Square martingale for independent centered sums

For independent centered square-integrable variables, if

S_{n} = \sum_{m \leq n} X_{m}

and

V_{n} = Var (S_{n})

, then

S_{n}^{2} - V_{n}

is a martingale.

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- Small-ball maximal bound for sums with bounded increments Square martingale for independent centered sums

Small-ball maximal bound for sums with bounded increments

If additionally

∣ X_{m} ∣ \leq K

, then

P (max_{m \leq n} ∣ S_{m} ∣ \leq x) \leq \frac{( x + K ) ^{2}}{Var ( S _{n} )} .

Stop the square martingale for independent centered sums at the first exit from

[- x, x]

or at time

n

.

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