Тема:
” Использование цепей Маркова в моделировании социально-экономических процессов ”
Содержание:
1. Основные понятия теории марковских цепей.
2. Теорема о предельных вероятностях.
3. Области применения цепей Маркова.
4. Управляемые цепи Маркова. Выбор стратегии.
Список использованной литературы.
§1. Основные понятия теории марковских цепей.
Пусть {
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAXCAYAAADpwXTaAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADdSURBVDjLY2hsbGSgFmYYNYw6huW4yRYzMDD8x4WNoxt8SHJZQ7SxDwOD7BuPvHpDmFhHRxqPh6zsSmQxogwDu07GY09aRwcPsniem1tYbH29JEmGRRkzLJR1yymGu8gjOpmsMKuvj5U0ZmC4CwsbkJdhBpNsGCS8iAt0goaBvMhgHLUQ5kV3OeNZuMIJr2HoXgThaA+PZPSIIMowbEkCEZPGKSBxcITIMOxhYDC+i+xi7F7EkiTQkwo27yO8l+dhKMvA8AZfykeOUbyGkYpHiGHgiAKFJVKkjJCSFgClh319MtmaXgAAAABJRU5ErkJggg==)
,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAAXCAYAAADk3wSdAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADzSURBVEjHY2hsbGSgNmYYNZQ+hua4yRYzMDD8x4WNoxt8yHJpQ7SxDwOD7BuPvHpDmFhHRxqPh6zsSmQxkgwFu1bGY09aRwcPsniem1tYbH29JFmGRhkzLJR1yymGu9AjOpmiMK2vj5U0ZmC4Cws7UFDALCDbUEh4khY5BA0FeZ3BOGohzOvucsazCIUjXkPRvQ7C0R4eyegRRpKh2JISIuaNU0DiMItBwYItrLF7HUtSQk5isGQFMtyIwWg3etAgvJ3nYSjLwPAGX07C5qooY9ledF9RnM+xhTdFBoLCH1tSo8hAWLKjiqHopRi6a0d4yQ8AeEHBn0lmgJIAAAAASUVORK5CYII=)
, ...,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAXCAYAAAALHW+jAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAD1SURBVEjHY2hsbGSgJmYYNZA2Bua4yRYzMDD8x4WNoxt8SHZhQ7SxDwOD7BuPvHpDmFhHRxqPh6zsSmQxog0Eu1LGY09aRwcPsniem1tYbH29JMkGRhkzLJR1yymGu8wjOpnsMKyvj5U0ZmC4CwsrkPdhhpNlICT8iI8IggaCvMtgHLUQ5l13OeNZ+MINr4Ho3gXhaA+PZPTIIdpAbMkFEcPGKfiSDFYDwd7FklxwJSOcBtbneRjKMjC8wZdDYDEN84Wbm3EtLKwpzsvRHqaFph7RhVQpHEARZyJrsiIyr16VKgaCvIwrXZLnXWPjelzpcgSW2AAEgKF2WIPNBgAAAABJRU5ErkJggg==)
} - множество возможных состояний некоторой физической системы. В любой момент времени система может находиться только в одном состоянии. С течением времени система переходит последовательно из одного состояния в другое. Каждый такой переход называется шагом процесса.
