A version of the strong law of large numbers universal under mappings (Q2702785)

Let \((\Omega,\mathcal A, P)\) stand for some probability space, \(\Theta \) for a separable topological space, and \((Y,\mathcal Y)\) for a measurable space. Furthermore, \(f\:Y\times \Theta \to \mathbb R\) is some function such that \(f_\theta\) is \(\mathcal Y\)-measurable for all \(\theta\in\Theta \) and \(f_y\), \(y\in\mathcal Y\), is pointwise equicontinuous. It is proved that for any sequence \(X_1,X_2,\dots \) of \(Y\)-valued random variables, which is i.i.d. relative to \(P\) such that \(E\bigl (|f(X_1,\theta)|\bigr) < \infty \) is valid for any \(\theta \in \Theta \), there exists some \(P\)-zero set \(N\) satisfying \(\frac 1{n}\sum_{i=1}^n f\bigl (X_i(\omega), \theta \bigr) \to E\bigl (f(X_1,\theta)\bigr)\), \(\omega \in \Omega \setminus N\), for all \(\theta\in\Theta\). This result is illustrated by examples and compared with known uniform versions of the SLLN.