The Expectation-Maximization (EM) algorithm is not guaranteed to converge to a global optimum. However, if the optimization target, $$L(\theta, \theta^j)$$, is a convex function, the EM algorithm is guaranteed to converge to the global optimum.

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Fortunately, EM algorithm will definitely converges. It's a great mathematical practice to prove why it must converges. But the basic intuition between this conclusion is fairly easy. We just need to prove that our log approximation fuctions continuously keep increasing in the EM Algorithm process. 

Convergence of EM Algorithm

Maximum likelihood from incomplete data via the EM algorithm which is proposed by Dempster, Laird and Rubin.
The link is : https://www.ece.iastate.edu/~namrata/EE527_Spring08/Dempster77.pdf

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