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- W204514329 abstract "1. Introduction. Let {X n n=1 ∞} be a sequence of independent symmetric random variables defined on the probability space (Ω,ℑ, P) and taking values in the real separable Banach space (B,∥ • ∥). In this note, we will investigate the validity of the bounded law of the iterated logarithm, in both the usual (BLIL) and self-normalized (SNBLIL) forms, for such variables, where the normalizer will often depend on the type of the Banach space in question. One of our goals will be to drop the standard boundedness assumption that has been made in such a situation ever since Kolmogorov’s fundamental LIL was proved; see, for example, (1977) and (1991). This will not be possible without self-normalization in the absence of an additional real-valued almost sure boundedness condition (Corollaries 1 and 2). Our main result (Theorem 2) also yields a characterization of a class of Banach spaces in a manner similar to that in (1976), (1975-76): A Banach space B is of “(LLn)p-1 — type p” if and only if each independent symmetric B-valued sequence X n n-1 ∞ satisfying a mild real-valued almost sure boundedness condition verifies a SNBLIL." @default.
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- W204514329 date "1992-01-01" @default.
- W204514329 modified "2023-09-25" @default.
- W204514329 title "Self-Normalized Bounded Laws of the Iterated Logarithm in Banach Spaces" @default.
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- W204514329 doi "https://doi.org/10.1007/978-1-4612-0367-4_20" @default.
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