Convolutions of heavy-tailed random variables and applications to portfolio diversification and \(\text{MA}(1)\) time series (Q2713154)

The asymptotic behaviour is studied for \(P(X_1 + X_2 > t)\), as \(t \to \infty\), where \(X_1\) and \(X_2\) are independent r.v.'s satisfying certain hypotheses on their tails, namely, the second-order regular variation condition and the balance condition. The last condition replaces the assumption of positivity of r.v.'s previously used by \textit{J. Geluk, L. de Haan, S. Resnick} and \textit{C. Stărică} [Stochastic Processes Appl. 69, No. 2, 139-159 (1997; Zbl 0913.60001)] and \textit{J. L. Geluk} and \textit{L. Peng} [Stat. Probab. Lett. 46, No. 3, 217-227 (2000; Zbl 0942.62060)]. The result is applied to the problem of risk diversification in portfolio analysis. Another application concerns the parameter \(\theta\) estimation in MA(1) model \(Y_i = \varepsilon_o - \theta \varepsilon_{i-1}\) where \(\{\varepsilon_i\}\) is a sequence of i.i.d. r.v.'s having mean zero and finite variance. Under specified conditions (involving the limiting behaviour for \(P(\varepsilon_i > t)\) and \(P(\varepsilon_i < -t)\) as \(t \to \infty\)) the asymptotic normality is proved for appropriate estimators \(\widehat{\theta}_n\).