Excursions of a random walk related to the strong law of large numbers (Q1293402)

Let \(X_i\), \(i=1,2,\dots\), be a sequence of independent normally distributed random variables with mean \(\mu\) and variance \(\sigma^2\). Let \(S_n= \sum^n_{i=1} X_i\), \(n=1,2, \dots\), for each \(n\) let \(A_n\) be the event \[ A_n= \bigl(S_n-n \mu>cn^\alpha,\;S_{n+1}-(n+1) \mu\leq c(n+1)^\alpha\bigr), \] where \(c>0\) and \(1/2<\alpha\leq 1\), and define the random variable \(X(c)\) by \(X(c)= \sum^\infty_{n=1} I(A_n)\), where \(I(\cdot)\) denotes the indicator function. \(X(c)\) represents the number of excursions of the random walk \(S_n\), and it is finite valued due to the strong law of large numbers (since \(\alpha> 1/2)\). The authors obtain upper and lower bounds for the expected value of \(X(c)\), which are asymptotically close as \(c\to 0\). This provides a characterization of the relationship between the fluctuations of the random walk and the strong law of large numbers. Several implications of the result are discussed, and tables for the upper and lower bounds are given for various values of \(c\) and \(\alpha\).