关于独立离散随机变量和的最大熵
On the Maximum Entropy of a Sum of Independent Discrete Random Variables
摘要：
令$ X_1, \ldots, X_n $为独立随机变量，取值字母表为$ \{0, 1, \ldots, r\} $和$ S_n = \sum_{i = 1}^n X_i $。Shepp-Olkin定理指出，在二进制情况下（$ r = 1 $），当所有$ X_i $均匀分布时，$ S_n $的香农熵最大化，即，Bernoulli（1/2）.为了将这一定理推广到任意有限字母表，我们得到了$ S_n $的最大熵的一个下界，并证明了在几种特殊情况下它是紧的。除了这些特殊情况外，还提出了一个论点，支持这样的猜想，即界代表所有$ n, r $的最佳值，即，当$ X_1, \ldots, X_{n-1} $在$ \{0, r\} $上均匀分布时，$ H(S_n) $最大化，而$ X_n $的概率质量函数是在$ \{0, r\} $和$ \{1, \ldots, r-1\} $上均匀分布的混合（具有明确定义的非零权重）。
英文摘要：
Let $ X_1, \ldots, X_n $ be independent random variables taking values in the alphabet $ \{0, 1, \ldots, r\} $, and $ S_n = \sum_{i = 1}^n X_i $. The Shepp--Olkin theorem states that, in the binary case ($ r = 1 $), the Shannon entropy of $ S_n $ is maximized when all the $ X_i $'s are uniformly distributed, i.e., Bernoulli(1/2). In an attempt to generalize this theorem to arbitrary finite alphabets, we obtain a lower bound on the maximum entropy of $ S_n $ and prove that it is tight in several special cases. In addition to these special cases, an argument is presented supporting the conjecture that the bound represents the optimal value for all $ n, r $, i.e., that $ H(S_n) $ is maximized when $ X_1, \ldots, X_{n-1} $ are uniformly distributed over $ \{0, r\} $, while the probability mass function of $ X_n $ is a mixture (with explicitly defined non-zero weights) of the uniform distributions over $ \{0, r\} $ and $ \{1, \ldots, r-1\} $.
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