对数凹分布的内禀熵
Intrinsic entropies of log-concave distributions
摘要：
众所周知，随机变量的熵等于其典型集的体积的指数增长率。本文证明了对任意对数凹随机变量$X$，在$n \geq 1$维上$X$的典型集的$\lfloor nθ\rfloor^{\text{th}}$内禀体积序列以一个定义好的速率指数增长.我们用$h_X(θ)$表示这个速率，并称之为$X$的$θ^{\text{th}}$内禀熵。我们表明，$h_X(θ)$是$θ$在范围$[0,1]$的连续函数，从而提供了一个平滑的插值值0和$h(X)$之间的端点0和1，分别。
英文摘要：
The entropy of a random variable is well-known to equal the exponential growth rate of the volumes of its typical sets. In this paper, we show that for any log-concave random variable $X$, the sequence of the $\lfloor nθ\rfloor^{\text{th}}$ intrinsic volumes of the typical sets of $X$ in dimensions $n \geq 1$ grows exponentially with a well-defined rate. We denote this rate by $h_X(θ)$, and call it the $θ^{\text{th}}$ intrinsic entropy of $X$. We show that $h_X(θ)$ is a continuous function of $θ$ over the range $[0,1]$, thereby providing a smooth interpolation between the values 0 and $h(X)$ at the endpoints 0 and 1, respectively.
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