Stochastic equicontinuity

Let ${\displaystyle \{H_{n}(\theta ):n\geq 1\}}$ be a family of random functions defined from ${\displaystyle \Theta \rightarrow \mathbb {R} }$, where ${\displaystyle \Theta }$ is any normed metric space. Here ${\displaystyle \{H_{n}(\theta )\}}$ might represent a sequence of estimators applied to datasets of size n, given that the data arises from a population for which the parameter indexing the statistical model for the data is θ. The randomness of the functions arises from the data generating process under which a set of observed data is considered to be a realisation of a probabilistic or statistical model. However, in ${\displaystyle \{H_{n}(\theta )\}}$, θ relates to the model currently being postulated or fitted rather than to an underlying model which is supposed to represent the mechanism generating the data. Then ${\displaystyle \{H_{n}\}}$ is stochastically equicontinuous if, for every ${\displaystyle \epsilon >0}$ and ${\displaystyle \eta >0}$, there is a ${\displaystyle \delta >0}$ such that:

${\displaystyle \limsup _{n\rightarrow \infty }\Pr \left(\sup _{\theta \in \Theta }\sup _{\theta '\in B(\theta ,\delta )}|H_{n}(\theta ')-H_{n}(\theta )|>\epsilon \right)<\eta .}$

Here B(θ, δ) represents a ball in the parameter space, centred at θ and whose radius depends on δ.

• Pollard, David (1984). "Stochastic Equicontinuity". Convergence of Stochastic Processes. New York: Springer. pp. 138–142. ISBN 0-387-90990-7.