematical modelling of a multidimensional random signal, which in the structure of its generation is the sum of a large number of random impulses that occur at random times. Examples of stochastic signals of this type can be, in particular, electroencephalographic and cardiographic signals, photoplethysmography signals, resource consumption processes (electricity, gas, water consumption), radar signals, vibrations of bearings of electric machines and others. A common mathematical model (especially in the multidimensional case) of this type of signal is a linear random process that allows the signal to be represented as the sum of a large number of stochastically independent random impulses that occur at Poisson moments. If the impulses are stochastically dependent (or the moments of time of their appearance are not Poisson), then the mathematical model is a conditional linear random process. The definition and analysis of the probabilistic properties of such processes for the multidimensional case have not been conducted. The paper defines a multidimensional conditional linear random process, each component of which is represented as a stochastic integral of a random kernel driven by a process with independent increments. Expressions for the characteristic function and moment functions of the specified process are obtained. The approach used was to use the mathematical apparatus of conditional characteristic functions, as well as the known representation of an infinitely divisible characteristic function of a linear random process as a functional of a process with independent increments. The obtained results provide a possibility for theoretical analysis of probabilistic properties of multichannel stochastic signals, the mathematical model of which is a multidimensional conditional linear random process. Justification of their properties of stationarity or cyclostationarity, which are the consequence of corresponding properties of the kernel and process with independent increments, can be carried out.

Continuous-time conditional linear random process is represented as a stochastic integral of a random kernel driven by a process with independent increments. Such processes are used in the problems of mathematical modelling, computer simulation, and processing of stochastic signals, the physical nature of which generates them to be represented as the sum of many random impulses that occur at Poisson moments. Impulses are stochastically dependent functions, in contrast to another well-known mathematical model which is a linear random process, that has a similar structure but is represented as the sum of a large amount of independent random impulses that occur at Poisson moments of time. The application areas of these models are mathematical modelling, computer simulation, and processing of electroencephalographic signals, cardio signals, resource consumption processes (such as electricity consumption, water consumption, gas consumption), radar signals, etc. A discrete-time conditional linear random process has been defined in the paper, the relationships with corresponding continuous-time model has been shown. According to the given definition the discrete-time conditional linear random process can be considered as an output of linear digital filter with random parameters on the input of the white noise which is infinitely divisible distributed. Moment functions of first and second order have been analyzed. In particular, the expressions for mathematical expectation, variance and covariance function have been obtained. The results can be utilized to study the probabilistic characteristics of the investigated information stochastic signals, which will depend on the properties of the corresponding kernel and white noise. In particular, the conditions for the process to be wide-sense stationary have been represented.