If I make a JVM in Java, for example, is it possible to make the implementation I made actually faster than the original implementation I used to build this implementation, even though my implementation is built on top of the original implementation and may even be dependant on that implementation?
( Confusing… )
Look at PyPy. It’s a JIT Compiler for Python made in Python. That’s alright, but how can it claim to be faster than the original implementation of Python which it is using and is dependent on?
You are confused between a language and the execution apparatus for that language.
One of the reasons why PyPy can be faster than CPython is because PyPy is compiled to a completely separate native executable, and does not depend on, nor execute in, CPython.
Nevertheless, it would be possible for an inefficient implementation of one language to be surpassed by an interpreter written in that same language, and hosted in the inefficient interpreter, if the higher-level interpreter made use of more efficient execution strategies.