Heavy traffic and heavy tails for the maximum of a random walk

Consider a family of random walks $S_n^{(a)}=X_1^{(a)}+\cdots+X_n^{(a)}$ with negative drift $\mathbf E X_1^{(a)}=-a<0$ and finite variance $\mbox{var}(X_1^{(a)})=\sigma^2<\infty$.Let $M^{(a)}=\max_{n\ge 0} S_n^{(a)}$ be the maximums of the random walks. The exponential asymptotics $\mathbf P(aM^{(a)}>x)\sim e^{-2x/\sigma^2}$, as $a\to 0$, were found by Kingman and are known as heavy traffic approximation in the queueing theory. For subexponential random variables the large deviation asymptotics for $\mathbf P(M^{(a)}>x)\sim \frac{1}{a}\overline F^I(x)$ hold for fixed $a$ as $x\to\infty$. In this paper we present asymptotics for $\mathbf P(M^{(a)}>x)$, which hold uniformly on the whole positive axis, as $a\to 0$. Thus, these uniform asymptotics include both the regime of normal and large deviations. We identify the regions where exponential or subexponential asymptotics hold. Our approach is based on construction of corresponding super/sub - martingales to obtain sharp upper and lower bounds.