14
votes

Accepted

### A curious martingale

No. see Martingale Convergence
Theorem 4 Let X be a continuous martingale. Then, almost surely, one of the following is satisfied
$X_\infty=\lim_{t\rightarrow\infty}X_t$ exists and is finite.
$\...

13
votes

Accepted

### Does there exist an almost surely differentiable martingale?

The answer is no.
Indeed, if a martingale is a.s. everywhere differentiable, then its quadratic variation is a.s $0$. So, by the Burkholder--Davis--Gundy inequality, the martingale is a.s. constant.
...

11
votes

Accepted

### Martingales converging in probability but not a.s

$\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\F}{\mathcal{F}}\newcommand{\Om}{\Omega}\newcommand{\Z}{\mathbb{Z}}$A counterexample can be obtained as follows. Let $T_1,T_2,\dots$ be ...

11
votes

Accepted

### Concentration bounds for martingales with adaptive Gaussian steps

Observe that $X_n=X_{n-1}(1+Z_n)$ where $\{Z_k\}_{k \ge 1}$ are i.i.d. standard normal. Hence to analyze the asymptotic distribution of $|X_n|$, pass to logarithms, to get $$\log(|X_n|)= \log(|X_1|) +\...

11
votes

Accepted

### On martingale convergence

You can construct a continuous local martingale $X$ such that $X(n) = n$ almost surely. Indeed, for $t < 1$, set $Y(t) = B(t/(1-t))$ for $B$ a Brownian motion and then $X(t) = Y(t \wedge \tau)$ ...

8
votes

Accepted

### Can we do better than Azuma-Hoeffding when the variance is small?

Exponential inequalities for sums of independent random variables (r.v.'s) can be extended to martingales in a standard and completely general manner; see Theorem 8.5 or Theorem 8.1 for real-valued ...

8
votes

Accepted

### Is a martingale conditioned to be large a submartingale?

No.
For a simple counterexample, let's work in discrete time. Consider the following gambling strategy: start with \$0 and bet \$1 on a fair coin flip. If you win, you take your dollar and go home. ...

7
votes

### Examples of discrete time martingales

$\newcommand{\bN}{\mathbb{N}}$ $\newcommand{\eF}{\mathscr{F}}$ $\newcommand{\si}{\sigma}$
Branching processes Set $\bN_0=\{0,1,2,\dotsc\}$. Fix a probability measure $\mu$ on $\bN_0$ such that
$$
m:=\...

7
votes

Accepted

### Prove an anti-concentration inequality for a martingale

Basically, the proof goes along the following lines:
(1) Take a small $\varepsilon>0$ and show that the expected exit time from the interval $[-\varepsilon\sqrt{vl},\varepsilon\sqrt{vl}]$ is less ...

7
votes

Accepted

### Show that this process is not a martingale

Here's an approach that comes from
Li, Xue-Mei, Strict local martingales: examples, Stat. Probab. Lett. 129, 65-68 (2017). ZBL1386.60159, https://arxiv.org/abs/1609.00935. Indeed, she mentions this ...

7
votes

Accepted

### Is there an i.i.d sequence in the unit cube $[-1,1]^d$ with $\mathbb E \left[ \Big \| \sum_{i=1}^N X_N \Big \|_\infty\right] = \sqrt {dN}$?

Let $X_i=(X_{i,1},\dots,X_{i,d})$, $S:=(S_1,\dots,S_d)$, $S_j:=\sum_{i=1}^d X_{i,j}/\sqrt n$. Then, by Hoeffding's inequality, for $s\ge0$
$$P(|S_j|\ge s)\le2e^{-s^2/2},$$
whence
$$E\|S\|_\infty=\...

7
votes

Accepted

### A comparison of diffusions

The inequality is not true in general â€” additional assumptions are needed. I think some kind of monotonicity of $a_1$ and $a_2$ should help, but this is merely a guess.
Here is a counterexample. ...

6
votes

### A curious martingale

I think Iosif's Fatou lemma argument can be fixed, as follows.
Assume without loss of generality that $X_0 = 0$.
Suppose to the contrary that $X_t \to +\infty$ a.s. Then it must be that $\inf_{t \ge ...

5
votes

Accepted

### Convergence of conditional second moments

Let us state Corollary 2.1 of these notes.
Let $p>1$, $X\in\mathbb L^p$ and let $\left(\mathcal F_n\right)_{n\geqslant 1}$ be a filtration. Denote by $\mathcal F$ the $\sigma$-algebra generated ...

5
votes

### Large deviation/concentration inequality for submartingale

This looks like a weak law of large numbers, and in fact a strong law holds: I claim that $\liminf_{t \to \infty} \frac{S_t}{t} \ge \Delta$ almost surely, which implies the desired result.
The key is ...

5
votes

Accepted

### Moment bounds on exponential martingale

There are a number of ways to bound moments of $Z$ in terms of exponential moments of $X$. For some sharp results, see Theorem 1.5 of Kazamaki's book, "Continuous exponential martingales and BMO," as ...

5
votes

### Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?

This is worked out in some detail in the paper of Fan, Grama and Liu,
J. Math Anal. Appl. 448 (2017), 538-566 (see in particular Theorem 2.1 there, and the references). Unfortunately I do not have an ...

5
votes

Accepted

### Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences

It appears you want to have the following:
Let $X_1,\dots,X_n$ be independent zero-mean random variables (r.v.'s ) (or, more generally, martingale-differences) with $S_n:=X_1+\dots+X_n$, $B^2:=...

5
votes

Accepted

### Proof of extended supermartingale convergence theorem

Here's one approach.
First notice that
$$
R_t: = Y_{t} + \sum_{i=1}^{t-1} X_i - \sum_{i=1}^{t-1} Z_i
$$
is a supermartingale, since
$$
R_{t+1} - R_t = Y_{t+1} - Y_t + X_t - Z_t,
$$
giving
$$
E(R_{...

5
votes

### Identity involving the probability that a random walk stays below a curve

Example 2 of arXiv:0704.2826 considers the analogous problem for the continuous-time random walk, in the more general case that the curve has the form $g(t)=a+b\sqrt{T-t}$ with $a+b\sqrt T\geq 0$. The ...

5
votes

Accepted

### Does a continuous martingale converge almost surely on the event that its quadratic variation is finite?

It is true even for local Martingales, see Proposition 1.26 page 124 in [1].
Here is an intuitive way to understand it:
The proof of the Dambis-Dubins-Schwarz theorem [2, 3] (see also [1,4]) implies ...

5
votes

Accepted

### On the convergence of a martingale

Here we study
$$M_{t}=B_{A_{t}}\stackrel{d}{=}\int_{0}^{t}\sqrt{1+e^{W_{s}}}dW_{s}.$$
First, as mentioned Martingale Convergence here
Theorem 4 Let X be a continuous martingale. Then, almost surely, ...

4
votes

### Examples of discrete time martingales

If you're interested in a somewhat off-beat example of a discrete martingale, you could look at
Rafe Jones, Iterated Galois towers, their associated martingales, and
the $p$-adic Mandelbrot set, ...

4
votes

### Examples of discrete time martingales

Change of probability
An old remembrance, and I don't daily practice maths for a long time, so please correct me if I'm wrong. Let ${(\mathcal{F}_n)}_{n \geq 0}$ be a filtration on $(\Omega, \mathcal{...

4
votes

### 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 ...

4
votes

### 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, ...

4
votes

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 ...

4
votes

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 ...

4
votes

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\...

4
votes

Accepted

### Do i.i.d. sums concentrate any faster than martingales?

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 ...

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