Fix any
\(n\ge 1\). Let
\(\tilde{X}_1,\ldots ,\tilde{X}_n\) be independent random variables. For each
\(1\le j \le n\),
\(\tilde{X}_j\) is transformed in a canonical manner into a random variable
\(X_j\). The
\(X_j\) inherit independence from the
\(\tilde{X}_j\). Let
\(s_y\) and
\(s_y^*\) denote the upper
\(\frac{1}{y}{\underline{\text{ th }}}\) quantile of
\(S_n=\sum _{j=1}^nX_j\) and
\(S^*_n=\sup _{1\le k\le n}S_k\), respectively. We construct a computable quantity
\(\underline{Q}_y\) based on the marginal distributions of
\(X_1,\ldots ,X_n\) to produce upper and lower bounds for
\(s_y\) and
\(s_y^*\). We prove that for
\(y\ge 8\) $$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$
where
$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$
and
\(w_y\) is the unique solution of
$$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$
for
\(w_y>\ln (\frac{y}{y-2})\), and for
\(y\ge 37\) $$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$
where
$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$
The distribution of
\(S_n\) is approximately centered around zero in that
\(P(S_n\ge 0) \ge \frac{1}{18}\) and
\(P(S_n\le 0)\ge \frac{1}{65}\). The results extend to
\(n=\infty \) if and only if for some (hence all)
\(a>0\) $$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$
(1)