Consider some probability distribution $D$ over non-negative reals with finite expectation $\mu$. Now for any positive $T$ consider sums of $T$ iid random variables drawn from $D$. A single sum of this sort would be $S(T) = \sum\_{i = 1}^T x\_i$ where each $x\_i$ is a iid random sample from $D$.


We will consider $n$ such sums $S\_1(T), S\_2(T) \ldots S\_n(T)$.


My question: Is it true that for any distribution $D$ and any finite positive $n$, there exists a finite positive $T$ (which may be a function of $n$ and $D$) such that $E(\max\_{1 \leq j \leq n}S\_j(T)) \leq 2 T \mu$?


This is true for, say, Bernoulli random variables, but I'd like to know the mildest condition under which a statement like this can be made. For example, is it true for all distributions with finite $4^{th}$ moments?



Maximum of a set of sums of iid random variables
It is always true. Split $x\_i=y\_i+z\_i$ where $y\_i$ are bounded and $Ez\_i\le \frac \mu{10n}$. You have no problems with $y\_i$ because if they were alone,$ES\_j$ would be concentrated in a very strong sense around $\nu T$ for large $T$ where $\nu=Ey\_i\le\mu$ (see Didier's argument for details or recall the Bernstein inequality for exponential moments of sums of bounded variables). But you also have no problems with $z\_i$ because even if you add them all up instead of taking the maximum at some point, you end up with mere $0.1 T\mu$.


