I think the Khazar hypothesis has been pretty much shredded by Behar:
http://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1040&context=humbiol_preprints
Also, importantly, it contradicts all the PCA plotting of numerous studies, which shows that after whatever amount of mixing has taken place, most Jews have been dragged in a northwestern direction not much further than Cyprus, and that includes a big chunk of the Ashkenazim. (The slight pull toward the Adriatic and the Balkans for some Ashkenazim can be explained through Greek gene flow into Jewish populations, I think. The IBD analyses may not be showing it because the Greeks themselves have experienced some new gene flows? Perhaps some minor input from Khazarians and eastern Europeans is also a factor. Or, it could even be some central or western European gene flow. Fwiw, I’ve seen a number of calculator results where half Ashkenazim/half British Isles people plot either near northern Italy or in Romania.)
And, as Semitic Duwa pointed out, the Sephardic Jews, who would have no possible connection to Khazaria, overlap with the Ashkenazim.
That isn’t to say that there wasn’t some impact from the Khazarian Jews, in my opinion, as I suggested above. I’ve wondered whether that persistent 2% Siberian that the Ashkenazim get in the calculators could be traced to them. However, it now strikes me that perhaps that’s present in northern Near Easterners as well.
As for the overall genetic similarities between Sicilians, Cypriots, the Maltese and the Ashkenazim, I think the IBD data pretty much establishes that most of the similarities result from ancient common ancestors rather than modern admixture, as proposed by Aberdeen.
The other important factor in all of this is the actual history of these communities. I don’t think there’s any doubt that the genesis of the Ashkenazi community was in France and the Rhineland. Those Jewish communities were decimated by the horrors and the barbarity inflicted upon them by the Crusaders, creating the famous bottleneck which has marked their subsequent genetic history. They then fled east, where they experienced the massive expansion, and practiced the endogamy that created the modern Ashkenazi community.
In fact, looking at the history of the Near East and at the genetics as well, I think that one could argue that the people of the Near East, through the invasions from Central Asia, the movement north of Bedouin tribes, and the importation of African slaves, have changed more than have the Jews in the last 2,000 years.
Oh, one other thought…in all those comparisons with Oetzi, the Ashkenazim also show up as a very closely related population, which supports the idea, in my opinion, that they have preserved the EEF signature better than have modern populations of the Near East.