We classify cycles of length $k$ into three subsets:
the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.
What can be said about $\mathcal{B}_{k}$ if $k$ is odd? Deduce that $\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = 0$ in this case.
We classify cycles of length $k$ into three subsets:
\begin{itemize}
\item the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
\item the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
\item the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.
\end{itemize}
What can be said about $\mathcal{B}_{k}$ if $k$ is odd? Deduce that $\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = 0$ in this case.