Для описания эволюции этой системы введем последовательность дискретных случайных величин
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAYAAAAcYhYyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjL7ZSxDYMwEEVvABZAwkOg6yktF9SRANEhKuSGOjro3DBEyuzAKtmAHZIYAgoKwQZaii9RoHd3/3+Auq7hqOCE2ENihBvjRbkbQlL4IqJAPxeclQCsE5L8zRDk/IoADxPgL6RKMGRM3FMi99A5DKADgKeWyZtlwHuLXClnVzr6DPBEuwUwgxClLmJCh3pCkQhMKQxx/3o0T2TFwN4rjJuhiKz5Hji9pFTuCA/aMZFJH4/0FphU4dJA470JIum+SI7ZOH0oo8ysINpszuVl9MN6kz7m6Rx8jK218sRG+steTef8PU56AWpcViM0mncIAAAAAElFTkSuQmCC)
,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAXCAYAAAAC9s/ZAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADTSURBVDjLY2hsbGSgBDOMGoDbgChjhoWybjnFZBlQn+dh6BFZbwdi57jJFjMwyL7xyKs3JMkAYze3WmMGhrv4NOM0oCHa2EdW1mNlbH29JNlekGVgeMPAwPAfhPGFBXbNQNvTOjp4SI4FkNMZZDz2EKsZxYD6+lhJY+PoerLTQX2khx2u0O7oSOPxkGHYw8BgfBc9YFFDHk9ggQxxlzOehdMAhC2QkIdjaJgQNAAXjjY2rgdpIssAUMC6ueWFEeUFlKiEewERaGR7ATl3IofJMCqRABbVG7lpbnFpAAAAAElFTkSuQmCC)
,...,
![](data:image/png;base64,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)
,... Индекс n играет роль времени. Если в момент времени n система находилась в состоянии
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAAZCAYAAADe1WXtAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEDSURBVEjH7VU9DoIwFH4H4ACSUO9A3u5YOzCbSMOEYSJdmE1hY/ECbozegat4A+6gKQYlTYEUiBPDW97w9ft7AEVRwNoDG+h/QFNKMgB4DQ3yPJjFNOcYAJCGCel3u7JMHEbIo7+zAm3ZeqxOytLp7wWlp0hKdxZoiFARmmZfhozHizyVMnIR4Nl5p6zoHpgN+vHTLpxJUCUdMKw66cc93qd8HAXVpavhjMV6YFagpir9kseLac8RZV+JWbqhSkMVG2UqBfMJQDN2SXoD2qp5UOshLr5zeWYH3ZLFoKYQl7FUlmF4W/XTp8IzHYU1kAqHIr2KXOyGjmIWw7Z2A13e/lHwBhPVAs+gHkxBAAAAAElFTkSuQmCC)
, то мы будем считать, что
![](data:image/png;base64,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)
= j. Таким образом, случайные величины являются номерами состояний системы.
Последовательность
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAYAAAAcYhYyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjL7ZSxDYMwEEVvABZAwkOg6yktF9SRANEhKuSGOjro3DBEyuzAKtmAHZIYAgoKwQZaii9RoHd3/3+Auq7hqOCE2ENihBvjRbkbQlL4IqJAPxeclQCsE5L8zRDk/IoADxPgL6RKMGRM3FMi99A5DKADgKeWyZtlwHuLXClnVzr6DPBEuwUwgxClLmJCh3pCkQhMKQxx/3o0T2TFwN4rjJuhiKz5Hji9pFTuCA/aMZFJH4/0FphU4dJA470JIum+SI7ZOH0oo8ysINpszuVl9MN6kz7m6Rx8jK218sRG+steTef8PU56AWpcViM0mncIAAAAAElFTkSuQmCC)
,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAXCAYAAAAC9s/ZAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADTSURBVDjLY2hsbGSgBDOMGoDbgChjhoWybjnFZBlQn+dh6BFZbwdi57jJFjMwyL7xyKs3JMkAYze3WmMGhrv4NOM0oCHa2EdW1mNlbH29JNlekGVgeMPAwPAfhPGFBXbNQNvTOjp4SI4FkNMZZDz2EKsZxYD6+lhJY+PoerLTQX2khx2u0O7oSOPxkGHYw8BgfBc9YFFDHk9ggQxxlzOehdMAhC2QkIdjaJgQNAAXjjY2rgdpIssAUMC6ueWFEeUFlKiEewERaGR7ATl3IofJMCqRABbVG7lpbnFpAAAAAElFTkSuQmCC)
,...,
![](data:image/png;base64,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)
,... образует цепь Маркова, если для любого n и любых
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAYAAAAcYhYyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjL7ZS9DYMwEIXfACyAhIeA2wG5YAIQVRAVcsMAho6GISizAxOwQQo2YIdENjJy/oGkpDjJluzv3r1nGXVd49fCAfkMkTJ1CRgBXFlYlLuVKFAANnAh/f0QwX3G+DlvGmc3pAhZCYq73Z40Te5wDz0lVaQVAZPy55tHz6MgGFIpXW0yJXKzEjOKDdsMiQmdkQ/QuBliRzsrYZfHmLXSF/4siyqhCB7vVbTGYHVYr3ly0mCK21kxa+0Gd6PYHUxXgE3qgtqr1ExD++zqtyBCykx3pYpCkW2G/EXJKk/WlHkCb9M5vselbhHxVO4LhJq9AAAAAElFTkSuQmCC)
