In this text we study, for positive random variables, the relation between the behaviour of the Laplace transform near infinity and the distribution near zero. A result of De Bruijn shows that E(e^-λX) _~ exp(rλ^α) for λ→∞ and P(X≤ε) _~ exp(s/ε^β) for ε↓0 are in some sense equivalent (for 1/α=1/β+1) and gives a relation between the constants r and s. We illustrate how this result can be used to obtain simple large deviation results. For use in more complex situations we also give a generalisation of De Bruijn's result to the case when the upper and lower limits are different from each other.

Tbilisi Mathematical Journal, Vol. 2 (2009), pp. 41-50