Please note that the explanation you are giving isn't accurate, but "is oversimplified to the point of being totally wrong" (Ethan Siegel). Although Hawking himself gave this explanation in "A brief history of time". But he knew better then.

https://medium.com/starts-with-a-bang/ask-ethan-how-do-black-holes-actually-evaporate-1f195c289aa1

As for "being wrong": the explanation, Hawking and you gave, would make one expect that:

- Hawking radiation is made of particles (it is not, it consists of photons)

- Hawking radiation originates at the event horizon (it does not, it does orignate over a volume as big as ten Schwarzschild radii of the black hole)

As you can see from Siegels article (and the Wikipedia article on Hawking radiaiton) the real cause of the radiation is way more complex to explain. I won't even try to describe it here, it is in the math and very hard to visualize.