Cycles of length $k$ are classified 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.
Cycles of length $k$ are classified 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.