Abstract
Let $A$ be a $C^2$ curve of length $L(A)$ in some Euclidean space. Let $P_n$ be a sequence of randomly chosen polygons with $n$ vertices which are inscribed in $A$. It is shown that with probability 1 $\lim n^2\lbrack L(A) - L(P_n)\rbrack = \frac{1}{4} \int_A \kappa^2(s) ds$ where $\kappa$ is the curvature.
Citation
Vladimir Drobot. "Probabilistic Version of a Curvature Formula." Ann. Probab. 10 (3) 860 - 862, August, 1982. https://doi.org/10.1214/aop/1176993798
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