# Tag Info

Accepted

• 2,550

### Examples of discrete time martingales

If $X$ is an integrable random variable and $\left(\mathcal F_n\right)_{n\geqslant 1}$ is a filtration, then $X_n:=\mathbb E\left[X\mid\mathcal F_n\right]$ is a martingale. It is worth mentioning that ...
• 3,938

### Uniform martingale convergence of Radon-Nikodym derivatives of a convex set of probabilities

(1) If you assumed that the $Y_{n}^{Q},Y_{\infty}^{Q}$ are all convex(thus continuous) functions along $\mathcal{C}$ being convex, then the uniform convergence follows easily from a classical result, ...
• 8,041
Accepted

### History of optional sampling/stopping theorem

"Optional stopping" and "optional sampling" refer to a strategy of peeking while you are sampling and then, based on what you find, exercise the option to quit sampling. Doob's theorem states that in ...
• 184k
Accepted

### Inequality for exponential sum in Dvoretzky 1972

First here, there is a typo in the Dvoretzky paper: there must be $-1+\frac{1}{2}t^2E[X_{n,k}^2|F_{n,k-1}]$ instead of $-1-\frac{1}{2}t^2E[X_{n,k}^2|F_{n,k-1}]$ there. Otherwise, the inequality will ...
• 122k
Accepted

### Concentration of a modified random walk

Let me try an answer. [Edit: simplified and (hopefully) corrected.] Let $\alpha$ be the only positive solution of $\mathrm{ch}\alpha = \exp(\varepsilon\alpha)$, so that  \exp(\alpha x) = \frac12\...
• 3,429
Other than the presence of an extra factor $D$ or $D^2$, depending on the context, the coefficients in the bounds for martingales in $(2,D)$-smooth spaces in the paper you cite are exactly the same as ...