How to prove the security of the PRNG? Are there any realties tests or criterias that prove the security of the PRNG?
What kind of tests or criteria?
We currently have no way to prove that a specific PRNG is cryptographically secure.  In fact, we currently cannot prove that there exists a cryptographically secure PRNG (!).

If you scale back the requirement from "mathematical proof" to "something we generally accept", there's still no way for an automated test to verify that a specific output is cryptographically secure.  Yes, it can run a series of statistical tests, and verify that, from that perspective, it looks like what we expect a random stream to be.  However, that is not sufficient; to be cryptographically secure, we require that someone cannot devise a clever test (using knowledge of how the PRNG works internally) that distinguishes it from random.  The hard word (for an automated test) is the word "clever"; automated tests have great difficulties with "cleverness".

Here's a little thought expirement that demonstrates one of the subtleties; suppose I generate a PRNG using AES in counter mode; that is, I pick an initial 128-bit value $N$ and an AES key $k$, and produce the sequence:


  $AES_k(N), AES_k(N+1), AES_k(N+2), AES_k(N+3), ...$


If I pick $k$ and $N$ randomly, and keep them secret (and keep the number of outputs within the above sequence to significantly below $2^{64}$), then this is, as far as we know, a cryptographically secure PRNG.

However, suppose I do the same, but instead of picking $k$ randomly, I use a fixed public value.  In that case, the sequence is known not to be a CSPRNG; it is easy to test (using two adjacent outputs), whether the output is from this specific PRNG.

Here's the point: publishing $k$ or keeping it secret does not actually change the output.  However, even though it doesn't, it does change whether we have a CSPRNG or not.

So, going back to your question: we do have things that we believe are CSPRNG (such as the above AES counter mode with a secret key); why do we believe those?  Well, that's because we have clever people (that is, people that have demonstrated skill in finding clever ways to break other CSPRNGs) analyze the PRNGs.  It's not an ideal solution (it's possible that there is a way to break it, and our clever people missed it); however it's better than any other way we have.