Let $X_1, \dots, X_n$ be a sequence of i.i.d. random variables with distribution function $G$, let

$$ G_t = \frac{\# \{ k : X_k \leq t \}}{n} $$

define the empirical distribution relative to the random variables. Set $A_t = \sqrt{n} (G_t - G(t))$,

$$ M_t = \frac{A_t}{1 - G(t)}\ \ \ \ \ B_t = A_t + \int_{-\infty}^t M_s\ dG(s)\ \ \ \ \ V_t = B_t^2 - G_t $$

It is rather simpler to verify that $M_t$ and $B_t$ are martingales with respect to the filtration $\Sigma_t = \sigma(G_s: s \leq t)$, but however I try to compute that $V_t$ is a martingale, I end up with an incredibly difficult calculation. Is there any easy way to see that $V_t$ is a martingale?

  • 1
    $\begingroup$ Maybe you could shot the computations for the proof that $(M_t)$ and $(B_t)$ are martingale (and also the beginning for $V_t$). This could be inspiring for other readers. $\endgroup$ – Davide Giraudo Nov 16 '17 at 12:38

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