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What has faster performance speed: Columnstore Indexes or SSAS Cubes? Are Columnstore indexes proven to be faster than SSAS cubes (SQL Analysis Services)? Have their been any performance studies conducted by Microsoft?

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There is no single right answer to your question.

Its worth noting they are not mutially exclusive technologies. You can have an OLAP cube reading from table using column store indexes.

The performance of each in various scenarios will differ depending upon the size, number, complexity of tables used and granularity/sparcity of data within them.

It will also depend upon the variability of queries used. Are they frequently aggregating? Or constantly seeking row level data?

The only way to know for sure in your case is to test your queries and scenarios with representative data.