The $\chi^2$ distribution describes the sum of squares of independent normal random variables. Is there an analogous distribution for the discrete case of sum of squares of independent (identical) binomial random variables?
I'm particularly interested in concentration bounds on the resulting sum. I know how to compute the expectation (and I suppose Chernoff bound from there), but is there anything stronger or more explicit to be said? I know that the binomial distribution approximates a normal distribution---does this imply that the sum of squares of binomials also approximates a $\chi^2$ distribution?
Thanks in advance.