Weighted Persistent Homology Sums of Random \v{C}ech Complexes

We study the asymptotic behavior of random variables of the form \begin{equation*} E_{\alpha}^i\left(x_1,\ldots,x_n\right)=\sum_{\left(b,d\right)\in \mathit{PH}_i\left(x_1,\ldots,x_n\right)} \left(d-b\right)^{\alpha} \end{equation*} where $\left\{x_j\right\}_{j\in\mathbb{N}}$ are i.i.d. samples from a probability measure on a triangulable metric space, and $\textit{PH}_i\left(x_1,\ldots,x_n\right)$ denotes the $i$-dimensional reduced persistent homology of the \v{C}ech complex of $\left\{x_1,\ldots,x_n\right\}.$ These quantities are a higher-dimensional generalization of the $\alpha$-weighted sum of a minimal spanning tree; we seek to prove analogues of the theorems of Steele (1988) and Aldous and Steele (1992) in this context. As a special case of our main theorem, we show that if $\left\{x_j\right\}_{j\in\mathbb{N}}$ are distributed independently and uniformly on the $m$-dimensional Euclidean sphere, $\alpha<m,$ and $0\leq i <n,$ then there are real numbers $\gamma$ and $\Gamma$ so that \begin{equation*} \gamma \leq \lim_{n\rightarrow\infty} n^{-\frac{m-\alpha}{m}} E_i^{\alpha}\left(x_1,\ldots,x_n\right) \leq \Gamma \end{equation*} in probability. More generally, we prove results about the asymptotics of the expectation of $E_\alpha^i$ for points sampled from a locally bounded probability measure on a space that is the bi-Lipschitz image of an $m-$dimensional Euclidean simplicial complex, as well as measures supported on sets of fractional dimension that respect box counting.