  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.