,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAXCAYAAAAC9s/ZAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADSSURBVDjLY2hsbGSgBDOMGoBqQH19rKQxA8NdBgaG/7JuOcVkuQBkiBGD7CmPvHpD8gzI8zCUlfVYmdbRwUOWATlussUMxlELyQqDjo40Hg8Zhj3G0Q0+YJcwMLwBhQe+MMF0PoPRqdj6eklwgBpH15PkApjzkQ0iyYAoY4aFMCczMBjfJckA5OiDuED2JnJUwsIH3WC4AQ3Rxj4MMh57QNEHUwwKODDbIzoZZoi7nPEsrAaAnI8c0uDwAHtF9g3MJXgNIAYPAwPg0QwN7GFSIgEAiYMahL4VLwoAAAAASUVORK5CYII=)
, ...,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAYCAYAAAAYl8YPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAD0SURBVEjH7ZS9DcIwEIXfAAwAEjdEuB0sFyyQWKmQqFAoGMBJF4kpUmaHTJANKNiAHUBnMIpCg21KCkuuPr0/G03T4FcHf1gazNpyxcAVwJ3U4ZSsTIAb0Kgrm6XDKp0R6X7ftotk2EHRCVx0yZm17X6h1xjY1FunELhJft9m+GkRm7G0duXKYGOjlXmLU2g0rGB03hbA12jYdBJPZXQJncf7UhveYq0HmYQvQkJ3d212QTCxOG3M5efs0s0r9FnmNs/kpcwbDtpRXVVLpdRRM58dcNZ2EExy1US97FBiSVM2ARhmO287COYB79/lVdj/244/D39dnkucqa5CAAAAAElFTkSuQmCC)
,...
P(
![](data:image/png;base64,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)
=j /
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAYAAAAcYhYyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjL7ZSxDYMwEEVvABZAwkOg6yktF9SRANEhKuSGOjro3DBEyuzAKtmAHZIYAgoKwQZaii9RoHd3/3+Auq7hqOCE2ENihBvjRbkbQlL4IqJAPxeclQCsE5L8zRDk/IoADxPgL6RKMGRM3FMi99A5DKADgKeWyZtlwHuLXClnVzr6DPBEuwUwgxClLmJCh3pCkQhMKQxx/3o0T2TFwN4rjJuhiKz5Hji9pFTuCA/aMZFJH4/0FphU4dJA470JIum+SI7ZOH0oo8ysINpszuVl9MN6kz7m6Rx8jK218sRG+steTef8PU56AWpcViM0mncIAAAAAElFTkSuQmCC)
=
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAYAAAAcYhYyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjL7ZS9DYMwEIXfACyAhIeA2wG5YAIQVRAVcsMAho6GISizAxOwQQo2YIdENjJy/oGkpDjJluzv3r1nGXVd49fCAfkMkTJ1CRgBXFlYlLuVKFAANnAh/f0QwX3G+DlvGmc3pAhZCYq73Z40Te5wDz0lVaQVAZPy55tHz6MgGFIpXW0yJXKzEjOKDdsMiQmdkQ/QuBliRzsrYZfHmLXSF/4siyqhCB7vVbTGYHVYr3ly0mCK21kxa+0Gd6PYHUxXgE3qgtqr1ExD++zqtyBCykx3pYpCkW2G/EXJKk/WlHkCb9M5vselbhHxVO4LhJq9AAAAAElFTkSuQmCC)
, ...,
![](data:image/png;base64,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)
=i)=P(
![](data:image/png;base64,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)
=j /
![](data:image/png;base64,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)
=i).
Для цепей Маркова вероятность в момент времени n попасть в состояние
![](data:image/png;base64,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)
, если известна вся предыдущая история изучаемого процесса, зависит только от того, в каком состоянии находился процесс в момент n-1. То есть при фиксированном "настоящем" "будущее" не зависит от "прошлого". Свойство независимости "будущего" от "прошлого" при фиксированном "настоящем" называется
марковским свойством.
Вероятности
![](data:image/png;base64,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)
(
![](data:image/png;base64,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)
=j /
![](data:image/png;base64,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)
=i), i, j=1,2,..., r называются вероятностями перехода из состояния
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAYCAYAAAAYl8YPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADtSURBVEjH7ZSxDYMwEEVvABZAirMDuh0cF9QpsKiIqBANdXTQMUM6yuyQVbIBOyQykRPLMgRblBTXXPHu3/9nQ9d1sFXBDtsGVnHWAMBrrlC2qZeyVmIKwEZRU6J7fV9GgrG72VsFm9QdxKPs+8js15yfc6LYC5YhDIxXzVeRkEWQZ0R5jABP7Y1aWYO9YR+/1pn+F6ZWBMwGveLpiLc5nxZh9oqqpBCFHcQqmOskfkniRfeVYo78ag5xr+g4iblTcSqjWiQMYFy6fDNR19Dgd6jssBMOhklEshMOAk2Jo6RNviB91Par2L9t/3oDj6SbWLI+VT4AAAAASUVORK5CYII=)
в состояние
![](data:image/png;base64,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)
за один шаг.
