Very few mathematical results have been derived about the randomness properties of these generators, so little is known about their theoretical properties. We are forced to rely on results of empirical statistical tests.
When the arithmetic operation used is XOR, the LFGs are more commonly referred to as generalized feedback shift register generators. These have often been used for simulation because they are extremely fast, since XOR can be done with simple bit operations. However the randomness properties of these generators are quite poor unless the lag is extremely large.
It has recently been shown that LFGs perform poorly for certain types of applications, such as random walks and percolation clusters, unless very large lags are used.