## Upper and Lower Bounds in Exponential Tauberian Theorems## Jochen Voss
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 |