= Solution
Write $A_k=N_{2k}-N_k$ and $B_k=N_k$. The <stationary increments> and <independent increments> of the <Poisson process> make $A_k$ and $B_k$ <independent random variables>, each with <Poisson distribution> of parameter $k$. Thus
$$
\frac{N_{2k}-2N_k}{\sqrt{k}}=\frac{A_k-B_k}{\sqrt{k}}.
$$
Multiplication of the two <moment-generating functions> gives
$$
M_k(t)=\exp\!\left\{k\left(e^{t/\sqrt{k}}+e^{-t/\sqrt{k}}-2\right)\right\}.
$$
The odd terms cancel in <Taylor's theorem>, so $\log M_k(t)=t^2+O(k^{-1})$. The limit $e^{t^2}$ is the <moment-generating function of a normal distribution> with <mean> zero and <variance> two. The same <continuity theorem for moment-generating functions> therefore yields
$$
\boxed{\frac{N_{2k}-2N_k}{\sqrt{k}}\xrightarrow{\mathrm d}N(0,2).}
$$
In particular, the two equal-length increments must be separated before using <independence>: $N_{2k}$ and $N_k$ themselves are dependent.
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