Logarithmic asymptotics for multidimensional extremes under non-linear   scalings
  Let $\boldsymbol W=\{\boldsymbol W_n:n\in\mathbb N\}$ be a sequence of random vectors in $\mathbb R^d$, $d\ge 1$. This paper considers the logarithmic asymptotics of the extremes of $\boldsymbol W$, that is, for any vector $\boldsymbol q>\boldsymbol 0$ in $\mathbb R^d$, we find $$\log\mathbb P\left(\exists{n\in\mathbb N}:\boldsymbol W_n> u \boldsymbol q\right), \quad\text{as} u\to\infty.$$ We follow the approach of the restricted large deviation principle introduced in Duffy et al. \textit{Logarithmic asymptotics for the supremum of a stochastic process} (Ann. Appl. Probab., 13:430--445, 2003). That is, we assume that, for every $\boldsymbol q\ge\boldsymbol 0$, and some scalings $\{a_n\},\{v_n\}$, $\frac{1}{v_n}\log\mathbb P\left(\boldsymbol W_n/a_n\ge u \boldsymbol q\right)$ has a, continuous in $\boldsymbol q$, limit $J_{\boldsymbol W}(\boldsymbol q)$. We allow the scalings $\{a_n\}$ and $\{v_n\}$ to be regularly varying with a positive index. This approach is general enough to incorporate sequences $\boldsymbol W$, such that the probability law of $\boldsymbol W_n/a_n$ satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The formula for these asymptotics agrees with the seminal papers on this topic. 
Kamil Marcin Kosinski, Michel Mandjes
math.PR