  We study the asymptotic behavior of a multidimensional random walk in a
general cone. We find the tail asymptotics for the exit time and prove integral
and local limit theorems for a random walk conditioned to stay in a cone. The
main step in the proof consists in constructing a positive harmonic function
for our random walk under minimal moment restrictions on the increments. For
the proof of tail asymptotics and integral limit theorems, we use a strong
approximation of random walks by Brownian motion. For the proof of local limit
theorems, we suggest a rather simple approach, which combines integral theorems
for random walks in cones with classical local theorems for unrestricted random
walks. We also discuss some possible applications of our results to ordered
random walks and lattice path enumeration.
