{"title":"随机信道的渐近容量","authors":"Tobias Sutter, David Sutter, J. Lygeros","doi":"10.3934/amc.2017060","DOIUrl":null,"url":null,"abstract":"We consider discrete memoryless channels with input and output alphabet size n whose channel transition matrix consists of entries that are independent and identically distributed according to some probability distribution v on (R≥0, B(R≥0)) before being normalized, where v is such that E[X log X)<sup>2</sup> 1 <; ∞, μ<sub>1</sub> := E[X] and μ<sub>2</sub> := E[X log X] for a random variable X with distribution v. We prove that in the limit as n → ∞, the capacity of such a channel converges to μ<sub>2</sub>/μ<sub>1</sub> - log μ<sub>1</sub> almost surely and in L<sup>2</sup>. We further show that the capacity of these random channels converges to this asymptotic value exponentially in n.","PeriodicalId":330880,"journal":{"name":"2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)","volume":"18 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Asymptotic capacity of a random channel\",\"authors\":\"Tobias Sutter, David Sutter, J. Lygeros\",\"doi\":\"10.3934/amc.2017060\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider discrete memoryless channels with input and output alphabet size n whose channel transition matrix consists of entries that are independent and identically distributed according to some probability distribution v on (R≥0, B(R≥0)) before being normalized, where v is such that E[X log X)<sup>2</sup> 1 <; ∞, μ<sub>1</sub> := E[X] and μ<sub>2</sub> := E[X log X] for a random variable X with distribution v. We prove that in the limit as n → ∞, the capacity of such a channel converges to μ<sub>2</sub>/μ<sub>1</sub> - log μ<sub>1</sub> almost surely and in L<sup>2</sup>. We further show that the capacity of these random channels converges to this asymptotic value exponentially in n.\",\"PeriodicalId\":330880,\"journal\":{\"name\":\"2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)\",\"volume\":\"18 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2014-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/amc.2017060\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/amc.2017060","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
摘要
我们考虑离散无记忆信道的输入和输出字母大小为n通道转移矩阵的条目包括独立且同分布根据一些概率分布v (R≥0,B (R≥0))在标准化之前,v在哪里,E (X日志)2 1 1:= E (X)和μ2:= E (X日志)与分布随机变量X诉我们证明极限n→∞,这样一个信道的容量收敛于日志μμ2 /μ1 - 1几乎肯定在L2。我们进一步证明了这些随机信道的容量在n上指数收敛于这个渐近值。
We consider discrete memoryless channels with input and output alphabet size n whose channel transition matrix consists of entries that are independent and identically distributed according to some probability distribution v on (R≥0, B(R≥0)) before being normalized, where v is such that E[X log X)2 1 <; ∞, μ1 := E[X] and μ2 := E[X log X] for a random variable X with distribution v. We prove that in the limit as n → ∞, the capacity of such a channel converges to μ2/μ1 - log μ1 almost surely and in L2. We further show that the capacity of these random channels converges to this asymptotic value exponentially in n.