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# 数学代写|概率论代考Probability Theory代写|MATH407

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## 数学代写|概率论代考Probability Theory代写|Characteristic functions

Given a random variable $X$, we define its characteristic function (or Fourier transform) by
$$\phi_X(t)=\mathbf{E}\left(e^{i t X}\right)=\mathbf{E}[\cos (t X)]+i \mathbf{E}[\sin (t X)], \quad t \in \mathbf{R} .$$
The characteristic function is thus a function from the real numbers to the complex numbers. Of course, by the Change of Variable Theorem (Theorem 6.1 .1$), \phi_X(t)$ depends only on the distribution of $X$. We sometimes write $\phi_X(t)$ as $\phi(t)$.

The characteristic function is clearly very similar to the moment generating function introduced earlier; the only difference is the appearance of the imaginary number $i=\sqrt{-1}$ in the exponent. However, this change is significant; since $\left|e^{i t X}\right|=1$ for any (real) $t$ and $X$, the triangle inequality implies that $\left|\phi_X(t)\right| \leq 1<\infty$ for all $t$ and all random variables $X$. This is quite a contrast to the case for moment generating functions, which could be infinite for any $s \neq 0$.

Like for moment generating functions, we have $\phi_X(0)=1$ for any $X$, and if $X$ and $Y$ are independent then $\phi_{X+Y}(t)=\phi_X(t) \phi_Y(t)$ by (4.2.7). We further note that, with $\mu=\mathcal{L}(X)$, we have
\begin{aligned} &\left|\phi_X(t+h)-\phi_X(t)\right|=\left|\int\left(e^{i(t+h) x}-e^{i t x}\right) \mu(d x)\right| \ & \leq \int\left|e^{i(t+h) x}-e^{i t x}\right| \mu(d x) \leq \int\left|e^{i t x}\right|\left|e^{i h x}-1\right| \mu(d x) \ &=\int\left|e^{i h x}-1\right| \mu(d x) . \end{aligned}

## 数学代写|概率论代考Probability Theory代写|The continuity theorem

In this subsection we shall prove the continuity theorem for characteristic functions (Theorem 11.1.14), which says that if characteristic functions converge pointwise, then the corresponding distributions converge weakly: $\mu_n \Rightarrow \mu$ if and only if $\phi_n(t) \rightarrow \phi(t)$ for all $t$. This is a very important result; for example, it is used to prove the central limit theorem in the next subsection. Unfortunately, the proof is somewhat technical; we must show that characteristic functions completely determine the corresponding distribution (Theorem 11.1.1 and Corollary 11.1.7 below), and must also establish a simple criterion for weak convergence of “tight” measures (Corollary 11.1.11).

We begin with an inversion theorem, which tells how to recover information about a probability distribution from its characteristic function.
Theorem 11.1.1. (Fourier inversion theorem) Let $\mu$ be a Borel probability measure on $\mathbf{R}$, with characteristic function $\phi(t)=\int_{\mathbf{R}} e^{i t x} \mu(d x)$. Then if $a<b$ and $\mu{a}=\mu{b}=0$, then
$$\mu([a, b])=\lim {T \rightarrow \infty} \frac{1}{2 \pi} \int{-T}^T \frac{e^{-i t a}-e^{-i t b}}{i t} \phi(t) d t .$$
To prove Theorem 11.1.1, we use two computational lemmas.
Lemma 11.1.2. For $T \geq 0$ and $a<b$,
$$\int_{\mathbf{R}} \int_{-T}^T\left|\frac{e^{-i t a}-e^{-i t b}}{i t} \phi(t)\right| d t \mu(d x) \leq 2 T(b-a)<\infty .$$

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|Characteristic functions

$$\phi_X(t)=\mathbf{E}\left(e^{i t X}\right)=\mathbf{E}[\cos (t X)]+i \mathbf{E}[\sin (t X)], \quad t \in \mathbf{R} .$$

\begin{aligned} &\left|\phi_X(t+h)-\phi_X(t)\right|=\left|\int\left(e^{i(t+h) x}-e^{i t x}\right) \mu(d x)\right| \ & \leq \int\left|e^{i(t+h) x}-e^{i t x}\right| \mu(d x) \leq \int\left|e^{i t x}\right|\left|e^{i h x}-1\right| \mu(d x) \ &=\int\left|e^{i h x}-1\right| \mu(d x) . \end{aligned}

## 数学代写|概率论代考Probability Theory代写|The continuity theorem

$$\mu([a, b])=\lim {T \rightarrow \infty} \frac{1}{2 \pi} \int{-T}^T \frac{e^{-i t a}-e^{-i t b}}{i t} \phi(t) d t .$$

$$\int_{\mathbf{R}} \int_{-T}^T\left|\frac{e^{-i t a}-e^{-i t b}}{i t} \phi(t)\right| d t \mu(d x) \leq 2 T(b-a)<\infty .$$

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## MATLAB代写

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