Цепь Маркова называется однородной, если вероятности перехода
![](data:image/png;base64,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)
не зависят от n, т.е. если вероятности перехода не зависят от номера шага, а зависят только от того, из какого состояния и в какое осуществляется переход. Для однородных цепей Маркова вместо
![](data:image/png;base64,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)
будем писать
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.
Вероятности перехода удобно располагать в виде квадратной матрицы
![](data:image/png;base64,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)
Матрица P называется матрицей вероятностей перехода однородной цепи Маркова за один шаг. Она обладает следующими свойствами:
а)
![](data:image/png;base64,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)
;
б) для всех i:
![](data:image/png;base64,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)
Квадратные матрицы, для которых выполняются условия а) и б), называются стохастическими.
Вектор
![](data:image/png;base64,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)
, где
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAYCAYAAADzoH0MAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADKSURBVDjL7ZOxDYMwEEX/ACyAhIcgt4NFwQLGchdRoTTU6HDnNdiDCbIDG2SHRCay5EhB4KSluMJfund3/86w1uKfwAk4AVuATooewBMQDylpQFHNrXPZLsC5NqsKzKBm8m9mkxOwCNn1hzpoCFNIjoGkx3oXMGqqAVoMcx40vlWlwOUea5uAreqxFnRJcgiefAWE2b2RZMxVKS4PjuCd90GLYlW+IZ9jrYWirSTv3ReKTU0GaCKOO0pKXr0hzT+fcvApPqzzO1u8ANh7NbswoMfBAAAAAElFTkSuQmCC)
=P(
![](data:image/png;base64,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)
), i=1,2...,r называется вектором начальных вероятностей.
Свойства однородных цепей Маркова полностью определяются вектором начальных вероятностей и матрицей вероятностей перехода.
Приведем пример: Завод выпускает телевизоры определенного типа. В зависимости от того, находит ли данный тип телевизора спрос у населения, завод в конце каждого года может находиться в одном из состояний: состояние 1 – спрос есть, состояние 2 – спроса нет. Пусть вероятность сохранить состояние 1 в в следующем году с учетом возможного изменения спроса равна
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAApCAYAAAA8hqkEAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADaSURBVEjHY2hsbGSgBDOMAAMaoo19GGQ89qR1dPCQbEB9faykMQPDXbINiDY2rvf0lO0hywCQ042jG3xy3GSLSTYA7HTj6HoQmywDQE6Pra+XJMuA+jwPQ4/IejsYn2QDoowZFjIwMPzHhkFhQnI6ICsMBpcBo9mZ3gbgSoHIeDQQ6WkAqGCRZWB4A48B46iFJBkAzoUEChOiClWyvACujQjYjNOAjo40Hg8Zhj0oKRCH/wnGArJhJIUB1pKa1FjACBNKDSDbC+AEJeuxkqSqDSUF4inSh0luBADUTsKoxPvDagAAAABJRU5ErkJggg==)
, а вероятность изменить состояние 2 с учетом мероприятий по улучшению выпускаемой модели равна
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAApCAYAAADqI3NhAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADxSURBVEjHY2hsbGQgFzMME8319bGSxgwMdxkYGP6DsHF0gw9RmkEajRiMdsfW10uC+DlussUMDMZ3YXy8mhvy8iTSOjp4UA2TPeWRV29Isp/r8zwMZY2jekkOMLBGWY+VyC4hSjPEr5AAI9rPuAyRdcspJllzR0caj4cMwx6yNINwlDHDQlxxTTi08QQaajxHG/sgAgqIZTz2kBzaIy1LDoxmlGjCgUcDjNqawbmJgeENPJSNoxaSWQzhLruxF/rG0fVkORs5T+OrLTA0w8oslFSFw794QxvZILLKMFgBSFJoY4QBJZppX/RipKzhWfQCACY3hWBJKj6gAAAAAElFTkSuQmCC)
. Тогда процесс производства на данном заводе можно описать цепью Маркова с матрицей переходов:
