A joint limit theorem on the complex plane for general Dirichlet series (Q2571497)

Denote by \[ f_j(s) = \sum_1^\infty a_{m,j} e^{-\lambda_m\cdot s}, \qquad 0<\lambda_m \nearrow \infty, \quad a_{m.j} \in \mathbb C, \quad j=1,2, \] general Dirichlet series, absolutely convergent in \(\sigma > \sigma_{a,j}\), meromorphically continuable to some halfplane \(\sigma >\sigma_{1,j}\) [where \(\sigma_{1,j} < \sigma_{a,j}\)], with all poles in this region contained in a compact set. These functions are assumed to satisfy, for \(\sigma >\sigma_{1,j}\), the conditions \[ f_j(s) = {\mathcal O}\left( |t|^\delta\right) \text{ for } t\geq t_0 \text{ (and some } \delta>0),\text{ and } \int_{-T}^T |f_j(\sigma + it)|^2 dt = {\mathcal O}(T). \] Extending earlier work of the author [``Limit theorems for general Dirichlet series'', Theory Stoch. Process. 8(24), No. 3--4, 356--268 (2002; Zbl 1036.11041)] on the weak convergence of the probability measure \(P_T(A)\) (defined on Borel sets on \(\mathbb C\)) \[ P_T(A) = \nu_T(f_1(\sigma+it) \in A) \; \Bigl[ = \frac1T \cdot \text{meas}\{t\in [0,T]\, ; f_1(\sigma + it)\in A\}\Bigr] \] to some [explicitly described] probability measure \(P\) (the distribution of some random variable \(f(\sigma,\omega)\)), the author proves corresponding results for the ``joint'' weak convergence of the probability measure \(Q_T(A)\), defined on Borel sets \(A \subset \mathbb C^2\), \[ Q_T(A) = \frac1T \cdot \text{meas}\Bigl\{ t\in [0,T]\, ;\; \Bigl( f_1(\sigma_1+it),\, f_2(\sigma_2 +it) \Bigr) \in A\Bigr\} \] to some limit measure \(Q\), described as the distribution of some random element \(( f_1((\sigma_1,\omega),\, f_2(\sigma_2,\omega) ).\) For the description of the limit measures \(P,\, Q\) it is assumed that the set \(\{\lambda_m\}\) is \(\mathbb Q\)-linearly independent and satisfies \(\lambda_m \geq c \cdot (\log m)^\delta\) for some positive constants \(c\) and \(\delta\). The proof starts with showing that the family of probability measures \(\{Q_T\}\) is tight and so (by Prokhorov's theorem) relatively compact. Then, arguments of the author's paper in Theory Stoch. Proc. 2002 (mentioned above) play an important rôle.