Excitation of the middle latitude ionospheric Alfv¿n resonator (IAR) due to global thunderstorm activity is considered. The lightning discharges in thunderstorm centers are modeled in terms of a stochastic Poisson process. The lightning onset time and the current moment amplitude are supposed to be random values, while the shape of current moment is assumed to be a given function. Electromagnetic TM mode radiated from the lightning discharges penetrates into the ionosphere via mode coupling to the shear Alfv¿n and magnetosonic wave modes in the E layer of the ionosphere. The ionospheric resonance cavity can trap and accumulate the shear Alfv¿n wave energy with periods from one second to a few tenths of seconds, thereby exciting the IAR resonances. Using a simplified model of the topside ionosphere, we have analyzed a possibility for the IAR excitation far from the tropical thunderstorm region. The low-frequency power spectra of the IAR on the ground surface is studied both analytically and numerically. It is shown that the IAR power spectra exhibit spectral resonance structure (SRS) only during the nighttime conditions. Furthermore, it is found that the calculated spectra are one or two orders of magnitude lower than that observed at Karimshino station in Kamchatka region. This fact suggests that the SRS is not simply due to stochastic lightning discharge process but that some generation mechanisms for the IAR excitation at middle latitudes might also be operative.