All Questions
Tagged with martingales st.statistics
9 questions
6
votes
0
answers
164
views
Stopping time on the sum of a moving window
(Cross post from MSE)
Let $\xi_i$ be iid random variables, and define:
$$S_{(k)} = \sum_{i=1}^N \xi_{i+k}$$
Now, define:
$$\tau = \min \left\{ k : S_{(k)} \notin (a,b) \right\}$$
How can I find $\...
- 193
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
139
views
Does a sequence that verifies the assumptions of a square integrable martingale on some event need to be convergent on this event?
I came across this claim by reading some literature on stochastic approximation.
Let $(\Omega, \mathcal{A}, \mathbb{P}$) be a probability space, $(\mathcal{F}_n)$ a filtration on it. Let $(\epsilon_{n}...
- 115
0
votes
2
answers
251
views
Martingale optional stopping before a stopping time
Here’s an easy one, I hope:
Suppose $\tau$ is a stopping time and $(M_t)$ is a martingale which together satisfy the hypotheses of the optional stopping theorem so that $\mathbb{E}[M_\tau]= \mathbb{E}...
- 3
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
7
votes
1
answer
466
views
Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?
It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds:
$$
ℙ\left(\left|
\sum_{i=1}^N a_i X_i
\right| \ge t \right)
\le
2\...
- 719
3
votes
2
answers
319
views
Concentration inequality of joint event over time of a submartingale
Consider a discrete time submartingale $X_n$ with bounded difference $|X_n-X_{n-1}|\leq c$. With Azuma inequality we have the concentration of a single time event as
$$
P(X_t-X_0 \leq -t) \leq exp\...
- 185
1
vote
2
answers
788
views
Uniform law of large numbers for martingale difference
Let $\xi_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for all fixed $\theta \in \...
- 13
4
votes
5
answers
2k
views
Martingales and Betting Strategies
Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots ...