  In this paper we construct high moment partial sum processes based on
residuals of a GARCH model when the mean is known to be 0. We consider partial
sums of $k$th powers of residuals, CUSUM processes and self-normalized partial
sum processes. The $k$th power partial sum process converges to a Brownian
process plus a correction term, where the correction term depends on the $k$th
moment $\mu_k$ of the innovation sequence. If $\mu_k=0$, then the correction
term is 0 and, thus, the $k$th power partial sum process converges weakly to
the same Gaussian process as does the $k$th power partial sum of the i.i.d.
innovations sequence. In particular, since $\mu_1=0$, this holds for the first
moment partial sum process, but fails for the second moment partial sum
process. We also consider the CUSUM and the self-normalized processes, that is,
standardized by the residual sample variance. These behave as if the residuals
were asymptotically i.i.d. We also study the joint distribution of the $k$th
and $(k+1)$st self-normalized partial sum processes. Applications to
change-point problems and goodness-of-fit are considered, in particular, CUSUM
statistics for testing GARCH model structure change and the Jarque--Bera
omnibus statistic for testing normality of the unobservable innovation
distribution of a GARCH model. The use of residuals for constructing a kernel
density function estimation of the innovation distribution is discussed.
