Weak convergence of sum implies convergence of summands Suppose $X_n$ and $Y_n$ are independent, $X_n \overset d \to X$ and $X_n + Y_n \overset d \to Z$ for some random variable $Z$.
Is it necessary, that $Y_n$ converges in distribution?

I tired using characteristic functions and Levy-Cramer theorem, but I'm missing one thing. Here is what I have:
$$
\forall t \in \mathbb R \quad
\varphi_{X_n}(t) \to \varphi_X(t), \quad \varphi_{X_n}(t)\varphi_{Y_n}(t) \to \varphi_Z(t)
$$
Since $\varphi_{Y_n}$ is bounded, it must converge for all $t$ such that $\varphi_{X}(t) \neq 0$. For those $t$ we would have
$$
\varphi_{Y_n}(t) \to \frac {\varphi_Z(t)}{\varphi_{X}(t)}
$$
Which is continuous at $0$ and equal to $1$ there. If $\varphi_{Y_n}$ converged in every point, then $Y_n$ would have to converge in distribution.
Can I show that $\varphi_{Y_n}$ converges everywhere? Or in other way show that $Y_n$ converges in distribution? Counterexamples are welcome too.
 A: To provide a counterexample, the following Proposition(c.f. Y. S. Chow & H. Teicher,
Probability Theory, 3rdEd, Springer Verlag, 1997, Prop. 8.4.3, p.299, or
K. L. Chung,  A Course in ProbabilityTheory, 3rdEd, Academic Press, 2003. Th.6.5.3, p.191.)
is useful:
Theorem(Pólya): A nonnegative, even function $\psi$ convex and decreasing on $ (0,\infty) $ with $ \psi(0+)=\psi(0)=1 $ is a c.f.(characteristic function).
From Pólya's Theorem, the following $ \psi $'s are c.f., for $ t\in\mathbb{R} $
\begin{align*}
 \psi_1(t)&=(1-|t|)^+,\\
 \psi_2(t)&=\Big(1-\frac{|t|}{2}\Big)^+,\\
 \psi_3(t)&=\Big(1-\frac{|t|}{2}\Big)^+\vee \frac12.
\end{align*}
Now define the distibutions of $ X_n, Y_n $ by c.f. as following:
\begin{align*}
 \phi_{X_n}(t)&=\psi_1(t),\\
 \phi_{Y_n}(t)&=\begin{cases}
  \psi_2(t), \quad & n=2k,\\ \psi_3(t), & n=2k-1.
 \end{cases}
\end{align*}
Then
\begin{align*}
 \phi_{X_n+Y_n}(t)&=\phi_{X_n}(t)\phi_{Y_n}(t)\\
                     &=\psi_1(t)\psi_2(t).
\end{align*}
Hence $ X_n+Y_n\overset d\to Z $, but the sequence of distributions of  $\{Y_{2k-1}, k\ge1\}$ and sequence of distributions of $\{Y_{2k}, k\ge 1\}$ have different limit.
