Summary Let X n, n?N d be a field of independent random variables taking values in a semi-normed measurable vector spaceF. For a broad class of fields an, ? Nd of positive numbers, the almost sure behaviour of Sk?nXk/an, n ? Nd is studied. The main result allows us to deduce some new and well-known theorems for fields of independentF random variables from related results for fields of independent real random variables.
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