The term real-time is meaningless unless you qualify it with a deadline and an acceptable miss rate. When you have these two pieces of information, you can determine whether or not Linux is a suitable candidate for the operating system and, if so, begin to tune your system to meet the requirements. Tuning Linux and your application to handle real-time events means making it more deterministic so that the real-time threads can meet their deadlines reliably. Determinism usually comes at the price of total throughput, so a real-time system is not going to be able to process as much data as a nonreal-time system.
It is not possible to provide mathematical proof that a complex operating system such as Linux will always meet a given deadline, so the only approach is through extensive testing using tools such as cyclictest and Ftrace and, more importantly, using your own benchmarks...
