{"id":13028,"date":"2022-07-01T12:44:14","date_gmt":"2022-07-01T09:44:14","guid":{"rendered":"http:\/\/journals.khnu.km.ua\/vestnik\/?p=13028"},"modified":"2022-07-21T14:26:40","modified_gmt":"2022-07-21T11:26:40","slug":"umovni-linijni-vypadkovi-proczesy-z-dyskretnym-chasom-ta-yih-vlastyvosti","status":"publish","type":"post","link":"https:\/\/journals.khnu.km.ua\/vestnik\/?p=13028","title":{"rendered":"\u0423\u043c\u043e\u0432\u043d\u0456 \u043b\u0456\u043d\u0456\u0439\u043d\u0456 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u0456 \u043f\u0440\u043e\u0446\u0435\u0441\u0438 \u0437 \u0434\u0438\u0441\u043a\u0440\u0435\u0442\u043d\u0438\u043c \u0447\u0430\u0441\u043e\u043c \u0442\u0430 \u0457\u0445 \u0432\u043b\u0430\u0441\u0442\u0438\u0432\u043e\u0441\u0442\u0456"},"content":{"rendered":"

<\/p>\n

\u0423\u041c\u041e\u0412\u041d\u0406 \u041b\u0406\u041d\u0406\u0419\u041d\u0406 \u0412\u0418\u041f\u0410\u0414\u041a\u041e\u0412\u0406 \u041f\u0420\u041e\u0426\u0415\u0421\u0418 \u0417 \u0414\u0418\u0421\u041a\u0420\u0415\u0422\u041d\u0418\u041c \u0427\u0410\u0421\u041e\u041c \u0422\u0410 \u0407\u0425 \u0412\u041b\u0410\u0421\u0422\u0418\u0412\u041e\u0421\u0422\u0406<\/p>\n

DISCRETE-TIME CONDITIONAL LINEAR RANDOM PROCESSES AND THEIR PROPERTIES<\/p>\n

\u0421\u0442\u043e\u0440\u0456\u043d\u043a\u0438: 7-12. \u041d\u043e\u043c\u0435\u0440: \u21163, 2022 (309)\u00a0\u00a0<\/strong> \"\"<\/a>
\n\u0410\u0432\u0442\u043e\u0440\u0438:<\/strong>
\n\u0424\u0420\u0418\u0417 \u041c. \u0404.
\n\u0422\u0435\u0440\u043d\u043e\u043f\u0456\u043b\u044c\u0441\u044c\u043a\u0438\u0439 \u043d\u0430\u0446\u0456\u043e\u043d\u0430\u043b\u044c\u043d\u0438\u0439 \u0442\u0435\u0445\u043d\u0456\u0447\u043d\u0438\u0439 \u0443\u043d\u0456\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u0456\u043c\u0435\u043d\u0456 \u0406\u0432\u0430\u043d\u0430 \u041f\u0443\u043b\u044e\u044f
\nhttps:\/\/orcid.org\/0000-0002-8720-6479<\/a>
\ne-mail: mykh.fryz@gmail.com<\/a>
\n\u041c\u041b\u0418\u041d\u041a\u041e \u0411. \u0411.
\n\u0422\u0435\u0440\u043d\u043e\u043f\u0456\u043b\u044c\u0441\u044c\u043a\u0438\u0439 \u043d\u0430\u0446\u0456\u043e\u043d\u0430\u043b\u044c\u043d\u0438\u0439 \u0442\u0435\u0445\u043d\u0456\u0447\u043d\u0438\u0439 \u0443\u043d\u0456\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u0456\u043c\u0435\u043d\u0456 \u0406\u0432\u0430\u043d\u0430 \u041f\u0443\u043b\u044e\u044f
\nhttps:\/\/orcid.org\/0000-0003-0780-5365<\/a>
\ne-mail: b.mlynko@gmail.com<\/a>
\nMykhailo FRYZ, Bogdana MLYNKO
\nTernopil Ivan Puluj National Technical University
\nDOI:<\/strong>\u00a0https:\/\/www.doi.org\/10.31891\/2307-5732-2022-309-3-7-12<\/a>
\n\u0410\u043d\u043e\u0442\u0430\u0446\u0456\u044f \u043c\u043e\u0432\u043e\u044e \u043e\u0440\u0438\u0433\u0456\u043d\u0430\u043b\u0443<\/strong>
\n\u0423\u043c\u043e\u0432\u043d\u0456 \u043b\u0456\u043d\u0456\u0439\u043d\u0456 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u0456 \u043f\u0440\u043e\u0446\u0435\u0441\u0438 \u0437 \u043d\u0435\u043f\u0435\u0440\u0435\u0440\u0432\u043d\u0438\u043c \u0447\u0430\u0441\u043e\u043c \u0437\u043e\u0431\u0440\u0430\u0436\u0430\u044e\u0442\u044c\u0441\u044f \u0443 \u0432\u0438\u0433\u043b\u044f\u0434\u0456 \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u0447\u043d\u043e\u0433\u043e \u0456\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u0430 \u0432\u0456\u0434 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u043e\u0433\u043e \u044f\u0434\u0440\u0430 \u0437\u0430 \u043f\u0440\u043e\u0446\u0435\u0441\u043e\u043c \u0456\u0437 \u043d\u0435\u0437\u0430\u043b\u0435\u0436\u043d\u0438\u043c\u0438 \u043f\u0440\u0438\u0440\u043e\u0441\u0442\u0430\u043c\u0438. \u0422\u0430\u043a\u0456 \u043f\u0440\u043e\u0446\u0435\u0441\u0438 \u0432\u0438\u043a\u043e\u0440\u0438\u0441\u0442\u043e\u0432\u0443\u044e\u0442\u044c\u0441\u044f \u0432 \u0437\u0430\u0434\u0430\u0447\u0430\u0445 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u043e\u0433\u043e \u0442\u0430 \u043a\u043e\u043c\u043f\u2019\u044e\u0442\u0435\u0440\u043d\u043e\u0433\u043e \u043c\u043e\u0434\u0435\u043b\u044e\u0432\u0430\u043d\u043d\u044f, \u043e\u043f\u0440\u0430\u0446\u044e\u0432\u0430\u043d\u043d\u044f \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u0447\u043d\u0438\u0445 \u0441\u0438\u0433\u043d\u0430\u043b\u0456\u0432, \u0444\u0456\u0437\u0438\u0447\u043d\u0430 \u043f\u0440\u0438\u0440\u043e\u0434\u0430 \u043f\u043e\u0440\u043e\u0434\u0436\u0435\u043d\u043d\u044f \u044f\u043a\u0438\u0445 \u0434\u043e\u043f\u0443\u0441\u043a\u0430\u0454 \u0457\u0445 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u043d\u044f \u0443 \u0432\u0438\u0433\u043b\u044f\u0434\u0456 \u0441\u0443\u043c\u0438 \u0432\u0435\u043b\u0438\u043a\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u0438\u0445 \u0456\u043c\u043f\u0443\u043b\u044c\u0441\u0456\u0432, \u044f\u043a\u0456 \u0432\u0438\u043d\u0438\u043a\u0430\u044e\u0442\u044c \u0443 \u043f\u0443\u0430\u0441\u0441\u043e\u043d\u0456\u0432\u0441\u044c\u043a\u0456 \u043c\u043e\u043c\u0435\u043d\u0442\u0438 \u0447\u0430\u0441\u0443. \u0406\u043c\u043f\u0443\u043b\u044c\u0441\u0438 \u043f\u0440\u0438 \u0446\u044c\u043e\u043c\u0443 \u0454 \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u0447\u043d\u043e \u0437\u0430\u043b\u0435\u0436\u043d\u0438\u043c\u0438 \u0444\u0443\u043d\u043a\u0446\u0456\u044f\u043c\u0438, \u043d\u0430 \u0432\u0456\u0434\u043c\u0456\u043d\u0443 \u0432\u0456\u0434 \u0456\u043d\u0448\u043e\u0457 \u043f\u043e\u0448\u0438\u0440\u0435\u043d\u043e\u0457 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u043e\u0457 \u043c\u043e\u0434\u0435\u043b\u0456 \u2013 \u043b\u0456\u043d\u0456\u0439\u043d\u043e\u0433\u043e \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u043e\u0433\u043e \u043f\u0440\u043e\u0446\u0435\u0441\u0443, \u044f\u043a\u0438\u0439 \u043c\u0430\u0454 \u043f\u043e\u0434\u0456\u0431\u043d\u0443 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u0443, \u0430\u043b\u0435 \u044f\u0432\u043b\u044f\u0454 \u0441\u043e\u0431\u043e\u044e \u0441\u0443\u043c\u0443 \u0432\u0435\u043b\u0438\u043a\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 \u043d\u0435\u0437\u0430\u043b\u0435\u0436\u043d\u0438\u0445 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u0438\u0445 \u0456\u043c\u043f\u0443\u043b\u044c\u0441\u0456\u0432, \u0449\u043e \u0432\u0438\u043d\u0438\u043a\u0430\u044e\u0442\u044c \u0443 \u043f\u0443\u0430\u0441\u0441\u043e\u043d\u0456\u0432\u0441\u044c\u043a\u0456 \u043c\u043e\u043c\u0435\u043d\u0442\u0438 \u0447\u0430\u0441\u0443. \u041f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u0438\u043c\u0438 \u043e\u0431\u043b\u0430\u0441\u0442\u044f\u043c\u0438, \u0434\u0435 \u0454 \u043f\u043e\u0448\u0438\u0440\u0435\u043d\u0438\u043c\u0438 \u0437\u0433\u0430\u0434\u0430\u043d\u0456 \u043c\u043e\u0434\u0435\u043b\u0456 \u0454 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u0435, \u043a\u043e\u043c\u043f\u2019\u044e\u0442\u0435\u0440\u043d\u0435 \u043c\u043e\u0434\u0435\u043b\u044e\u0432\u0430\u043d\u043d\u044f \u0442\u0430 \u043e\u043f\u0440\u0430\u0446\u044e\u0432\u0430\u043d\u043d\u044f \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u0435\u043d\u0446\u0435\u0444\u0430\u043b\u043e\u0433\u0440\u0430\u0444\u0456\u0447\u043d\u0438\u0445 \u0441\u0438\u0433\u043d\u0430\u043b\u0456\u0432, \u043a\u0430\u0440\u0434\u0456\u043e\u0441\u0438\u0433\u043d\u0430\u043b\u0456\u0432, \u043f\u0440\u043e\u0446\u0435\u0441\u0456\u0432 \u0440\u0435\u0441\u0443\u0440\u0441\u043e\u0441\u043f\u043e\u0436\u0438\u0432\u0430\u043d\u043d\u044f, \u0440\u0430\u0434\u0456\u043e\u043b\u043e\u043a\u0430\u0446\u0456\u0439\u043d\u0438\u0445 \u0441\u0438\u0433\u043d\u0430\u043b\u0456\u0432 \u0442\u0430 \u0456\u043d.
\n\u0423 \u0434\u0430\u043d\u0456\u0439 \u0440\u043e\u0431\u043e\u0442\u0456 \u043d\u0430\u0432\u0435\u0434\u0435\u043d\u043e \u043e\u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f \u0443\u043c\u043e\u0432\u043d\u043e\u0433\u043e \u043b\u0456\u043d\u0456\u0439\u043d\u043e\u0433\u043e \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u043e\u0433\u043e \u043f\u0440\u043e\u0446\u0435\u0441\u0443 (\u0423\u041b\u0412\u041f) \u0437 \u0434\u0438\u0441\u043a\u0440\u0435\u0442\u043d\u0438\u043c \u0447\u0430\u0441\u043e\u043c, \u043f\u043e\u043a\u0430\u0437\u0430\u043d\u043e \u0439\u043e\u0433\u043e \u0437\u0432\u2019\u044f\u0437\u043e\u043a \u0456\u0437 \u0432\u0456\u0434\u043f\u043e\u0432\u0456\u0434\u043d\u043e\u044e \u043c\u043e\u0434\u0435\u043b\u043b\u044e \u0437 \u043d\u0435\u043f\u0435\u0440\u0435\u0440\u0432\u043d\u0438\u043c \u0447\u0430\u0441\u043e\u043c. \u0412\u0456\u0434\u043f\u043e\u0432\u0456\u0434\u043d\u043e \u0434\u043e \u043d\u0430\u0432\u0435\u0434\u0435\u043d\u043e\u0433\u043e \u043e\u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f \u0423\u041b\u0412\u041f \u0456\u0437 \u0434\u0438\u0441\u043a\u0440\u0435\u0442\u043d\u0438\u043c \u0447\u0430\u0441\u043e\u043c \u043c\u043e\u0436\u043d\u0430 \u0442\u0440\u0430\u043a\u0442\u0443\u0432\u0430\u0442\u0438 \u044f\u043a \u0432\u0456\u0434\u0433\u0443\u043a \u043b\u0456\u043d\u0456\u0439\u043d\u043e\u0433\u043e \u0446\u0438\u0444\u0440\u043e\u0432\u043e\u0433\u043e \u0444\u0456\u043b\u044c\u0442\u0440\u0430 \u0437 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u0438\u043c\u0438 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0430\u043c\u0438 \u043d\u0430 \u0432\u043f\u043b\u0438\u0432 \u0431\u0456\u043b\u043e\u0433\u043e \u0448\u0443\u043c\u0443 \u0437 \u0431\u0435\u0437\u043c\u0435\u0436\u043d\u043e \u043f\u043e\u0434\u0456\u043b\u044c\u043d\u0438\u043c \u0440\u043e\u0437\u043f\u043e\u0434\u0456\u043b\u043e\u043c. \u041f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u043e \u0430\u043d\u0430\u043b\u0456\u0437 \u043c\u043e\u043c\u0435\u043d\u0442\u043d\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0456\u0439 \u043f\u0435\u0440\u0448\u043e\u0433\u043e \u0456 \u0434\u0440\u0443\u0433\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0443 \u0423\u041b\u0412\u041f \u0456\u0437 \u0434\u0438\u0441\u043a\u0440\u0435\u0442\u043d\u0438\u043c \u0447\u0430\u0441\u043e\u043c, \u0437\u043e\u043a\u0440\u0435\u043c\u0430, \u043e\u0442\u0440\u0438\u043c\u0430\u043d\u043e \u0432\u0438\u0440\u0430\u0437\u0438 \u0434\u043b\u044f \u0439\u043e\u0433\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u043e\u0433\u043e \u0441\u043f\u043e\u0434\u0456\u0432\u0430\u043d\u043d\u044f, \u0434\u0438\u0441\u043f\u0435\u0440\u0441\u0456\u0457 \u0442\u0430 \u043a\u043e\u0440\u0435\u043b\u044f\u0446\u0456\u0439\u043d\u043e\u0457 \u0444\u0443\u043d\u043a\u0446\u0456\u0457.
