Large deviations for trajectories of sums of independent random variables
Suppose \(S_ 0 = 0\) and for \(k = 1,2, \dots ,\) let \(S_ k\) be the partial sums of i.i.d. random variables \(Y_ 1, Y_ 2 \dots\); define \(\widetilde{S} : [0,\infty) \to \mathbb{R}\) as \(\widetilde{S}(k) = S_ k\), \(\forall k \in \mathbb{N}\), with \(\widetilde{S}\) being linear on intervals \([k - 1, k]\). The trajectories \(\widetilde{S}_ n\) are defined as \(\widetilde{S}_ n(t) := \widetilde{S}(nt)\), \(t \in [0,1]\). Under fairly general conditions, there exists a deterministic function \(\widetilde {m}\) such that \(\sup_{t \in [0,1]} | {1\over n} \widetilde{S}_ n - \widetilde{m} | \to 0\) in probability. The author studies the large sample principle associated with the above when the component random variables \(Y_ i\) are independent, but not necessarily identically distributed.
scientific article; zbMATH DE number 501129
scientific article