multinomial sum upper bound

Question

I am looking for an upper bound on the following sum, where $p_1,\dots,p_r > 0$ and $\sum_i p_i\leq 1$.

$$ \sum_{0\leq i_1,\dots,i_r\leq n} \binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots p_{r}^{i_r} $$

I tried the upper bound in MSE 2589433 for the multinomial coefficient but I had no idea how to control things like $$ \exp\left(i_1\log\frac{i_1+\cdots+i_r}{i_1}\right) $$ in a nice way. A trivial bound such as $\log\frac{i_1+\cdots+i_r}{i_1}\leq \log\frac{rn}{i_1} \leq \log(rn)$ is too cruel.

Answer 1

Note that $$\sum_{0\leq i_1,\dots,i_r\leq n} \binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots p_{r}^{i_r}\leq\sum_{0\leq i_1,i_2,\ldots,i_r\atop\sum i_j\leq nr }\binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots p_{r}^{i_r}=\\ \sum_{l=0}^{nr}\sum_{i_1+\ldots+i_r=l}\binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots p_{r}^{i_r}=\sum_{l=0}^{nr}(p_1+\ldots+p_r)^l $$ This expresion is always $\leq nr+1$ and for $p_1+p_2+\ldots+p_r<1$ we have also the better estimate $$ \leq \frac{1-(p_1+\ldots+p_r)^{nr+1}}{1-(p_1+\ldots+p_r)} $$