\n\u0420\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0438 \u043c\u043e\u0436\u0443\u0442\u044c \u0431\u0443\u0442\u0438 \u0432\u0438\u043a\u043e\u0440\u0438\u0441\u0442\u0430\u043d\u0456 \u0434\u043b\u044f \u0434\u043e\u0441\u043b\u0456\u0434\u0436\u0435\u043d\u043d\u044f \u0439\u043c\u043e\u0432\u0456\u0440\u043d\u0456\u0441\u043d\u0438\u0445 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043a \u0434\u043e\u0441\u043b\u0456\u0434\u0436\u0443\u0432\u0430\u043d\u0438\u0445 \u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0456\u0439\u043d\u0438\u0445 \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u0447\u043d\u0438\u0445 \u0441\u0438\u0433\u043d\u0430\u043b\u0456\u0432, \u044f\u043a\u0456 \u0434\u043b\u044f \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u0438\u0445 \u043f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u0438\u0445 \u0437\u0430\u0434\u0430\u0447 \u0431\u0443\u0434\u0443\u0442\u044c \u0437\u0430\u043b\u0435\u0436\u0430\u0442\u0438 \u0432\u0456\u0434 \u0432\u043b\u0430\u0441\u0442\u0438\u0432\u043e\u0441\u0442\u0435\u0439 \u0432\u0456\u0434\u043f\u043e\u0432\u0456\u0434\u043d\u043e\u0433\u043e \u044f\u0434\u0440\u0430 \u0442\u0430 \u043f\u043e\u0440\u043e\u0434\u0436\u0443\u044e\u0447\u043e\u0433\u043e \u0431\u0456\u043b\u043e\u0433\u043e \u0448\u0443\u043c\u0443 \u0432 \u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u043d\u0456 \u0423\u041b\u0412\u041f. \u0417\u043e\u043a\u0440\u0435\u043c\u0430, \u0432 \u0440\u043e\u0431\u043e\u0442\u0456 \u043d\u0430\u0432\u0435\u0434\u0435\u043d\u043e \u0443\u043c\u043e\u0432\u0438, \u044f\u043a\u0438\u043c \u043f\u043e\u0432\u0438\u043d\u043d\u0456 \u0437\u0430\u0434\u043e\u0432\u043e\u043b\u044c\u043d\u044f\u0442\u0438 \u0441\u043a\u043b\u0430\u0434\u043e\u0432\u0456 \u0423\u041b\u0412\u041f \u0456\u0437 \u0434\u0438\u0441\u043a\u0440\u0435\u0442\u043d\u0438\u043c \u0447\u0430\u0441\u043e\u043c \u0434\u043b\u044f \u0442\u043e\u0433\u043e, \u0449\u043e\u0431 \u0446\u0435\u0439 \u043f\u0440\u043e\u0446\u0435\u0441 \u0431\u0443\u0432 \u0441\u0442\u0430\u0446\u0456\u043e\u043d\u0430\u0440\u043d\u0438\u043c \u0443 \u0448\u0438\u0440\u043e\u043a\u043e\u043c\u0443 \u0440\u043e\u0437\u0443\u043c\u0456\u043d\u043d\u0456.
