We present a method for upper and lower bounding the right and the left tail probabilities of continuous random variables (RVs). For the right tail probability of RV $X$ with probability density function $f (x)$, this method requires first setting a continuous, positive, and strictly decreasing function $g (x)$ such that $-f (x)/g' (x)$ is a decreasing and increasing function, $\forall x>x_0$, which results in upper and lower bounds, respectively, given in the form $-f (x) g (x)/g' (x)$, $\forall x>x_0$, where $x_0$ is some point. Similarly, for the upper and lower bounds on the left tail probability of $X$, this method requires first setting a continuous, positive, and strictly increasing function $g (x)$ such that $f (x)/g' (x)$ is an increasing and decreasing function, $\forall x<x_0$, which results in upper and lower bounds, respectively, given in the form $f (x) g (x)/g' (x)$, $\forall x<x_0$. We provide some examples of good candidates for the function $g (x)$. We also establish connections between the new bounds and Markov's inequality and Chernoff's bound. In addition, we provide an iterative method for obtaining ever tighter lower and upper bounds, under certain conditions. As an application, we use the proposed method to derive a novel closed-form asymptotic expression of the converse bound on the capacity of the additive white Gaussian noise (AWGN) channel in the finite-blocklength regime, which is tighter than the closed-form asymptotic expression by Polyanskiy-Poor-Verdú. Finally, we provide numerical examples where we show the tightness of the bounds obtained by the proposed method.

翻译：本文提出了一种对连续随机变量（RVs）的右尾概率和左尾概率进行上界与下界界定的方法。对于概率密度函数为 $f (x)$ 的随机变量 $X$ 的右尾概率，该方法首先要求设定一个连续、正值且严格递减的函数 $g (x)$，使得 $-f (x)/g' (x)$ 在 $\forall x>x_0$ 时分别为递减和递增函数，从而分别产生上界和下界，其形式为 $-f (x) g (x)/g' (x)$, $\forall x>x_0$，其中 $x_0$ 为某一点。类似地，对于 $X$ 的左尾概率的上界与下界，该方法首先要求设定一个连续、正值且严格递增的函数 $g (x)$，使得 $f (x)/g' (x)$ 在 $\forall x<x_0$ 时分别为递增和递减函数，从而分别产生上界和下界，其形式为 $f (x) g (x)/g' (x)$, $\forall x<x_0$。我们提供了一些函数 $g (x)$ 的优良候选示例。我们还建立了新界与马尔可夫不等式及切尔诺夫界之间的联系。此外，我们在特定条件下提供了一种迭代方法，用于获得不断收紧的下界和上界。作为应用，我们使用所提方法推导了有限块长体制下加性高斯白噪声（AWGN）信道容量逆界的一种新颖的闭式渐近表达式，该表达式比Polyanskiy-Poor-Verdú的闭式渐近表达式更紧。最后，我们提供了数值示例，展示了所提方法所得界值的紧致性。