On the supremum of the tails of normalized sums of independent Rademacher random variables (Q2344878)

Let \(\varepsilon = ({\varepsilon _1},{\varepsilon _2},\dots)\) be a sequence of independent identically distributed random variables, \(P({\varepsilon _1} = 1) = P({\varepsilon _1} = - 1) = \frac{1}{2}\). Put \(M(x): = \sup \{ P(a \cdot \varepsilon \geq x):a \in \Sigma \} \), where \(x \in R\), \(a \cdot \varepsilon = {a_1}{\varepsilon _1} + {a_2}{\varepsilon _2} + \dots\), \(\Sigma : = \{ a = ({a_1},{a_2},\dots) \in {R^n}: a_1^2 + a_2^2 + \dots = 1\} \) is the unit sphere in \({l^2}\). It follows from the central limit theorem that \(M(x) \geq \bar \Phi (x)\), where \(\bar \Phi (x)\) is the standard normal tail function. An upper bound for \(M(x)\) is equivalent for \(x \to \infty \) to \(\bar \Phi (x)\). This situation generated a conjecture that for all \(x\) \(M(x)\mathop = \limits^? {M^ = }(x)\), where \({M^ = }(x) = \sup_n P({e^{(n)}} \cdot \varepsilon \geq x)\) and \({e^{(n)}} = \left( {\frac{1}{{\sqrt n }},\dots,\frac{1}{{\sqrt n }},0,\dots,0\dots} \right) \in \Sigma \) . The paper shows that some of the sequences \({f^{(n,t)}}: = \left( {\sqrt {\frac{{1 - {t^2}}}{n}} ,\dots,\sqrt {\frac{{1 - {t^2}}}{n}} ,t,0,0,\dots} \right) \in \Sigma \) can be used to disprove this conjecture.