I am not an expert on the Khazarian empire, I had to do some research while I say this, I used a picture of the Khazarian empire to help show the expanse of it. I have to leave some open ended answers to things concerning it, as I have not studied much about this - but I do know it is significant history of the Eastern European Jews and the Ashkenazi - which make up the largest ethnic group of the Jews. I encourage you to do more research on the Khazars and the Eastern European Jews if this field interests you, as it will unveal alot about the make-up of Israel and Israeli politics, who is behind them.

What I am not sure about is whether the Khazars were converted to Judaism (Hebrewism mostly did not do conversion, as it was an ethnic religion - but Judaism changed laws of Hebrewism to make it more open to outsiders to convert to Judaism - hence one of the differences between the now dead Hebrewism and Judaism), or whether it started out as a small group of Jews that expanded as a result of intermarriages with heathens to expand the number of Jews.

As I have heard, the Khazarian Jews were a very strong kingdom of Jews that were hard as steel - enemies of the Khazars just could not compete. The Khazars were responsible for stopping the spread of Islam to Europe and may have been largely responsible to shielding Russia from Islam (except a bit like Chechnya). As shown on the map, the Khazars were centred in the Caucasus region, more specifically in the Black sea region of the Caucasus, roughly around present day Russia and Ukraine, whoch is why we see a significant presence of Jews in this part of the world, they have a long history here.