  We derive convenient uniform concentration bounds and finite sample
multivariate normal approximation results for quadratic forms, then describe
some applications involving variance components estimation in linear
random-effects models. Random-effects models and variance components estimation
are classical topics in statistics, with a corresponding well-established
asymptotic theory. However, our finite sample results for quadratic forms
provide additional flexibility for easily analyzing random-effects models in
non-standard settings, which are becoming more important in modern applications
(e.g. genomics). For instance, in addition to deriving novel non-asymptotic
bounds for variance components estimators in classical linear random-effects
models, we provide a concentration bound for variance components estimators in
linear models with correlated random-effects. Our general concentration bound
is a uniform version of the Hanson-Wright inequality. The main normal
approximation result in the paper is derived using Reinert and R\"{o}llin's
(2009) embedding technique and multivariate Stein's method with exchangeable
pairs.
