Accuracy for main class of Simultaneous Perturbation Stochastic Approximation (SPSA) procedures is being researched. The model of observation is considered to be one of the most general among SPSA research. The power of moments of expectation for which the estimates of the procedure do converge is lowerized from 2 to 1 (not including lower bound). The conditions for the convergence are presented, with additional generalisations made about heavy-tailed noise and trial perturbation properties.