  This paper addresses the problem of deriving the asymptotic performance of
adaptive Low Rank (LR) filters used in target detection embedded in a
disturbance composed of a LR Gaussian noise plus a white Gaussian noise. In
this context, we use the Signal to Interference to Noise Ratio (SINR) loss as
performance measure which is a function of the estimated projector onto the LR
noise subspace. However, although the SINR loss can be determined through
Monte-Carlo simulations or real data, this process remains quite time
consuming. Thus, this paper proposes to predict the SINR loss behavior in order
to not depend on the data anymore and be quicker. To derive this theoretical
result, previous works used a restrictive hypothesis assuming that the target
is orthogonal to the LR noise. In this paper, we propose to derive this
theoretical performance by relaxing this hypothesis and using Random Matrix
Theory (RMT) tools. These tools will be used to present the convergences of
simple quadratic forms and perform new RMT convergences of structured quadratic
forms and SINR loss in the large dimensional regime, i.e. the size and the
number of the data tend to infinity at the same rate. We show through
simulations the interest of our approach compared to the previous works when
the restrictive hypothesis is no longer verified.
