ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary In deriving his strong invariance principles, Strassen used a construction of Skorokhod: if the univariate d. f. F has first, second, and fourth moments 0, 1, and Β〈t8, respectively, then there is a probability space on which are defined a standard Brownian motion {ξ(t), t≧0} and a sequence of nonnegative i.i.d. Skorokhod random variables {T i i〉0} such that $$\left\{ {\xi \left( {\sum\limits_1^{n + 1} {T_i } } \right) - \xi \left( {\sum\limits_1^n {T_i } } \right),n \geqq 0} \right\}$$ are i. i. d. with d. f. F. Let $$Z = \mathop {\lim \sup }\limits_{n \to \infty } \pm {{\left[ {\xi \left( {\sum\limits_1^n {T_i } } \right) - \xi (n)} \right]} \mathord{\left/ {\vphantom {{\left[ {\xi \left( {\sum\limits_1^n {T_i } } \right) - \xi (n)} \right]} {[n(\log n)^2 \log \log n]^{\tfrac{1}{4}} }}} \right. \kern-\nulldelimiterspace} {[n(\log n)^2 \log \log n]^{\tfrac{1}{4}} }}$$ Strassen showed Z=O(1) wp 1. We prove Z=(2Β)1/4 wp 1. Consequently Z=0 wp 1 implies F is Gaussian, answering a special case of a question of Strassen. Analogous results hold for cases where $$\xi \left( {n\sum\limits_1^n {T_i } } \right)$$ is not a sum of independent random variables.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00539208
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