On a problem of estimation of an infinite-dimensional parameter

Let $X$ be a random variable taking the positive integer values and let $\mathbf P\{X=k\}=\theta(k)$. We consider the problem of estimation of the parameter $\theta=(\theta(1),\theta(2),\dots)$ on the base of the sample $X_1,X_2,\dots,X_n$ where the observations $X_j$ are independent copies of $X$.