On the Distribution of Integer Random Variables Satisfying Two Linear Relations

We consider the multiplicative (in the sense of Vershik) probability measure corresponding to an arbitrary real dimension $d$ on the set of all collections $\{N_j\}$ of integer nonnegative numbers $N_j$, $j=l_0,l_0+1,\dots$, satisfying the conditions
$$\sum_{j=l_0}^\infty jN_{j}\le M, \qquad \sum_{j=l_0}^\infty N_j=N,$$
where $l_0,M,N$ are natural numbers. If $M,N\to\infty$ and the rates of growth of these parameters satisfy a certain relation depending on $d$, and $l_0$ depends on them in a special way (for $d\ge2$ we can take $l_0=1$), then, in the limit, the “majority” of collections (with respect to the measure indicated above) concentrates near the limit distribution described by the Bose–Einstein formulas. We study the probabilities of the deviations of the sums $\sum_{j=l}^{\infty} N_j$ from the corresponding cumulative integrals for the limit distribution. In an earlier paper (see [6]), we studied the case $d=3$.