\n\u041a\u043b\u044e\u0447\u043e\u0432\u0456 \u0441\u043b\u043e\u0432\u0430:<\/strong> \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u0430 \u043c\u043e\u0434\u0435\u043b\u044c, \u0443\u043c\u043e\u0432\u043d\u0438\u0439 \u043b\u0456\u043d\u0456\u0439\u043d\u0438\u0439 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u0438\u0439 \u043f\u0440\u043e\u0446\u0435\u0441, \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u0435 \u0441\u043f\u043e\u0434\u0456\u0432\u0430\u043d\u043d\u044f, \u043a\u043e\u0440\u0435\u043b\u044f\u0446\u0456\u0439\u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0456\u044f, \u044f\u0434\u0440\u043e, \u0431\u0456\u043b\u0438\u0439 \u0448\u0443\u043c, \u0441\u0442\u0430\u0446\u0456\u043e\u043d\u0430\u0440\u043d\u0438\u0439 \u043f\u0440\u043e\u0446\u0435\u0441.<\/p>\n

\u0420\u043e\u0437\u0448\u0438\u0440\u0435\u043d\u0430 \u0430\u043d\u043e\u0442\u0430\u0446\u0456\u044f \u0430\u043d\u0433\u043b\u0456\u0439\u0441\u044c\u043a\u043e\u044e \u00a0\u043c\u043e\u0432\u043e\u044e<\/strong><\/p>\n

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.
\nA 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.
\nThe 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.
\nKeywords:<\/strong> mathematical model, conditional linear random process, mathematical expectation, covariance function, kernel, white noise, stationary process.<\/p>\n

\u041b\u0456\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u0430<\/strong><\/p>\n

    \n
  1. Babak V. P. Methods and Models for Information Data Analysis \/ V. P. Babak, S. V. Babak, A. O. Zaporozhets, M. V. Myslovych, V. M. Zvaritch \/\/ Diagnostic Systems For Energy Equipments, volume 281 of Studies in Systems, Decision and Control. \u2013 Springer, Cham, 2020. \u2013 P. 23\u201370. \u2013 DOI: doi.org\/10.1007\/978-3-030-44443-3_2<\/li>\n
  2. \u0424\u0440\u0438\u0437 \u041c. \u0415. \u042d\u0440\u0433\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043b\u0438\u043d\u0435\u0439\u043d\u044b\u0445 \u043f\u0440\u043e\u0446\u0435\u0441\u0441\u043e\u0432 \u0432 \u0437\u0430\u0434\u0430\u0447\u0430\u0445 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0438 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u043d\u0430\u043b\u0438\u0437\u0430 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u044b\u0445 \u0441\u0438\u0433\u043d\u0430\u043b\u043e\u0432 \/ \u041c. \u0415. \u0424\u0440\u0438\u0437, \u041b. \u041d. \u0429\u0435\u0440\u0431\u0430\u043a \/\/ \u042d\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u043e\u0435 \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435. \u2014 \u041a. : \u0418\u043d.-\u0442 \u043f\u0440\u043e\u0431\u043b\u0435\u043c \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0432 \u044d\u043d\u0435\u0440\u0433\u0435\u0442\u0438\u043a\u0435 \u0438\u043c. \u0413. \u0415. \u041f\u0443\u0445\u043e\u0432\u0430 \u041d\u0410\u041d \u0423\u043a\u0440\u0430\u0438\u043d\u044b, 2010. \u2014 \u0422. 32. \u2014 \u2116 1. \u2014 C. 3\u201314.<\/li>\n
  3. \u0424\u0440\u0438\u0437 \u041c. \u0404. \u0412\u043b\u0430\u0441\u0442\u0438\u0432\u0456\u0441\u0442\u044c \u043f\u0435\u0440\u0435\u043c\u0456\u0448\u0443\u0432\u0430\u043d\u043d\u044f \u0442\u0430 \u0435\u0440\u0433\u043e\u0434\u0438\u0447\u043d\u0456\u0441\u0442\u044c \u043b\u0456\u043d\u0456\u0439\u043d\u0438\u0445 \u043f\u0440\u043e\u0446\u0435\u0441\u0456\u0432 \u0443 \u0437\u0430\u0434\u0430\u0447\u0430\u0445 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u043e\u0433\u043e \u043c\u043e\u0434\u0435\u043b\u044e\u0432\u0430\u043d\u043d\u044f \u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u043d\u043e\u0433\u043e \u0430\u043d\u0430\u043b\u0456\u0437\u0443 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u0438\u0445 \u0441\u0438\u0433\u043d\u0430\u043b\u0456\u0432 \/ \u041c. \u0404. \u0424\u0440\u0438\u0437, \u041b. \u041c. \u0429\u0435\u0440\u0431\u0430\u043a \/\/ \u041c\u043e\u0434\u0435\u043b\u044e\u0432\u0430\u043d\u043d\u044f \u0442\u0430 \u0456\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0456\u0439\u043d\u0456 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0456\u0457 : \u0437\u0431\u0456\u0440\u043d\u0438\u043a \u043d\u0430\u0443\u043a\u043e\u0432\u0438\u0445 \u043f\u0440\u0430\u0446\u044c \u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442\u0443 \u043f\u0440\u043e\u0431\u043b\u0435\u043c \u043c\u043e\u0434\u0435\u043b\u044e\u0432\u0430\u043d\u043d\u044f \u0432 \u0435\u043d\u0435\u0440\u0433\u0435\u0442\u0438\u0446\u0456 \u0456\u043c. \u0413. \u0404. \u041f\u0443\u0445\u043e\u0432\u0430 \u041d\u0410\u041d \u0423\u043a\u0440\u0430\u0457\u043d\u0438. \u2014 \u041a., 2009. \u2014 \u0412\u0438\u043f. 51. \u2014 \u0421. 53\u201357.<\/li>\n
  4. Pierre P. A. Central Limit Theorems for Conditionally Linear Random Processes \/ P. A. Pierre \/\/ SIAM J. of Applied Math. \u2014 1971. \u2014 Vol. 20, \u2116 3. \u2014 P. 449\u2013461. \u2013 DOI: http:\/\/doi.org\/10.1137\/0120048<\/li>\n
  5. \u0424\u0440\u0438\u0437 \u041c. \u0404. \u0412\u043b\u0430\u0441\u0442\u0438\u0432\u043e\u0441\u0442\u0456 \u0443\u043c\u043e\u0432\u043d\u0438\u0445 \u043b\u0456\u043d\u0456\u0439\u043d\u0438\u0445 \u043f\u0440\u043e\u0446\u0435\u0441\u0456\u0432 \u0442\u0430 \u0457\u0445 \u0437\u0430\u0441\u0442\u043e\u0441\u0443\u0432\u0430\u043d\u043d\u044f \u0432 \u043f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u0438\u0445 \u0437\u0430\u0434\u0430\u0447\u0430\u0445 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u043e\u0433\u043e \u043c\u043e\u0434\u0435\u043b\u044e\u0432\u0430\u043d\u043d\u044f \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u0447\u043d\u0438\u0445 \u0441\u0438\u0433\u043d\u0430\u043b\u0456\u0432 \/ \u041c. \u0404. \u0424\u0440\u0438\u0437 \/\/ \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u0435 \u0442\u0430 \u043a\u043e\u043c\u043f\u2019\u044e\u0442\u0435\u0440\u043d\u0435 \u043c\u043e\u0434\u0435\u043b\u044e\u0432\u0430\u043d\u043d\u044f. \u0421\u0435\u0440\u0456\u044f: \u0422\u0435\u0445\u043d\u0456\u0447\u043d\u0456 \u043d\u0430\u0443\u043a\u0438: \u0437\u0431\u0456\u0440\u043d\u0438\u043a \u043d\u0430\u0443\u043a\u043e\u0432\u0438\u0445 \u043f\u0440\u0430\u0446\u044c. \u2013 2012. \u2013 \u0412\u0438\u043f. 6. \u2013 \u0421. 228\u2013238.<\/li>\n
  6. Fryz M. Characteristic Function of Conditional Linear Random Process \/ M. Fryz, L. Scherbak, M. Karpinski, B. Mlynko \/\/ Proceedings of the 1st International Workshop on Information Technologies: Theoretical and Applied Problems. \u2013 Ternopil, Ukraine, 2021. \u2013 C. 129\u2013135.<\/li>\n
  7. \u0424\u0440\u0438\u0437 \u041c. \u0404. \u0421\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u043d\u0438\u0439 \u0430\u043d\u0430\u043b\u0456\u0437 \u043f\u0435\u0440\u0456\u043e\u0434\u0438\u0447\u043d\u043e\u0457 \u0430\u0432\u0442\u043e\u0440\u0435\u0433\u0440\u0435\u0441\u0456\u0457 \u0437 \u0432\u0438\u043f\u0430\u0434\u043a\u043e\u0432\u0438\u043c\u0438 \u043a\u043e\u0435\u0444\u0456\u0446\u0456\u0454\u043d\u0442\u0430\u043c\u0438 \u0432 \u0437\u0430\u0434\u0430\u0447\u0430\u0445 \u043e\u043f\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0433\u043d\u043e\u0437\u0443\u0432\u0430\u043d\u043d\u044f \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u0441\u043f\u043e\u0436\u0438\u0432\u0430\u043d\u043d\u044f \u043f\u0456\u0434\u043f\u0440\u0438\u0454\u043c\u0441\u0442\u0432 \/ \u041c. \u0404. \u0424\u0440\u0438\u0437, \u041b. \u041c. \u0429\u0435\u0440\u0431\u0430\u043a \/\/ \u0422\u0435\u0445\u043d\u0456\u0447\u043d\u0430 \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u0434\u0438\u043d\u0430\u043c\u0456\u043a\u0430. \u2013 \u041a. : \u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u0434\u0438\u043d\u0430\u043c\u0456\u043a\u0438 \u041d\u0430\u0446\u0456\u043e\u043d\u0430\u043b\u044c\u043d\u043e\u0457 \u0430\u043a\u0430\u0434\u0435\u043c\u0456\u0457 \u043d\u0430\u0443\u043a \u0423\u043a\u0440\u0430\u0457\u043d\u0438, 2019. \u2013 \u0412\u0438\u043f. 2. \u2013 \u0421.\u00a038\u201347. \u2013 DOI: https:\/\/doi.org\/10.15407\/techned2019.02.038<\/li>\n
  8. Fryz M. Conditional linear random process and random coefficient autoregressive model for EEG analysis \/ M. Fyz \/\/ Proceedings of the 2017 IEEE First Ukraine Conference on Electrical and Computer Engineering. \u2013 Kyiv Ukraine, 2017. \u2013 P. 305\u2013309. \u2013 DOI:10.1109\/UKRCON.2017.8100498<\/li>\n
  9. Iwankiewicz R. Dynamic response of mechanical systems to impulse process stochastic excitations: Markov approach \/ R. Iwankiewicz \/\/ Journal of Physics: Conference Series, 2016.<\/li>\n
  10. Fryz M., Mlynko B. Properties of Stationarity and Cyclostationarity of Conditional Linear Random Processes \/ M. Fryz, B. Mlynko \/\/ Proceedings of the 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET). \u2013 Lviv, Slavske, Ukraine, 2020. \u2013 P. 166\u2013170. \u2013 DOI: 10.1109\/TCSET49122.2020.235415<\/li>\n
  11. Chen P. Strong laws for randomly weighted sums of random variables and applications in the bootstrap and random design regression \/ P. Chen, T. Zhang, S. H. Sung \/\/ Statistica Sinica. \u2013 2019. \u2013 Vol. 29, No. 4. \u2013 P.\u00a01739\u20131749. \u2013 DOI: 10.5705\/ss.202017.0106<\/li>\n
  12. Vasudeva R. Limit Theorems for randomly weighted sums of random variables \/ R. Vasudeva \/\/ ProbStat Forum. \u2013 2018. \u2013 Volume 11. \u2013 P. 8\u201318.<\/li>\n
  13. Kevei P. The asymptotic distribution of randomly weighted sums and self-normalized sums \/ P. Kevei, D. Mason \/\/ Electron. J. Probab. \u2013 2012. \u2013 Vol. 17. \u2013 P. 1\u201321. \u2013 DOI: 10.1214\/EJP.v17-2092<\/li>\n
  14. Krizmani\u0107 D. Maxima of linear processes with heavy-tailed innovations and random coefficients \/ D. Krizmani\u0107 \/\/ J. Time Ser. Anal. \u2013 \u2013 Volume 43, Issue2. \u2013 P. 238\u2013262.<\/li>\n
  15. Marek T. On invertibility of a random coefficient moving average model \/ T. Marek \/\/ Kybernetika. \u2013 2005. \u2013 Vol. 41, No. 6. \u2013 P. 743\u2013756.<\/li>\n<\/ol>\n

    References<\/strong><\/p>\n

      \n
    1. Babak V. P. Methods and Models for Information Data Analysis \/ V. P. Babak, S. V. Babak, A. O. Zaporozhets, M. V. Myslovych, V. M. Zvaritch \/\/ Diagnostic Systems For Energy Equipments, volume 281 of Studies in Systems, Decision and Control. \u2013 Springer, Cham, 2020. \u2013 P. 23\u201370. \u2013 DOI: doi.org\/10.1007\/978-3-030-44443-3_2<\/li>\n
    2. Fryz M. Ergodic properties of linear processes in problems of random signal mathematical modelling and statistical analysis \/ M. Fryz, L. Scherbak \/\/ Electronic Modeling. – 2010. – V. 32. \u2013 No.1. \u2013 PP. 3 \u2013 14.<\/li>\n
