Topological data analysis extracts topological features by examining the shape of the data through persistent homology to produce topological summaries, such as the persistence landscape. While the persistence landscape makes it easier to conduct statistical analysis, the Strong Law of Large Numbers and a Central Limit Theorem for the persistence landscape applies to independent and identically distributed copies of a random variable. Therefore, we developed a Strong Law of Large Numbers and a Central Limit Theorem for the persistence landscape when the stochastic component of our series is driven by an autoregressive process of order one. Theoretical results for the persistence landscape are demonstrated computationally and applied to financial time series.