科研之友
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摘要
Fix any n >= 1. Let X(1),...,X(n) be independent random variables such that S(n)=X(1)+...+X(n), and let S(n)*=sup(1 <= k <= n)S(k). We construct upper and lower bounds for s(y) and s(y)*, the upper 1/yth quantiles of S(n) and S(n)*, respectively. Our approximations rely on a computable quantity Q(y) and an explicit universal constant gamma(y), the latter depending only on y, for which we prove that s(y) <= s(y)* <= Q(y) for y > 1, gamma 3y/16Q3y/16 - Q(1) <= s(y)* for y > 32/3, and gamma u(y) Q(u)(y) - 2Q(1) <= s(y) for y > 64/3, where u(y) = 3y/16(1+root 1-64/3y/2) and gamma(y) -> 1/3 as y -> infinity.
- 出版日期2010-12
