Stability analysis of lattice Boltzmann equations (LBEs) on initial conditions for one-dimensional diffusion is performed. Stability of the solution of the Cauchy problem for the system of linear Bhatnaghar-Gross-Krook kinetic equations is demonstrated for the cases of D1Q2 and D1Q3 lattices. Stability of the scheme for D1Q2 lattice is analytically analyzed by the method of differential approximation. Stability of parametrical scheme is numerically investigated by von Neumann method in parameter space. As a result of numerical analysis, the correction of the hypothesis on transfer of stability conditions of the scheme for macroequation to the system of LBEs is demonstrated.