We study a standard two-source model for common randomness (CR) generation in which Alice and Bob generate a common random variable with high probability of agreement by observing independent and identically distributed (i.i.d.) samples of correlated sources on countably infinite alphabets. The two parties are additionally allowed to communicate as little as possible over a noisy memoryless channel. In our work, we give a single-letter formula for the CR capacity for the proposed model and provide a rigorous proof of it. This is a challenging scenario because some of the finite alphabet properties, namely of the entropy can not be extended to the countably infinite case. Notably, it is known that the Shannon entropy is in fact discontinuous at all probability distributions with countably infinite support.
翻译:我们研究的是通用随机(CR)生成的标准双源模型,其中爱丽丝和鲍勃通过观测独立且分布相同的(i.d.)可计算到无限字母的相关来源样本,生成了一个共同随机(CR)概率高的随机变量。另外允许双方在一个吵闹的无记忆频道上尽可能少地进行沟通。在我们的工作中,我们为拟议模型的 CR 容量给出一个单字母公式,并提供严格的证明。这是一个具有挑战性的设想,因为一些限定字母属性,即酶的特性不能扩大到可计算到无限的情况。值得注意的是,众所周知,香农的酶在各种概率分布上实际上都是不连贯的,而且有无限的支持。
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