{"title":"具有独立开关时间的非对称开关过程的特征","authors":"Henrik Bengtsson, Krzysztof Podgorski","doi":"arxiv-2409.05641","DOIUrl":null,"url":null,"abstract":"The asymmetric switch process is a binary stochastic process that alternates\nbetween the values one and minus one, where the distribution of the time in\nthese states may differ. In this sense, the process is asymmetric, and this\npaper extends previous work on symmetric switch processes. Two versions of the\nprocess are considered: a non-stationary one that starts with either the one or\nminus one at time zero and a stationary version constructed from the\nnon-stationary one. Characteristics of these two processes, such as the\nexpected values and covariance, are investigated. The main results show an\nequivalence between the monotonicity of the expected value functions and the\ndistribution of the intervals having a stochastic representation in the form of\na sum of random variables, where the number of terms follows a geometric\ndistribution. This representation has a natural interpretation as a model in\nwhich switching attempts may fail at random. From these results, conditions are\nderived when these characteristics lead to valid interval distributions, which\nis vital in applications.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"5 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Characteristics of asymmetric switch processes with independent switching times\",\"authors\":\"Henrik Bengtsson, Krzysztof Podgorski\",\"doi\":\"arxiv-2409.05641\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The asymmetric switch process is a binary stochastic process that alternates\\nbetween the values one and minus one, where the distribution of the time in\\nthese states may differ. In this sense, the process is asymmetric, and this\\npaper extends previous work on symmetric switch processes. Two versions of the\\nprocess are considered: a non-stationary one that starts with either the one or\\nminus one at time zero and a stationary version constructed from the\\nnon-stationary one. Characteristics of these two processes, such as the\\nexpected values and covariance, are investigated. The main results show an\\nequivalence between the monotonicity of the expected value functions and the\\ndistribution of the intervals having a stochastic representation in the form of\\na sum of random variables, where the number of terms follows a geometric\\ndistribution. This representation has a natural interpretation as a model in\\nwhich switching attempts may fail at random. From these results, conditions are\\nderived when these characteristics lead to valid interval distributions, which\\nis vital in applications.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"5 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.05641\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05641","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Characteristics of asymmetric switch processes with independent switching times
The asymmetric switch process is a binary stochastic process that alternates
between the values one and minus one, where the distribution of the time in
these states may differ. In this sense, the process is asymmetric, and this
paper extends previous work on symmetric switch processes. Two versions of the
process are considered: a non-stationary one that starts with either the one or
minus one at time zero and a stationary version constructed from the
non-stationary one. Characteristics of these two processes, such as the
expected values and covariance, are investigated. The main results show an
equivalence between the monotonicity of the expected value functions and the
distribution of the intervals having a stochastic representation in the form of
a sum of random variables, where the number of terms follows a geometric
distribution. This representation has a natural interpretation as a model in
which switching attempts may fail at random. From these results, conditions are
derived when these characteristics lead to valid interval distributions, which
is vital in applications.