In this paper, a comprehensive performance analysis of a distributed intelligent reflective surfaces (IRSs)-aided communication system is presented. First, the optimal signal-to-noise ratio (SNR), which is attainable through the direct and reflected channels, is quantified by controlling the phase shifts of the distributed IRS. Next, this optimal SNR is statistically characterized by deriving tight approximations to the exact probability density function (PDF) and cumulative distribution function (CDF) for Nakagami-$m$ fading. The accuracy/tightness of this statistical characterization is investigated by deriving the Kullback-Leibler divergence. Our PDF/CDF analysis is used to derive tight approximations/bounds for the outage probability, achievable rate, and average symbol error rate (SER) in closed-form. To obtain useful insights, the asymptotic outage probability and average SER are derived for the high SNR regime. Thereby, the achievable diversity order and array gains are quantified. Our asymptotic performance analysis reveals that the diversity order can be boosted by using distributed passive IRSs without generating additional electromagnetic (EM) waves via active radio frequency chains. Our asymptotic rate analysis shows that the lower and upper rate bounds converge to an asymptotic limit in large reflective element regime. Our analysis is validated via Monte-Carlo simulations. We present a rigorous set of numerical results to investigate the performance gains of the proposed system model. Our analytical and numerical results reveal that the performance of single-input single-output wireless systems can be boosted by recycling the EM waves generated by a transmitter through distributed passive IRS reflections to enable constructive signal combining at a receiver.