![](data:image/png;base64,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)
В конкретных случаях для описания эволюции цепи Маркова вместо явного выписывания матрицы P используют граф, вершинами которого являются состояния цепи, а стрелка, идущая из состояния
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAYCAYAAAAYl8YPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADtSURBVEjH7ZSxDYMwEEVvABZAirMDuh0cF9QpsKiIqBANdXTQMUM6yuyQVbIBOyQykRPLMgRblBTXXPHu3/9nQ9d1sFXBDtsGVnHWAMBrrlC2qZeyVmIKwEZRU6J7fV9GgrG72VsFm9QdxKPs+8js15yfc6LYC5YhDIxXzVeRkEWQZ0R5jABP7Y1aWYO9YR+/1pn+F6ZWBMwGveLpiLc5nxZh9oqqpBCFHcQqmOskfkniRfeVYo78ag5xr+g4iblTcSqjWiQMYFy6fDNR19Dgd6jssBMOhklEshMOAk2Jo6RNviB91Par2L9t/3oDj6SbWLI+VT4AAAAASUVORK5CYII=)
в состояние
![](data:image/png;base64,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)
с числом
![](data:image/png;base64,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)
над ней показывает, что из состояния
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAYCAYAAAAYl8YPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADtSURBVEjH7ZSxDYMwEEVvABZAirMDuh0cF9QpsKiIqBANdXTQMUM6yuyQVbIBOyQykRPLMgRblBTXXPHu3/9nQ9d1sFXBDtsGVnHWAMBrrlC2qZeyVmIKwEZRU6J7fV9GgrG72VsFm9QdxKPs+8js15yfc6LYC5YhDIxXzVeRkEWQZ0R5jABP7Y1aWYO9YR+/1pn+F6ZWBMwGveLpiLc5nxZh9oqqpBCFHcQqmOskfkniRfeVYo78ag5xr+g4iblTcSqjWiQMYFy6fDNR19Dgd6jssBMOhklEshMOAk2Jo6RNviB91Par2L9t/3oDj6SbWLI+VT4AAAAASUVORK5CYII=)
в состояние
![](data:image/png;base64,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)
возможен переход с вероятностью
![](data:image/png;base64,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)
. В том случае, когда
![](data:image/png;base64,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)
, соответствующая стрелка не проводится.
Можно показать, что матрица вероятностей перехода цепи Маркова за n шагов равняется n-ой степени матрицы P вероятностей перехода за один шаг. Для однородной цепи Маркова при любом m выполняется равенство
P(
![](data:image/png;base64,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)
)=P(
![](data:image/png;base64,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)
).
Но последняя вероятность есть вероятность перехода из состояния
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAYCAYAAAAYl8YPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADtSURBVEjH7ZSxDYMwEEVvABZAirMDuh0cF9QpsKiIqBANdXTQMUM6yuyQVbIBOyQykRPLMgRblBTXXPHu3/9nQ9d1sFXBDtsGVnHWAMBrrlC2qZeyVmIKwEZRU6J7fV9GgrG72VsFm9QdxKPs+8js15yfc6LYC5YhDIxXzVeRkEWQZ0R5jABP7Y1aWYO9YR+/1pn+F6ZWBMwGveLpiLc5nxZh9oqqpBCFHcQqmOskfkniRfeVYo78ag5xr+g4iblTcSqjWiQMYFy6fDNR19Dgd6jssBMOhklEshMOAk2Jo6RNviB91Par2L9t/3oDj6SbWLI+VT4AAAAASUVORK5CYII=)
в состояние
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAAZCAYAAADe1WXtAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEDSURBVEjH7VU9DoIwFH4H4ACSUO9A3u5YOzCbSMOEYSJdmE1hY/ECbozegat4A+6gKQYlTYEUiBPDW97w9ft7AEVRwNoDG+h/QFNKMgB4DQ3yPJjFNOcYAJCGCel3u7JMHEbIo7+zAm3ZeqxOytLp7wWlp0hKdxZoiFARmmZfhozHizyVMnIR4Nl5p6zoHpgN+vHTLpxJUCUdMKw66cc93qd8HAXVpavhjMV6YFagpir9kseLac8RZV+JWbqhSkMVG2UqBfMJQDN2SXoD2qp5UOshLr5zeWYH3ZLFoKYQl7FUlmF4W/XTp8IzHYU1kAqHIr2KXOyGjmIWw7Z2A13e/lHwBhPVAs+gHkxBAAAAAElFTkSuQmCC)
за n шагов.
