This paper presents an adaptive beamforming application based on the capture of far-field speech data from a real single speaker in a real meeting room. After the position of a speaker is estimated by a speaker tracking system, we construct a subband-domain beamformer in generalized sidelobe canceller (GSC) configuration. In contrast to conventional practice, we then optimize the active weight vectors of the GSC so that the distribution of an output signal is as non-Gaussian as possible. We consider kurtosis in order to measure the degree of non-Gaussianity. Our beamforming algorithms can suppress noise and reverberation without the signal cancellation problems encountered in conventional beamforming algorithms. We demonstrate the effectiveness of our proposed techniques through a series of far-field automatic speech recognition experiments on the Multi-Channel Wall Street Journal Audio Visual Corpus (MC-WSJ-AV). The beamforming algorithm proposed here achieved a 13.6\% WER, whereas the simple delay-and-sum beamformer provided a WER of 17.8\%.