Do cyclic oxygen molecules exist and their theoretical stability?
Unlike the likes of carbon, oxygen-oxygen single bonds are highly unstable. Because oxygen is such an electronegative compound, it can only become satisfied with bonds where it hoards a majority of the electrons. Because of this, compounds such as peroxides ($\ce{RO-OR}$) will readily decompose into two radicals. This is why hydrogen peroxide is used as a radical initiator for chemical reactions. Since a single $\ce{O-O}$ bond is so unstable, it would be extremely unlikely for a larger chain to exist. Peroxides by themselves are highly unstable. Not only do they decompose into radicals, but they also rapidly decompose into water. Peroxides have a single $\ce{O-O}$ bond, and are somewhat stabilized by donating groups on either side of the peroxide. Now, imagine a chain that looks like this: $\ce{O-O-O}$. The central oxygen is not only bonded one inductively withdrawing oxygen as in peroxides, but two of them. In addition, it is not stabilized by electron donation from any source. Despite this, it is predicted that small amounts of cyclic ozone might exist in the atmosphere. Cyclic ozone is so much less stable than regular ozone for the reasons I listed above, but it should exist. Any larger oxygen ring will most likely not form.

Carbon does not have this same problem. Since carbon has a mid range electro-negativity, it is stable forming bonds with itself. In addition, carbon can form four covalent bonds, normally two of which are with hydrogens (as in a cycloalkane). Carbon is perfectly stable with this electron density, leading to the stability of cyclic alkanes.