Let $L$ and $S$ integer valued random variables $(0,1,2, \ldots).$ $R$ is another random variable which can be a floating numbers.
The RV $S$ is the least required number of $R$ to be equal or greater than $R$. Mathematically expressed as:
$$ \sum_{j = 1}^{S[m]-1} R[j] < L[m] \leq \sum_{j = 1}^{S[m]} R[j] $$
Since $R$ is floating point, $U_m$ is defined to express the excessive part of $m$th block the inequality above is expressed as:
$$L[m] +U[m] = \sum_{j = 1}^{S[m]} R[j] + U[m-1] $$
For large number of $m$ the problem can be expressed as: $$ \begin{array}{l} L[1] + U[1] = {\mkern 1mu} U[0] + \sum_{j = 1}^{S[1]} R[j] \\ L[2] + U[2] = {\mkern 1mu} {U}[1] + \sum_{j = 1}^{S[2]} R[j] \\ L[3] + U[3] = {\mkern 1mu} {U}[2] + \sum_{j = 1}^{S[3]} R[j] \\ \qquad\vdots\\ L[m] + U[m] = {\mkern 1mu} U[m - 1] + \sum_{j = 1}^{S[m]} R[j] \end{array} $$ Note that $U[0] = 0$ at the initial state.
Are the following equations correct?
$$E[L] = {\mkern 1mu} E\left[ \sum_{j = 1}^S R[j] \right],\tag 1$$ and
$$E[L^2] = {\mkern 1mu} E\left[\left( \sum_{j = 1}^S R[j] \right)^2 \right] \tag 2$$
Note: I use law of large number and approximations to prove (1). But I am open to other suggestions for both (1) and (2).