On the Maximum Entropy of a Sum of Independent Discrete Random Variables
| Autor: | Mladen Kovacevic |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
FOS: Computer and information sciences
Statistics and Probability Computer Science - Information Theory 02 engineering and technology Upper and lower bounds 01 natural sciences Combinatorics 010104 statistics & probability FOS: Mathematics 0202 electrical engineering electronic engineering information engineering 0101 mathematics Mathematics Conjecture Mathematics::Commutative Algebra Information Theory (cs.IT) Principle of maximum entropy Probability (math.PR) 010102 general mathematics 020206 networking & telecommunications Condensed Matter::Mesoscopic Systems and Quantum Hall Effect Binomial distribution 94A17 (Primary) 60C05 60G50 (Secondary) Statistics Probability and Uncertainty Alphabet Binary case Random variable Mathematics - Probability |
| Zdroj: | Theory of Probability & Its Applications. 66:482-487 |
| ISSN: | 1095-7219 0040-585X |
| DOI: | 10.1137/s0040585x97t99054x |
| Popis: | 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\} $. Comment: 8 pages, 1 figure |
| Databáze: | OpenAIRE |
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