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Suppose $X_1, ... X_n$ are i.i.d. random vectors in $d$-dimensional space (i.e., $R^d$) with continuous centrally symmetric density function $f(\cdot)$ (i.e., symmetric with respect to the origin). For concretenes's sake, assume $X_i$ is just a multivariate normal distribution with covariance matrix equal to the identity.

What is the expectation (as a function of $d$, $k$ and $n$) of the maximum value of $||X_{i_1}+...+X_{i_k}||^2$ where the maximum is taken over all possible $1 \leq i_{1} < i_{2} < \ldots < i_{k} \leq n$ and $||\cdot||$ denotes the Euclidean norm?

Are large deviation bounds known? In general, what is the relation between $f(\cdot)$ and the expectation of the maximum?

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Sounds like you want a 'maximal inequality'. Not exactly sure how to do what you are asking by I would start at Section 3 of stat.yale.edu/~pollard/Papers/Pollard89StatSci.pdf. –  Robby McKilliam Jun 11 '12 at 23:12
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