  In this paper, as a study of reinforcement learning, we converge the Q
function to unbounded rewards such as Gaussian distribution. From the central
limit theorem, in some real-world applications it is natural to assume that
rewards follow a Gaussian distribution , but existing proofs cannot guarantee
convergence of the Q-function. Furthermore, in the distribution-type
reinforcement learning and Bayesian reinforcement learning that have become
popular in recent years, it is better to allow the reward to have a Gaussian
distribution. Therefore, in this paper, we prove the convergence of the
Q-function under the condition of $E[r(s,a)^2]<\infty$, which is much more
relaxed than the existing research. Finally, as a bonus, a proof of the policy
gradient theorem for distributed reinforcement learning is also posted.
