To answer this question you should know about electronegativities of different elements.
Other than the first couple of chapters in most chemistry books, bonds are not usually considered completely ionic or covalent, but "more" ionic or covalent.
with AsI3 the electronegativites will be subtracted to get a difference (2.5 for I and 2.0 for As, so a difference of .5)
Typically on a scale of 0 - 3.3 (0 being completely covalent, 3.3 being completely ionic) you will be looking for differences of greater than around 1.7-1.9 for more Ionic bonds and less than that for more covalent bonds.
So for AsI3 (a difference of .5) we would consider this a more covalent bond.
For Sb2Te3 we take 2.1 - 1.9 which equals about .2
Another covalent bond.
If you're still having trouble try to find an electronegativity table in your chem book. The explanation can't be far away.
Typically you'll find that Ionic bonds have to do with anions and cations (usually listed in the front or back of your book). Typically the reason that they define ionic bonds as a bond between a metal and a non-metal is because their electronegativities differ because of their placement on the periodic table (this will make more sense when you look at an electronegativity table).
If I missed something, or didn't explain it properly, please let me know.
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