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2 - Characteristic Functions, Independence, and Weak Convergence in $\mathbb{R}^d$

1. Multivariate Inversion Formula

For a random vector $X \in \mathbb{R}^d$ with characteristic function
\(\varphi_X(t) = E[e^{i\, t \cdot X}], \quad t \in \mathbb{R}^d,\) the Fourier inversion theorem states that if $X$ has a “nice” density $f_X$, then \(f_X(0) = \lim_{T \to \infty} (2\pi)^{-d} \int_{[-T,T]^d} \varphi_X(t)\, dt.\)

This extends the one‑dimensional inversion formula to higher dimensions.
The integral is over a $d$-dimensional cube.

2. Computing $P(X \in A)$ Using the Uniform-Subtraction Trick

Let
\(A = \prod_{i=1}^d [a_i, b_i]\) be a $d$-dimensional rectangle with Lebesgue volume $\lambda(A)$.

Let:

  • $U \sim \mathrm{Unif}(A)$ independent of $X$,
  • define $Y = X - U$.

Then: \(f_Y(0) = \frac{P(X \in A)}{\lambda(A)}.\)

Because $U$ has density
\(f_U(u) = \frac{1}{\lambda(A)} \quad \text{for } u \in A,\) its characteristic function factorizes: \(\varphi_U(t) = \prod_{k=1}^d \varphi_{U_k}(t_k).\)

The characteristic function of $Y$ is: \(\varphi_Y(t) = E[e^{i t \cdot (X-U)}] = \varphi_X(t)\, \varphi_U(-t).\)

Applying inversion: \(P(X \in A) = \lambda(A)(2\pi)^{-d} \lim_{T \to \infty} \int_{[-T,T]^d} \varphi_X(t)\, \varphi_U(-t)\, dt.\)

This works because $Y$ is bounded and thus integrable.

3. Independence and Characteristic Functions

Let $Z = (Z_1, \dots, Z_d)$.
Then the following are equivalent:

  • The components $Z_1,\dots,Z_d$ are independent.
  • The characteristic function factorizes: \(\varphi_Z(t) = \prod_{k=1}^d \varphi_{Z_k}(t_k).\)

This is one of the most powerful and useful characterizations of independence.

4. Example: A Joint Distribution That Is Not Independent

If a joint distribution table has marginals that look uniform but entries do not factor as
\(P(Z_1 = i, Z_2 = j) \neq P(Z_1 = i)P(Z_2=j),\) then independence fails.

Characteristic functions detect this automatically because the factorization will fail.

5. Weak Convergence in $\mathbb{R}^d$

Weak convergence (convergence in distribution) of random vectors: \(X_n \Rightarrow X\) means: \(E[f(X_n)] \to E[f(X)] \quad \forall f \in C_b(\mathbb{R}^d).\)

This condition is equivalent to several others.
These equivalences form the Portmanteau Theorem.

Portmanteau Theorem (Multivariate Version)

The following are equivalent:

  1. $X_n \Rightarrow X$.

  2. For every closed set $F \subset \mathbb{R}^d$, \(\limsup_{n \to \infty} P(X_n \in F) \le P(X \in F).\)

  3. For every open set $O \subset \mathbb{R}^d$, \(\liminf_{n \to \infty} P(X_n \in O) \ge P(X \in O).\)

  4. For every continuity set $A\subset \mathbb{R}^d$ satisfying
    \(P(X \in \partial A)=0,\) we have
    \(P(X_n \in A) \to P(X \in A).\)

Boundary of a set

\(\partial A = \overline{A} \setminus A^\circ.\)

This is exactly the set where convergence is “unstable”—the probability mass must not accumulate there.

6. Why the Quantile (Skorokhod) Representation Fails in $d > 1$

In one dimension:

  • You can take $U \sim {\rm Uniform}(0,1)$,
  • Define $Y_n = F_{X_n}^{-1}(U)$ and $Y = F_X^{-1}(U)$.

Then:

  • $Y_n \stackrel{d}{=} X_n$,
  • $Y \stackrel{d}{=} X$,
  • $Y_n \to Y$ almost surely.

This is because 1‑dimensional distributions are totally ordered.

In higher dimensions:

  • There is no multivariate quantile function with these monotonicity properties.
  • There is no canonical order on $\mathbb{R}^d$.
  • Thus the simple quantile construction cannot be extended.

Skorokhod’s Representation Theorem still works in $\mathbb{R}^d$, but it uses a much more sophisticated construction (not quantiles).

7. Key Takeaways

Characteristic Functions

  • $\varphi_X(t) = E[e^{i t\cdot X}]$ determines the law of $X$.
  • Independence ⇔ factorization.

Inversion

  • Recover density at 0 via Fourier inversion.
  • Compute $P(X \in A)$ using the uniform shift trick.

Convergence in Distribution

  • Portmanteau Theorem gives all equivalent formulations.
  • Continuity sets matter.
  • Quantile tricks fail in $\mathbb{R}^d$ because no natural order exists.

8. Suggested Items for Your Prelim Survival Guide

  • Independence via characteristic function factorization.
  • Uniform-subtraction trick for computing probabilities of rectangles.
  • Full multivariate Portmanteau theorem.
  • Statement of why 1D quantile coupling does not generalize.

If you want, I can add diagrams, examples, or a micro-cheat-sheet summarizing only the formulas you must memorize for exam speed.

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