In recent years, there is a rapid increase in the population of elderly people. However, elderly people may suffer from the consequences of cognitive decline, which is a mental health disorder that primarily affects cognitive abilities such as learning, memory, etc. As a result, the elderly people may get dependent on caregivers to complete daily life tasks. Detecting the early indicators of dementia before it gets worsen and warning the caregivers and medical doctors would be helpful for further diagnosis. In this paper, the problem of activity recognition and abnormal behaviour detection is investigated for elderly people with dementia. First of all, the paper presents a methodology for generating synthetic data reflecting on some behavioural difficulties of people with dementia given the difficulty of obtaining real-world data. Secondly, the paper explores Convolutional Neural Networks (CNNs) to model patterns in activity sequences and detect abnormal behaviour related to dementia. Activity recognition is considered as a sequence labelling problem, while abnormal behaviour is flagged based on the deviation from normal patterns. Moreover, the performance of CNNs is compared against the state-of-art methods such as Naïve Bayes (NB), Hidden Markov Models (HMMs), Hidden Semi-Markov Models (HSMM), Conditional Random Fields (CRFs). The results obtained indicate that CNNs are competitive with those state-of-art methods.