---
tag: 3HFP9ZQ580
kind: statement
role: theorem
name: Cramér–Wold theorem
msc: ["60E10", "60F05", 60B10]
uses: [0SRYJGXKR4, 0S52TNNTH9, 0V5B6D6BQH, 0FPWF7TTMB]
about: [0SRYJGXKR4]
sources:
  - url: https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Wold_theorem
same_as:
  wikidata: Q966837
  wikipedia: Cramér–Wold theorem
---

## Statement

Let $k \ge 1$, and let $X, X_1, X_2, \dots$ be random vectors in $\R^k$.
Then $X_n$ [[converges in distribution|0SRYJGXKR4]] to $X$ if and only if, for every $t \in \R^k$, the real random variables $t \cdot X_n$ converge in distribution to $t \cdot X$.
Here $t \cdot x$ is the [[dot product|0S52TNNTH9]] of $t$ and $x$.

## Remarks

Proof idea: the [[characteristic function|0V5B6D6BQH]] of $X_n$ at $t$ is $E[e^{i t \cdot X_n}]$, which is the characteristic function of $t \cdot X_n$ at $1$.
So the condition gives pointwise convergence of the characteristic functions of $X_n$, and [[Lévy's continuity theorem|0FPWF7TTMB]] concludes.
The converse holds because $x \mapsto t \cdot x$ is continuous.
A special case: the law of a random vector $X$ is determined by the laws of all its one-dimensional projections $t \cdot X$.
Convergence of the coordinates alone is not enough.
If $Z$ is standard normal, then $(Z, Z)$ and $(Z, -Z)$ have the same marginals but different laws.
So the constant sequence $(Z, Z)$ has coordinates that converge in law to those of $(Z, -Z)$, but it does not converge in law to $(Z, -Z)$.

## Explanations

### L1 human

To check that random vectors settle down to a limit law, it is enough to look at every straight-line projection of them.
This reduces questions about many dimensions to questions about one dimension.

### L2 human

The standard use is the multivariate [[central limit theorem|0EPZC4Z1FC]]: for i.i.d. random vectors with finite covariance matrix, each projection $t \cdot X_i$ obeys the one-dimensional theorem, and the Cramér–Wold theorem lifts this to a [[multivariate normal|0XN3K7RSZY]] limit.
It is also called the Cramér–Wold device.