A theory of midlatitude Ionospheric Alfv¿n Resonator (IAR) excitation due to random cloud-to-ground lightning discharges is developed. Electromagnetic wave radiated from the lightning discharges penetrates into the ionosphere, thereby exciting the shear Alfv¿n and magnetosonic waves in the F region of ionosphere. The IAR arises due to wave reflection from the Alfv¿n velocity gradients in the topside ionosphere. Typically, the ionospheric resonance cavity accumulates the shear Alfv¿n wave energy with periods from 1 s to a few tenths of seconds. To proceed analytically, a suitably idealized plane-stratified model of the medium was used that ignores the magnetic field line curvature and dip angle but includes plasma conductivity variations with altitude. The thunderstorm centers distributed around a ground-recording station is assumed to be statistically independent sources of the lightning activity, which is a stochastic Poisson process. The lightning onset time and the current moment is supposed to be a random value, while the shape and duration of return strokes are deterministic. Model calculations of the IAR spectrum due to nearby thunderstorm activity were applied to interpret ULF observation made at Karimshino station (52.94¿N, 158.25¿E) in Kamchatka peninsula. It is shown that the sharp impulses which are in one-to-one correspondence with the appearance of the spectral resonance structure (SRS) in dynamic spectrograms can be the result from nearby lightning discharges followed by impulse IAR excitation. The correlation functions and power spectra of the IAR due to random lightning discharge process is studied both analytically and numerically. We found that the nearby thunderstorms in the range of 1000--2000 km make a main contribution to the SRS signature of the midlatitude IAR, whereas the remote/tropic thunderstorm activity is of minor importance. It is not inconceivable that there may exist other permanent mechanisms of the midlatitude IAR excitation, for example, the high-altitude neutral wind in the E region might also be operative.