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Not all JREs are - Jikes RVM for example is implemented in Java and yet has decent performance -

http://jikesrvm.org/FAQ#FAQ-GeneralWhatIs

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The speed-critical parts (like HotSpot) are most certainly written in C++.

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From TFA I linked -

Jikes RVM is unique in that it is the first self-bootstrapped virtual machine written entirely in the Java programming language, i.e., its Java code runs on itself, without requiring a second virtual machine.