Cognitive dysfunction in the aging Veteran population is a growing health concern in the Veterans Health System. It is not known whether OSA coexisting with COPD will enhance the risk for cognitive dysfunction. The investigators sought to investigate whether these two highly prevalent diseases, that often co-exist as the 'Overlap Syndrome', combine to enhance cognitive impairment in the elderly Veteran population. Thus, the investigators will study whether elderly patients with Overlap syndrome have increased cognitive deficits compared with OSA or COPD alone. Additionally, treatment of OSA with positive airway pressure (PAP) has been shown to improve neurocognitive function in moderate-to-severe OSA while cognitive decline in COPD may be reversible through treatment with long-term oxygen therapy. The investigators will also study whether treatment with positive airway pressure (PAP) and supplemental oxygen vs PAP alone will improve cognitive function and improve quality of life of elderly Veterans.