An almost sure invariance principle for trimmed sums of random vectors. (English) Zbl 1271.60051
Summary: Let \(\{X_n;n\geq 1\}\) be a sequence of independent and identically distributed random vectors in \(\mathbb{R}^p\) with Euclidean norm \(|\cdot|\), and let \(X_n^{(r)}=X_m\) if \(| X_m|\) is the \(r\)-th maximum of \(\{| X_k|;k\leq n\}\). Define \(S_n=\sum_{k\leq n}X_k\) and \(^{(r)}S_n=S_n(X_n^{(1)}+\dotsb+X_n^{(r)})\).
In this paper, a generalized strong invariance principle for the trimmed sums \(^{(r)}S_n\) is derived.
In this paper, a generalized strong invariance principle for the trimmed sums \(^{(r)}S_n\) is derived.
MSC:
| 60F17 | Functional limit theorems; invariance principles |
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