Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 28 / 6

For $\epsilon >0$, the triangle inequality and the union bound giveThus $\boxed{X_n+Y_n⟶X+Y\text{ in probability}}$. This proves closure of convergence in probability under addition. No independence is needed; the random variables are compared on their given common probability space.

Take a random variable $Z$ with the Rademacher distribution, and setBoth marginal distributions equal the law of $Z$ for every $n$. Hence $X_n$ and $Y_n$ each have weak convergence of random variables to $Z$; one may take $X=Y=Z$ on this probability space. But their sums alternate:Indeed, at $t=\pi /2$ their characteristic functions are $-1$ for even $n$ and $1$ for odd $n$, so the sums have no weak limit at all. Separate marginal weak convergence does not imply convergence of the sum. The marginal weak convergence does not control sums counterexample changes the dependence while leaving the marginals fixed. Unlike convergence in probability, separate marginal weak convergence of random variables contains no information about the joint coupling.

Write the constant limit as $c$. First, convergence in distribution to a constant implies convergence in probability: the Portmanteau theorem applied to the closed set $\{y:|y-c|\ge \epsilon \}$ givesNow compare characteristic functions without imposing independence. For fixed $t\in \mathbb{R}$,The second bound uses $|e^{iu}-e^{iv}|\le \min \{2,|u-v|\}$. By convergence in probability, the limiting upper bound is $|t|\epsilon$; then let $\epsilon \downarrow 0$. Meanwhile $X_n$ has weak convergence of random variables to $X$, so $\mathbb{E}{e}^{it{X}_n}\to \mathbb{E}{e}^{itX}$. Thus the characteristic function of the sum tends to $e^{itc}\mathbb{E}{e}^{itX}$, the characteristic function of $X+c$, which is continuous at zero. The Lévy continuity theorem provesThis is the additive case of the Slutsky theorem, with its proof supplied. The constant limiting marginal removes the dependence obstruction in part (b).