Strong Invariance Principle For Step-reinforced Random Walks

The rapid development of the internet has sparked interest in the study of long memory effects.In the real world,there are many examples of analysis based on long memory effects,such as user behavior pattern prediction,user interest preference analysis,and analysis of the correlation coefficient of stock price indices.Analysis of long memory effects plays an important role in an increasing number of disciplines,such as applied mathematics,theoretical physics,computer science,and econometrics.Under the background of deep cross-disciplinary fusion,scholars have established many longmemory random walk models.The elephant random walk model is a long-memory random walk model that has been widely studied in recent years,but the elephant random walk model only considers random walks with integer values on a straight line.With the deepening of research on long memory effects,scholars have proposed more general random walk models.In this paper,we will study positive and negative reinforced random walks.Under the condition of the existence of the second moment,we give the limit properties of the variance of the positive and negative reinforced random walks.Using martingale methods,we also prove the weak law of large numbers for the related process of positive reinforced random walks and the strong law of large numbers for negative reinforced random walks.Under the bounded condition,using the Skorohod representation theorem of martingales,we establish the strong invariance principle for positive and negative reinforced random walks.Using the properties of Brownian motion,we also obtain the law of iterated logarithm and functional central limit theorem of the positive and negative reinforced random walk.