A new method to prove strassen type laws of invariance principle. 1

SummaryA new method is developed to produce strong laws of invariance principle without making use of the Skorohod representation. As an example, it will be proved that $${{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} \mathord{\left/ {\vphantom {{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }}} \right. \kern-\nulldelimiterspace} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }} = 0$$ with probability 1, for any g3>0, where Sn=X1 + ... +Xn, Xi is a sequence of i.i.d.r.v.'s with P(Xi<t)=F(t), and F(t) is a distribution function obeying (i), (ii) and W(n) is a suitable Wiener-process. Strassen in [1], proved (under weaker conditions): $$S_n - W\left( n \right) = O\left( {\sqrt[4]{{n{\text{ log log }}n}}\sqrt {{\text{log }}n} {\text{ }}} \right)$$ with probability one. He conjectured that if $$S_n - W\left( n \right) = o\left( {\sqrt[4]{{n{\text{ log log }}n}}\sqrt {{\text{log }}n} {\text{ }}} \right)$$ then F(x) = Φ(x) where Φ(.) is the unit normal distribution function. (See also [2], [6] and [7].) Our result above is a negative answer to this question.