Weak Consistency

The quantile function of a cumulative distribution function is the generalized inverse given by

It is a left-continuous function with range equal to the support of F and hence is often unbounded. The following lemma records some useful properties.

Proof. The proofs of the inequalities in (i) through (iv) are best given by a picture. The equalities (v) follow from (ii) and (iv) and the monotonicity of by (iv). This proves the first statement in (ii); the second is immediate from the inequalities in (ii) and (iii). Statement (vi) follows from (i) and the definition of Consequences of (ii) and (iv) are that is strictly increasing i.e., has no. Thusis a proper inverse if and only if F is both continuous and strictly increasing, as one would expect.

By (i) the random variable has distribution function is uniformly distributed on [0, 1]. This is called the quantile transformation. On the other hand, by (i) and (ii) the variable is uniformly distributed on [0, 1] if and only if X has a continuous distribution function This is called the probability integral transformation.

A sequence of quantile functions is defined to converge weakly to a limit quantile function, denoted is continuous. This type of convergence is not only analogous in form to the weak convergence of distribution functions, it is the same.