Reference request: Strong Law of Large Numbers for V-statistics

Question

I'm requesting a reference for a Strong Law of Large Numbers theorem for V-statistics (similar to Hoeffding's 1961 paper for U-statistics). That is, I am searching for an almost sure convergence theorem, saying $$ V_n = \frac{1}{n^k}\sum_{i_1=1}^n \cdots \sum_{i_k=1}^n h\left(X_{i_1},...,X_{i_k}\right) \stackrel{\mbox{a.s.}}{\longrightarrow}_n \quad? $$ where $(X_n)$ is an i.i.d. sequence of random elements with values in some space $\mathcal{X}$ and $h:\mathcal{X}^k \to \mathbb{R}$ is a symmetric kernel with some moment requirements.

If Hoeffdings SSLN for U-statistics can be used to derive this result, an explanation of how this is done, would also more than suffice.

Answer 1

I found four references:

1. Giné, Evarist, and Joel Zinn. "Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics." Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Birkhäuser Boston, 1992.
2. Kondo, Masao, and Hajime Yamato. "Almost sure convergence of a linear combination of U-statistics." Scientiae Mathematicae japonicae 55.3 (2002): 605-613.
3. Borovskikh, Yuri Vasilevich. U-statistics in Banach Spaces. VSP, 1996.
4. Korolyuk, Vladimir S., and Yuri V. Borovskich. Theory of U-statistics. Vol. 273. Springer Science & Business Media, 2013.