  We investigate the stability of a Sequential Monte Carlo (SMC) method applied
to the problem of sampling from a target distribution on $\mathbb{R}^d$ for
large $d$. It is well known that using a single importance sampling step one
produces an approximation for the target that deteriorates as the dimension $d$
increases, unless the number of Monte Carlo samples $N$ increases at an
exponential rate in $d$. We show that this degeneracy can be avoided by
introducing a sequence of artificial targets, starting from a `simple' density
and moving to the one of interest, using an SMC method to sample from the
sequence. Using this class of SMC methods with a fixed number of samples, one
can produce an approximation for which the effective sample size (ESS)
converges to a random variable $\varepsilon_N$ as $d\rightarrow\infty$ with
$1<\varepsilon_{N}<N$. The convergence is achieved with a computational cost
proportional to $Nd^2$. If $\varepsilon_N\ll N$, we can raise its value by
introducing a number of resampling steps, say $m$ (where $m$ is independent of
$d$). In this case, ESS converges to a random variable $\varepsilon_{N,m}$ as
$d\rightarrow\infty$ and $\lim_{m\to\infty}\varepsilon_{N,m}=N$. Also, we show
that the Monte Carlo error for estimating a fixed dimensional marginal
expectation is of order $\frac{1}{\sqrt{N}}$ uniformly in $d$. The results
imply that, in high dimensions, SMC algorithms can efficiently control the
variability of the importance sampling weights and estimate fixed dimensional
marginals at a cost which is less than exponential in $d$ and indicate that, in
high dimensions, resampling leads to a reduction in the Monte Carlo error and
increase in the ESS.
