HARCH processes are heavy tailed (Q1979093)

A \(\text{HARCH}(k)\) process \(R_n\) is defined by the recursive relations \[ R_n=\sigma_n\varepsilon_n, \qquad \sigma_n^2=c_0+\sum_{j=1}^kc_j \left( \sum_{i=1}^j R_{n-i} \right)^2 \] where the \(c_j\) are some nonrandom constants, \(\varepsilon_n\) are i.i.d., \(E\varepsilon=0\), \(E\varepsilon^2<\infty\). Such models are used to describe the log-return process in financial asset market price modeling. Denote by \(R_\infty\) the distributional limit of \(R_n\) (if it exists). It is shown that if \(c_1>0\) and \(\Pr(\varepsilon^2>c_1^{-1})>0\), then \[ \liminf_{r\to\infty} { \log\Pr\{|R_\infty|>r\} \over \log r } >-\infty, \] i.e. \(R_\infty\) is a heavy tailed distribution.