  We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser.
Fiz.--Mat. Nauk 6 (1969) 17--22, Theory Probab. Appl. 14 (1969) 51--64,
193--208] on large deviations for sums of i.i.d. regularly varying random
variables to partial sum processes of i.i.d. regularly varying vectors. The
results are stated in terms of a heavy-tailed large deviation principle on the
space of c\`{a}dl\`{a}g functions. We illustrate how these results can be
applied to functionals of the partial sum process, including ruin probabilities
for multivariate random walks and long strange segments. These results make
precise the idea of heavy-tailed large deviation heuristics: in an asymptotic
sense, only the largest step contributes to the extremal behavior of a
multivariate random walk.
