Sharp asymptotics for the Laplace transform of the compound renewal process and related problems

Sharp asymptotics for the Laplace transform of the compound renewal process (CRP) are found under the Cramer moment condition on the jumps of the process. This result allowed us to obtain asymptotic inequalities for the distribution of the maximum value of the CRP on increasing time intervals and also to find the asymptotic behavior of all the moments of the CRP. The asimptotics of the two first moments of CRP are found under conditions close to minimal ones.