Integral limit theorem for the first passage time for the level of random walk, described with \(AR (1)\) sequences (Q2870873)

Let \(\{X_n\}\) be an autoregressive time series model given by the equation \(X_n=\beta X_{n-1}+\xi_n\), \(n\geq 1\), \(X_0=x\), where \(x\geq 0\) and \(|\beta|<1\) are given constants. In this paper, the authors prove the integral limit theorem for the first passage time \(\tau_a=\inf\{n\geq 1:T_n>a\}\) of the process \(\{T_n\}\) defined as \(T_n=\sum_{k=1}^n X_{k-1} X_k\), \(n\geq 1\). The paper begins with definitions and some properties of a sequence of random variables such as uniform continuity in probability. Then the main result is proved using some known statements such us almost sure convergence of the estimate \(T_n/n\), the uniform continuity in probability of the sequence \(\{T^\ast_n\}\) given by \(T_n^\ast=(T_n-\lambda_n)/\sqrt{n(1-\beta^2)}\), and properties of convergence of a sequence of random variables in probability, in distribution and almost surely.