A theory is given for (i) the linear Stark effect of the substates of the first excited level of the hydrogen atom and (ii) the linear Stark broadening of the Lyman &agr; line, in a three‐dimensional stochastic electric field. The latter is represented in the form of a three‐dimensional Fourier series with different random amplitudes and random phases for each vector component of the wave modes, which simulates a perturbing vector field varying with respect to intensity and direction in a random manner during an atomic transition. The line‐shape function, which is a moment of the distribution function of the random amplitudes and random phases of the stochastic field, is evaluated by statistical methods. It is shown that the Lyman &agr; line consists of a sequence of discrete spectra (Fourier analysis) at the frequencies &ohgr;=n&ohgr;p,n=0,±1,±2, etc., where &ohgr;pis the plasma frequency. The line intensity decreases exponentially with the square of the order numbern. The half‐width of the spectral line is proportional to the square root of the mean square stochastic electric field. In the case of electron acoustic turbulence, the half‐width is related to the unperturbed electron pressure. These results permit a quantitative determination of the intensity of the stochastic electric field and the average electron pressure in turbulent plasmas.