论文标题
对数符号总和的近似离散熵单调性
Approximate Discrete Entropy Monotonicity for Log-Concave Sums
论文作者
论文摘要
事实证明，Tao（2010）的猜想对于整数上的log-conconcave随机变量是正确的：对于$ n \ geq 1 $，如果$ x_1，\ ldots，x_n $是i.i.d. integer-valued，log-concave随机变量，然后$$ h（x_1+\ cdots+x_ {n+1}）\ geq h（x_1+\ cdots+x_ {n}）+\ \ \ \\ frac {1} {2} {2} {2} {2} \ logl { $$ as $ h（x_1）\ to \ infty $，其中$ h $表示（离散的）香农熵。通过表明$ u_1，\ ldots，u_n $是$（0,1）$上的独立连续制度，然后$$ H（x_1+\ cdots+x_n+x_n+x_n+u_1+cdots+cdots+u_n）= h（x_1+\ cdots+x_n）+o（x_1+x_n）+o（x_1+cdots+o_n） $ H $代表差分熵。提供了$ O（1）$ - 条款的明确界限。
It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{1}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} - o(1) $$ as $H(X_1) \to \infty$, where $H$ denotes the (discrete) Shannon entropy. The problem is reduced to the continuous setting by showing that if $U_1,\ldots,U_n$ are independent continuous uniforms on $(0,1)$, then $$ h(X_1+\cdots+X_n + U_1+\cdots+U_n) = H(X_1+\cdots+X_n) + o(1) $$ as $H(X_1) \to \infty$, where $h$ stands for the differential entropy. Explicit bounds for the $o(1)$-terms are provided.
