Convergence in distribution for nonnegative R.V.s

Question

Problem. Let $Z, Z_1, Z_2, \cdots, Z_n$ be non-negative R.V.s with integer values. Prove that $Z_n \to Z$ in distribution, i.f.f $Pr(Z_n=t) \to Pr(Z=t)$ for all non-negative $t \in N$.

I have read about the Portmanteau theorem which states that if $ E_n f \to Ef$, then $P_n \to P$, P being the probability measure. Does this theorem apply in this case? (I was thinking convergence in distribution implies convergence in expectation.) thanks in advance.

Answer 1

Indeed you can utilize the Portnamteau theorem. Let me prove that

• $\mathsf{P}[Z_n = t] \to \mathsf{P}[Z = t]$ for all $t \in \mathbb{N}_0$ implies $Z_n \stackrel{d}{\to} Z$.

To this end, we prove the following simple lemma:

Lemma. If $p_n, q_n \geq 0$ and $\sum_n p_n = \sum_n q_n < \infty$, then $$ \sum_n \left| p_n - q_n \right| = 2\sum_n (p_n - q_n)_+, $$ where $x_+ = \max\{x, 0\}$.

Indeed, by using the identity$|x| = 2x_+ - x$, we have

$$ \sum_{t=0}^{\infty} \left| p_n - q_n \right| = \sum_{t=0}^{\infty} 2\left( p_n - q_n \right)_+ - \underbrace{\sum_{t=0}^{\infty} \left( p_n - q_n \right)}_{=0}. $$

Now we apply this lemma to our problem. Let $f$ be an arbitrary bounded continuous function on $\mathbb{R}$. Then with $\|f\|_{\sup} := \sup\{ |f(x)| : x \in \mathbb{R} \}$, we have

\begin{align*} \left| \mathsf{E}[f(Z)] - \mathsf{E}[f(Z_n)] \right| &= \left| \sum_{t=0}^{\infty} f(t) \left( \mathsf{P}[Z = t] - \mathsf{P}[Z_n = t] \right) \right| \\ &\leq \| f\|_{\sup} \sum_{t=0}^{\infty} \left| \mathsf{P}[Z = t] - \mathsf{P}[Z_n = t] \right| \\ &= 2 \| f\|_{\sup} \sum_{t=0}^{\infty} \left( \mathsf{P}[Z = t] - \mathsf{P}[Z_n = t] \right)_+ . \end{align*}

Now the summand, as function of $t \in \mathbb{N}_0$, is dominated by the integrable function $t \mapsto \mathsf{P}[Z = t]$ (with respect to the counting measure on $\mathbb{N}_0$, of course) and hence by the dominated convergence theorem,

$$ \lim_{n\to\infty} \sum_{t=0}^{\infty} \left( \mathsf{P}[Z = t] - \mathsf{P}[Z_n = t] \right)_+ = \sum_{t=0}^{\infty} \lim_{n\to\infty} \left( \mathsf{P}[Z = t] - \mathsf{P}[Z_n = t] \right)_+ = 0 $$

by the assumption. Therefore $\mathsf{E}[f(Z_n)] \to \mathsf{E}[f(Z)]$ as required, and since this is true for any bounded continuous $f$ on $\mathbb{R}$, the conclusion follows from the Portmanteau theorem.