Let $a_1, \ldots, a_n \in \mathbb{R}$ satisfy $\sum_i a_i^2 = 1$, and let $\varepsilon_1, \varepsilon_n$ be uniformly random $\pm 1$ signs $X \sum_{i=1}^{n} a_i \varepsilon_i$. It is conjectured that \varepsilon_i$ has $\Pr[X \geq 1] 7/64$. The best lower bound so far $1/20$, due to Oleszkiewicz. In this paper we improve 6/64$.

independence of observations is a fundamental assumption in designing an x chart for individual observations. but unfortunately, the assumption of existence of independent data is not even approximately satisfied. in fact, there is one kind of autocorrelation in the data. this paper demonstrates, through simulating an ar(1) process, that the autocorrelation in the data has deep effect on x char...