  We consider the problem of estimating an expectation $ \mathbb{E}\left[
h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded
smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed
random variable. To study rates of convergence for QMC on unbounded integrands,
we use a smoothed projection operator to project the output of $W$ to a bounded
region, which differs from the strategy of avoiding the singularities along the
boundary of the unit cube $ [0,1]^d $ in 10.1137/S0036144504441573. The error
is then bounded by the quadrature error of the transformed integrand and the
projection error. If the function $h(\boldsymbol{x})$ and its mixed partial
derivatives do not grow too fast as the Euclidean norm $|\boldsymbol{x}|$ goes
to infinity, we obtain an error rate of $O(n^{-1+\epsilon})$ for QMC and
randomized QMC (RQMC) with a sample size $n$ and an arbitrarily small
$\epsilon>0$. However, the rate turns out to be $O(n^{-1+2M+\epsilon})$ if the
functions grow exponentially with a rate of $O(\exp\{M|\boldsymbol{x}|^2\})$
for a constant $M\in(0,1/2)$. Superisingly, we find that using importance
sampling with t distribution as the proposal can improve the root mean squared
error of RQMC from $O(n^{-1+2M+\epsilon})$ to $O( n^{-3/2+\epsilon})$ for any
$M\in(0,1/2)$.
