In a non-parametric framework, we establish some non-asymptotic bounds for self-normalized sums and quadratic forms in the multivariate case for symmetric and general random variables. This bounds are entirely explicit and essentially depends in the general case on the kurtosis of the Euclidean norm of the standardized random variables.
"Exponential bounds for multivariate self-normalized sums." Electron. Commun. Probab. 13 628 - 640, 2008. https://doi.org/10.1214/ECP.v13-1430