I am interested in concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables.


Let $X\_1,..., X\_n$ be i.i.d random variables, $S\_n$ their centered sum and $M\_n$ the maximum of the partial sums: 
$$ S\_n = \sum\_{k=1}^{n}{(X\_k - \mu)} \ , \quad M\_n = \max\_{1 \leq k \leq n}{S\_k}$$
In this case, we can easily derive concentration inequalities on $M\_n$ using the fact that $S\_n$ is a martingale and applying Azuma-Hoeffding's inequality (I don't know whether there are better techniques, if they are, please tell me).


Now, my question is how to derive concentration inequalities for the maximum of the rescaled or normalized sums:
 $$\max\_{1 \leq k \leq n}{\frac{S\_k}{k}} = \max\_{1 \leq k \leq n}{\hat{\mu}\_k-\mu}, \quad \max\_{1 \leq k \leq n}{\frac{S\_k}{\sqrt{k}}} \quad \text{or more generally} \quad \max\_{1 \leq k \leq n}{\frac{S\_k}{f(k)}}$$
The problem is we don't have martingales anymore. Of course, it is always possible to derive bounds by summing concentration bounds for each different value of $k$: $$\mathbb{P}(\max\_{1 \leq k \leq n}{\hat{\mu}\_k-\mu} \geq \varepsilon) \leq \sum\_{k=1}^{n}{\mathbb{P}(\hat{\mu}\_k-\mu \geq \varepsilon)}$$
but I'm looking for smarter things. In my community (statistics/machine learning, so no pure probabilists), the best technique used is "peeling": decomposing not for all values of $k$ but in slices of exponential length. But since the question is quite natural (at least it looks quite natural to me), I'm sure it must already have been studied. So I'm looking for any results or techniques that could be used to answer my question.


PS : You're free to assume whatever you need on $X\_k$, like existence of moments, of the moment generating function, boundedness...


PS2 : I'm looking for finite-time results, not purely asymptotic ones like the Hartman-Wintner theorem/law of the iterated logarithm.



Concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables
One can use Birnbaum and Marshall inequality:


**Theorem(Theorem 2.1. in [1](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177704964)).** If $\left(S\_k,k\geqslant 1\right)$ is a non-negative sub-martingale and $(c\_k,k\geqslant 1)$ a non-decreasing sequence of positive numbers, then for each $p\geqslant 1$:
$$\mathbb P\left\{\max\_{1\leqslant k\leqslant n}\frac{S\_k}{c\_k}\geqslant 1\right\}\leqslant \frac{\mathbb E\left[S\_n^p\right]}{c\_n^p}+\sum\_{i=1}^{n-1}\left(\frac 1{c\_i^p}-\frac 1{c\_{i+1}^p}\right)\mathbb E\left[S\_i^p\right].$$


**Reference:**


[1](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177704964) [Some Multivariate Chebyshev Inequalities with Extensions to Continuous Parameter Processes][1](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177704964)
Z. W. Birnbaum and Albert W. Marshall, Ann. Math. Statist. Volume 32, Number 3 (1961), 687-703. 


