Cramér–Wold theorem

Remarks

Proof idea: the characteristic function 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 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 $(Z, -Z)$.

Going deeper

For a student

The standard use is the multivariate central limit theorem: 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 limit. It is also called the Cramér–Wold device.