This is a continuation of the previous post available here. This section is quite different from the previous post. Here we present a series of propositions. The proofs of the propositions are not present here. I may include them later. However the proofs of the propositions (except proposition 4) are not necessary in our proof.
In the previous post, we introduced Hermite polynomials and wave functions. We noticed the Hermite polynomials are orthogonal polynomials. There are a plethora of results involving orthonormal polynomials. We merely state only few of them that we will be needing in the our proof.
Proposition 1: Let be the Hermite polynomial of degree . Let be the corresponding wave function. Let . We have the following results.
Proof: Not included
Let us recall Theorem 3 once again.
Theorem 3: For any measurable subset of ,
Let us replace by and rewrite the above result as
Now it is right time to see what exactly are we trying to show if we want to prove weak convergence. For that we need some basic idea about Airy function.