§2. Теорема о предельных вероятностях.
В 1930 году Дж.Биркгофом и Дж.фон Нейманом была сформулирована и доказана одна из основных эргодических теорем – теорема о предельных вероятностях:
Если при некотором
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAYCAYAAAAYl8YPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADkSURBVEjHY2hsbGSgFmYYNWzUMHoaVp/nYSjLwPCGgYHhv3F0g09HRxqPhwzDHhBf1i2nmGSX1dfHShoxGO2OrI80NGKQPeWRV2+Y4yZbTJZhDdHGPgzGUQujjY3rY+vrJcGuk5VdCTKUZMOijBkWysjIHAV5E+51WY+VaR0dPKSFGdCLxgwMd5G9hM2LIDFc4YgaAUiugIQfMNyiPZKN3fJS4GqMo3ohvpDtRfc+Ay5XwGNXxmMPzAKQGlgQgMIX3XUkpaM8N+MUmGtAlsFcTJZhVHUZ0WFGLAYlH4KxOVoEEY0Bj6eglSNdKocAAAAASUVORK5CYII=)
все элементы матрицы
![](data:image/png;base64,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)
=[
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADAAAAAZCAYAAAB3oa15AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAHdSURBVFjD7ZcxbsIwFIZ9AA5QpLh3gDe3UxU8cAESMYGYkBfmyrBl4QJsjJyhrF24QQduwB1aXtIXGdcmcTEIpAweUMzL+7/3v2eHLRYL9sjroZNvBDQC7lmAUsN2l3U/hkq1rxnjKsnPU+gzBvtLki9FSNHhjH8JqTo3ERCCvF1Ed2eLGVxAAmzN4+ksdNxpzGcMkvVZAUXp2be+Ef9YN6GCPt/p5S7osQPGhXTez7JJS0Rsi799hLqqcJr8MXEz4QT40uW/ut4nWw3UoEMCfcBQDGBsjxCcFTAJnvOes8wWAQQnBVD4LK8C55u6YHQBpui/BCOxnWRZi/xs852vAIwTRdEn0cvBcLGh9wQToDcgitFfSgvpxRC/08uJqstCthfb7FOId/dFpYV0+yChZ+ites+wQt8CpOqiJtZol3tSMYJYjss9kCzP9VxlE5PCfAodbSTl20sxPU6J5rbC53P5hNPEJGKOUZN2OZU0q+IeioNVtFWh1hitO2lKLw/Eq0nLVoWqJWMY64ODKlPncPQWoHs+FWJka8Sq499G11UBl/f/JSAP9tsPum8vvVK4eiD4ZY5OaiSkUwt1BfE9nb0F0AjFBsYJFfLCdrPvgYIUP/g0afNF1gi44/UDCrDima4jy+gAAAAASUVORK5CYII=)
] положительны, то существуют пределы
![](data:image/png;base64,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)
, i,j =1,2,...,r.
Предельные вероятности
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAZCAYAAADXPsWXAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAD1SURBVDjL7ZS7DYMwEIZvAAYIEs4OcH1Kx0UWiBFVIipEQx0ZOhoWSMccYZEUbMAOiQ6EhJB5GLoohTvu43+cDVmWwd4Df8g8JM9DSzhQAcCH8SjZpUQiK0Ss3M0QpQLbA+8VKGVvhqQ+XgBluStYiVBiENwRoKZswBFVmOfWaghZ6YaxJjsqFi4DaNBPL6shYys91AjSWhkMdErYe6mp2VYIqgvZR1TD7yatRJwlfTZG7bR/pkboaBT0Gz3OyPieqKs4jTMyhvhC3MZ7Y6aC2kJZ7HoKKGzdziwOUpgc+SNO48P5iE9dW6sUdK2xZmrpfuyN/QIsyXIlM6UZ7AAAAABJRU5ErkJggg==)
не зависят от начального состояния
![](data:image/png;base64,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)
и являются единственным решением системы уравнений
,
, j=1, 2, ..., r. Физический смысл этой теоремы заключается в том, что вероятности нахождения системы в состоянии
практически не зависят от того, в каком состоянии она находилась в далеком прошлом. Цепь Маркова, для которой существуют пределы
, называется эргодической. Решение (
,
,...,
) написанной выше системы (1) называется стационарным распределением вероятностей для марковской цепи с матрицей перехода P = [
].