    3. Fryz M. Mixing property and ergodicity of linear processes in the problems of random signal mathematical modelling and statistical analysis \/ M. Fryz, L. Scherbak \/\/ Modeliuvannia ta informatsiini tekhnolohii : zbirnyk naukovykh prats Instytutu problem modeliuvannia v enerhetytsi im. H. Ye. Pukhova NAN Ukrainy. \u2014 \u041a., 2009. \u2014 Vol. 51. \u2014 PP. 53\u201357.<\/li>\n
    4. Pierre P. A. Central Limit Theorems for Conditionally Linear Random Processes \/ P. A. Pierre \/\/ SIAM J. of Applied Math. \u2014 1971. \u2014 Vol. 20, \u2116 3. \u2014 P. 449\u2013461. \u2013 DOI: http:\/\/doi.org\/10.1137\/0120048<\/li>\n
    5. Fryz M. Properties of conditional linear random processes and their applications in the applied problems of stochastic signal mathematical modelling \/ M. Fryz \/\/ Matematychne ta kompiuterne modeliuvannia. Seriia: Tekhnichni nauky: zbirnyk naukovykh prats, 2012. – Vol. 6. – P.228\u2013238. doi: https:\/\/doi.org\/10.32626\/2308-5916.2012-6.228-238<\/li>\n
    6. Fryz M. Characteristic Function of Conditional Linear Random Process \/ M. Fryz, L. Scherbak, M. Karpinski, B. Mlynko \/\/ Proceedings of the 1st International Workshop on Information Technologies: Theoretical and Applied Problems. \u2013 Ternopil, Ukraine, 2021. \u2013 C. 129\u2013135.<\/li>\n
    7. Fryz M. Statistical analysis of random coefficient periodic autoregression and its application for short-term electricity consumption forecasting \/ M. Fryz, L. Scherbak \/\/ Tekhnichna elektrodynamika. \u2013 \u041a. : Institute of Electrodynamics National Academy of Science of Ukraine, 2019. \u2013 Vol. 2. \u2013 PP. 38 \u2013 47. doi: https:\/\/doi.org\/10.15407\/techned2019.02.038<\/li>\n
    8. Fryz M. Conditional linear random process and random coefficient autoregressive model for EEG analysis \/ M. Fyz \/\/ Proceedings of the 2017 IEEE First Ukraine Conference on Electrical and Computer Engineering. \u2013 Kyiv Ukraine, 2017. \u2013 P. 305\u2013309. \u2013 DOI:10.1109\/UKRCON.2017.8100498<\/li>\n
    9. Iwankiewicz R. Dynamic response of mechanical systems to impulse process stochastic excitations: Markov approach \/ R. Iwankiewicz \/\/ Journal of Physics: Conference Series, 2016.<\/li>\n
    10. Fryz M., Mlynko B. Properties of Stationarity and Cyclostationarity of Conditional Linear Random Processes \/ M. Fryz, B. Mlynko \/\/ Proceedings of the 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET). \u2013 Lviv, Slavske, Ukraine, 2020. \u2013 P. 166\u2013170. \u2013 DOI: 10.1109\/TCSET49122.2020.235415<\/li>\n
    11. Chen P. Strong laws for randomly weighted sums of random variables and applications in the bootstrap and random design regression \/ P. Chen, T. Zhang, S. H. Sung \/\/ Statistica Sinica. \u2013 2019. \u2013 Vol. 29, No. 4. \u2013 P.\u00a01739\u20131749. \u2013 DOI: 10.5705\/ss.202017.0106<\/li>\n
    12. Vasudeva R. Limit Theorems for randomly weighted sums of random variables \/ R. Vasudeva \/\/ ProbStat Forum. \u2013 2018. \u2013 Volume 11. \u2013 P. 8\u201318.<\/li>\n
    13. Kevei P. The asymptotic distribution of randomly weighted sums and self-normalized sums \/ P. Kevei, D. Mason \/\/ Electron. J. Probab. \u2013 2012. \u2013 Vol. 17. \u2013 P. 1\u201321. \u2013 DOI: 10.1214\/EJP.v17-2092<\/li>\n
    14. Krizmani\u0107 D. Maxima of linear processes with heavy-tailed innovations and random coefficients \/ D. Krizmani\u0107 \/\/ J. Time Ser. Anal. \u2013 \u2013 Volume 43, Issue2. \u2013 P. 238\u2013262.<\/li>\n
    15. Marek T. On invertibility of a random coefficient moving average model \/ T. Marek \/\/ Kybernetika. \u2013 2005. \u2013 Vol. 41, No. 6. \u2013 P. 743\u2013756.<\/li>\n<\/ol>\n

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