All Questions
12 questions
3
votes
0
answers
80
views
Seeking strong bounds on KL-divergence and martingales for a hypothesis-testing inequality
Let's say we have a finite set $\mathcal{O}$ of observations, and let $\mathcal{C}(\Delta\mathcal{O})$ denote the space of closed convex sets of probability distributions.
We have two hypotheses which ...
- 353
1
vote
1
answer
129
views
A martingale puzzle about sum of expected squared bounds
I'm trying to get one of those "with $1-\delta$ probability, the following holds"-style bounds, and the following martingale problem looks solvable by some Freedman or Bernstein-style bound, ...
- 353
3
votes
2
answers
636
views
Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$
Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that
$$
X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2.
$$
Question 1: Does the following hold?
$$...
- 33
5
votes
1
answer
208
views
Expected supremum of normalised random walk
Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$.
Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix.
Define $S^k=...
2
votes
1
answer
287
views
Bernstein Inequality for continous local martingale
I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time.
Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then :
$$P\left(\sup_{t\in [0,...
- 245
3
votes
1
answer
235
views
Inequality for exponential sum in Dvoretzky 1972
I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...
- 115
4
votes
1
answer
1k
views
Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences
A classic formulation of the Bernstein inequality (from Wikipedia) is as follow:
Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$...
- 53
4
votes
1
answer
225
views
concentration inequality for $d$-dimensional martingale
Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...
- 317
1
vote
1
answer
229
views
Tail inequality for orthomartingales/martingale difference random fields
It is known that if $(S_i= \sum_{j \leqslant i }X_i, \mathcal F_i)$ is a martingale,
then for each
$
\beta>1$, $\delta\in (0,\beta-1)$ and $\lambda>0$, and each integer $N \geqslant 1$, the ...
- 3,958
4
votes
1
answer
555
views
Conditional Form of Rosenthal's Inequality
Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following:
If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and $2\...
- 289
10
votes
3
answers
4k
views
Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability)
Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is
$$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$
...
- 223
9
votes
3
answers
868
views
Rosenthal like inequality for weak $\mathbb L^p$-norms
Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if $(X_i)_{i=...
- 3,958