All Questions

Consider a discrete time submartingale $X_n$ with bounded difference $|X_n-X_{n-1}|\leq c$. With Azuma inequality we have the concentration of a single time event as $$ P(X_t-X_0 \leq -t) \leq exp\...

Let $\{ Z_n \mid n = 0,1,..\}$ be a non-negative super-martingale and assume that it is of bounded difference i.e $\exists ~c_i >0$ s.t $\vert Z_{i+1} - Z_i \vert \leq c_i$. Then we know (Azuma-...

Suppose $X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $\|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae). The simplest concentration inequality I know ...

I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time. Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then : $$P\left(\sup_{t\in [0,...

Let $S_t = M_t + D_t$ be the sum of a martingale $\left(M_t\right)_{t=1,2,\ldots}$ and a predictable process $(D_t)_{t=1,2,\ldots}$ such that the variance of the increments of $M$ is uniformly bounded ...