在概率论学习中, 联合连续随机变量 X, Y 的联合概率密度函数 f 以及联合分布函数 F 具有如下关系:
\[\begin{align} F(x, y) &= \int_{-\infty}^x \int_{-\infty}^y f(s,t)\mathrm{d}t \mathrm{d}s\\ f(x, y) &= \frac{\partial^2 F}{\partial x \partial y}(x, y) \end{align}\]其非常类似于 Tao analysis 定理 11.9.1 介绍的微积分第一基本定理的多元版本, 但史济怀数学分析教程/陶哲轩 analysis 都没有介绍过类似定理, 我感觉还挺奇怪的. 这里研究一下.
定理 zy1: 已知 $f(x, t)$ 以及其偏导数 $\frac{\partial f}{\partial x} = f_x$ 在某个闭矩形 B 上连续, 如下 $u_0, a, b$ 均位于该闭矩形 B 中. 求证:
\[\lim_{h \to 0} \int_a^b \frac{1}{h} (f(u_0 + h, t) - f(u_0, t)) \mathrm{d}t = \int_a^b \lim_{h \to 0} \frac{1}{h} (f(u_0 + h, t) - f(u_0, t)) \mathrm{d}t\]证明: 令 $G(h, t) =\frac{1}{h} (f(u_0 + h, t) - f(u_0, t))$. 根据 Tao analysis 定理 9.3.9 可知我们只需证 $\forall a_n, \lim_{n \to \infty}a_n = 0, \lim_{n \to \infty} \int_a^b G(a_n, t) \mathrm{d}t$ 为上述公式中右侧即可. 定义 $H_n(t) = G(a_n, t)$. 易证 $H_n(t)$ 逐点收敛 $f_x(u_0, t)$. 根据 Tao analysis 推论 10.2.9 可知 $H_n(t) = f_x(u_0 + \xi, t),\xi \in (0, a_n)$. 考虑到 $f_x$ 连续由 Tao analysis 命题 13.3.2 可知 $f_x$ 有界, 即 $\vert H_n(t) \vert \le M, F(t) = M$. 代入 Tao analysis 定理 19.3.4 勒贝格控制收敛定理可得 $\lim_{n \to \infty} \int_a^b H_n(t) \mathrm{d}t = \int_a^b f_x(u_0, t) \mathrm{d}t$. 证明完毕!
P.S. 从如上证明中可以看到当 $a, b$ 为无穷时, $\int_a^b F(t) \mathrm{d}t$ 将不一定绝对可积, 无法应用控制收敛定理. 针对这种情况我们需要得加条件:
定理 zy2: Suppose that the function $h(x,y)$ is continuous at $y_0$ for each x, and there exists a function $g(x)$ satisfying $\vert h(x, y) \vert \le g(x)$ for all x and y in a neighborhood of $y_0$; $\int_{-\infty}^{\infty} g(x) \mathrm{d}x \lt \infty$. 则有:
\[\lim_{y \to y_0} \int_{-\infty}^{\infty} h(x, y) \mathrm{d}x = \int_{-\infty}^{\infty} \lim_{y \to y_0}h(x, y) \mathrm{d}x\]证明: 参考定理 zy1 证明结合控制收敛定理即可.
定理 zy3 Leibniz integral rule: Let $ f(x,t) $ be a function such that both $ f(x,t) $ and its partial derivative $ f_{x}(x,t) $ are continuous in $ t $ and $ x $ in some region of the $ xt $-plane, $x\in [x_0, x_1], t \in [t_1, t_2]$, including $ a(x) \leq t \leq b(x), x_0 \leq x \leq x_1 $. Also suppose that the functions $ a(x) $ and $ b(x) $ are both continuous and both have continuous derivatives for $ x_0 \leq x \leq x_1 $. Then, for $ x_0 \leq x \leq x_1 $,
\[\frac{\mathrm{d}}{\mathrm{d}x} \left( \int_{a(x)}^{b(x)} f(x,t) \mathrm{d}t \right) = f(x, b(x)) \cdot \frac{\mathrm{d}}{\mathrm{d}x} b(x) - f(x, a(x)) \cdot \frac{\mathrm{d}}{\mathrm{d}x} a(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t) \mathrm{d}t\]证明: 首先证明 $a(x), b(x)$ 为常数 $a, b$ 情况:
设 $h(x, y): \mathbb{R}^2 \to \mathbb{R}$ 为连续函数, 则由 定理 10.3.4, g(x) 连续 易证 $F(x) = \int_c^d h(x, y)\mathrm{d}y$ 连续, 即 $\int_a^b F(x) \mathrm{d}x$ 可积, 符合史数分定理 10.3.2 条件, 即积分顺序可交换:
\[\int_x^{x+h} \int_a^b f_x(x, t) \mathrm{d}t\mathrm{d}x = \int_a^b \int_x^{x+h} f_x(x, t) \mathrm{d}x \mathrm{d}t = \int_a^b (f(x + h, t) - f(x, t)) \mathrm{d}t.\]定义 $F(u) = \int_{x_0}^u \int_a^b f_x(x, t) \mathrm{d}t\mathrm{d}x$, 则:
\[\begin{align} F'(u_0) &= \lim_{h \to 0} \frac{F(u_0 + h) - F(u_0)}{h} = \lim_{h \to 0} \frac{1}{h} \int_{u_0}^{u_0+h} \int_a^b f_x(x, t) \mathrm{d}t\mathrm{d}x \\ &= \lim_{h \to 0} \frac{1}{h} \int_a^b (f(u_0 + h, t) - f(u_0, t)) \mathrm{d}t \\ &= \int_a^b \lim_{h \to 0} \frac{1}{h} (f(u_0 + h, t) - f(u_0, t)) \mathrm{d}t \quad \leftarrow zy1 \\ &= \int_a^b f_x(u_0, t) \mathrm{d}t \end{align}\]定义 $G(u) = \int_a^b f(u, t) \mathrm{d}t$, 易知 $F(u) = G(u) - \int_a^b f(x_0, t) \mathrm{d}t$, 这里 $\int_a^b f(x_0, t) \mathrm{d}t$ 可以视为常数, 所以 $G’(u) = F’(u)$. 易得结论:
\[\frac{\mathrm{d}}{\mathrm{d}x}\int_a^b f(x, t) \mathrm{d}t = G'(x) = F'(x) = \int_a^b f_x(x,t) \mathrm{d}t\]之后再考虑 $a(x), b(x)$ 不为常数的情况: 令 $F(x, y) = \int_{t_1}^y f(x, t) \mathrm{d}t$:
\[\begin{align} G(x) &= \int_{a(x)}^{b(x)} f(x,t) \mathrm{d}t = \int_{t_1}^{b(x)} f(x,t) \mathrm{d}t - \int_{t_1}^{a(x)} f(x,t) \mathrm{d}t \\ &= F(x, b(x)) - F(x, a(x)) \end{align}\]再证 F 是可微的, 考虑 $\frac{\partial F}{\partial x}(x, y) = \int_{t_1}^y f_x(x, t) \mathrm{d}t$, 这里 $f_x$ 是连续的易证 $\frac{\partial F}{\partial x}$ 连续. 由 Tao analysis 定理 11.9.1 可知 $\frac{\partial F}{\partial y}(x, y) = f(x,y)$ 由 f 连续可知 $\frac{\partial F}{\partial y}$ 也是连续的. 由 Tao analysis 定理 17.3.8 可知 F 可微.
定义 $h(x) = (x, a(x))$, 由 $a’(x)$ 连续易证 h 连续可微, 即 h 在每一点全导数都是存在的. $F(x, a(x)) = F \circ h$. 由 Tao analysis 定理 17.4.1 链式法可求:
\[\begin{align} G'(x) &= \left(\frac{\partial F}{\partial x}(x, b(x)) + \frac{\partial F}{\partial y}(x, b(x))b'(x)\right) - \left(\frac{\partial F}{\partial x}(x, a(x)) + \frac{\partial F}{\partial y}(x, a(x))a'(x)\right) \\ &= \left( \int_{t_1}^{b(x)} f_x(x,t) \mathrm{d}t + f(x, b(x))b'(x) \right) - \left( \int_{t_1}^{a(x)} f_x(x,t) \mathrm{d}t + f(x, a(x))a'(x) \right) \\ &= f(x,b(x))b'(x) - f(x, a(x))a'(x) + \int_{a(x)}^{b(x)} f_x(x,t) \mathrm{d}t, \end{align}\]定理 zy4: 设 $f(u, v)$ 是连续函数, 定义 $F(x, y) = \int_{a}^x \int_{c}^y f(u, v)\mathrm{d}v \mathrm{d}u$. 求证:
\[\frac{\partial^2 F}{\partial x \partial y}(x, y) = f(x,y)\]证明: 定义 $G(u, y) = \int_c^y f(u, v) \mathrm{d}v, F(x,y) = \int_a^x G(u, y) \mathrm{d}u$, 这里 f 连续易证 $G, G_y = f(u,y)$ 是连续的. 符合 Leibniz integral rule 条件, 可知:
\[\begin{align} \frac{\partial F}{\partial y}(x, y) &= \int_a^x G_y(u, y) \mathrm{d}u = \int_a^x f(u,y) \mathrm{d}u \\ \frac{\partial^2 F}{\partial x \partial y}(x, y) &= \frac{\partial }{\partial x}\int_a^x f(u,y) \mathrm{d}u = f(x, y) \end{align}\]P.S. 这里我们要求 f 是连续的, 不像 Tao 微积分第一基本定理中只要求 f 是单点连续的.
