There is a common argument used when investigating the concentration of the maximally loaded bin (say $X$ is the maximum load) when $m$ balls are thrown into $n$ bins under the uniform distribution. I give the argument for $m=n,$ showing that $X$ is approximately $\ln n/\ln \ln n$ with high probability. Using the union bound, and letting $X_i$ be the number of balls in the $i^{th}$ bin $$ \mathbb{P}(X_i=k)=\binom{n}{k}\left(\frac{1}{n}\right)^k \left(1-\frac{1}{n}\right)^{n-k}\leq \binom{n}{k} \left(\frac{1}{n}\right)^k\leq \left(\frac{ne}{k}\right)^k\left(\frac{1}{n}\right)^k= \left(\frac{e}{k}\right)^k $$ yielding $$ \mathbb{P}(X_i\geq k)\leq \sum_{j=k}^n \left(\frac{e}{j}\right)^j \leq \left(\frac{e}{k}\right)^k \left(1+\frac{e}{k}+\frac{e^2}{k^2}+\cdots\right). $$ Now let $k^{\ast}=\lceil e \ln n/\ln\ln n\rceil,$ giving $$ \mathbb{P}(X_i\geq k)\leq \left(\frac{e}{k^{\ast}}\right)^{k^{\ast}} \left[\frac{1}{1-e/k^{\ast}}\right]\leq n^{-2}, $$ and using the union bound, since there are $n$ bins $$ \mathbb{P}\left(\bigcup_{i=1}^n X_i\geq k\right)\leq \frac{1}{n},\quad (1) $$ giving the concentration. What if we now have $p=(p_1,\ldots,p_n)$ with $p_i$ the probability of each ball falling into bin $i$, in an independent manner.

As far as I can tell (sort the bins so $p_1\geq p_2\geq \cdots\geq p_1>0$) as long as the maximum probability obeys $p_1\leq \frac{\ln n}{n}$, a version of this argument works.

What about distributions with larger $p_1$? What can we say? Say we allow the quantity on the RHS of (1) to be $\frac{1}{\sqrt{n}}$, for example.

I am most interested in $m=n,$ or slightly larger $m$ say $m=n (\log n)^a.$

I suppose for $p_1$ large enough wrt the other probabilities its load will highly likely be the maximum. So a kind of convex combination argument is needed...

**Edit:** As far as lower bounds for the uniform case, it can be addressed in a number of ways, including Lemma 5.12 from Mitzenmacher and Upfal's book *Probability and Computing* which shows that the maximum load is at least $$\ln n/\ln \ln n$$ with probability at least $1-(1/n)$ for $n$ large.

**Remark:** This related question here was unanswered