All Questions
17 questions
7
votes
1
answer
736
views
How is the Gronwall lemma used in this paper?
Let $(X_t, t \ge 0)$ be a $\mathbb R^d$-valued stochastic process. Let $\lambda>0$. Assume we have $\mathbb E [|X_0|^2] < \infty$ and
$$
\mathbb E [|X_t|^2] - \mathbb E [|X_0|^2] \le -2 \lambda \...
- 835
6
votes
1
answer
375
views
Deviation bound for the maximum of the norm of Wiener process
Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof:
$$
{...
- 61
5
votes
2
answers
185
views
Density near at $0$ for the integral of the positive part of the Brownian motion
This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...
- 128k
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=...
5
votes
1
answer
445
views
A two-point inequality
Let $M(p,q) = (2p-\sqrt{p^{2}+q^{2}})\sqrt{p+\sqrt{p^{2}+q^{2}}}$ and set $B(t) = M(x+t, \sqrt{t^{2}+(y+bt)^{2}})$. Given any real $x,y,b$ is it true that $\varphi(t) = B(t)+B(-t)$ is decreasing in $...
- 3,946
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
3
votes
1
answer
364
views
Can anyone give a reference to the proof of this concentration inequality?
The following concentration inequality for the supremum of a Gaussian process indexed by a separable metric space appears here: http://math.iisc.ac.in/~manju/GP/6-Concentration%20and%20comparison%...
- 123
3
votes
0
answers
86
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
2
votes
1
answer
78
views
Existence of stationary stochastic processes with very high correlation
A question was recently asked by a new user, SomeoneHAHA, and then deleted by the user, after receiving an answer. I think the question and the answer (QA) to it may be of interest to some users. ...
- 128k
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 :
$$P\left(\sup_{t\in [0,...
- 245
2
votes
2
answers
161
views
Determine the affine envelope of a random process's MGF
Suppose that a stationary random process $S(t)$ can be characterized as the figure below, which for most of the time is a straight line $S(t)=c\cdot t$, but occasionally would "stall" for a ...
- 265
2
votes
1
answer
150
views
Normalized concentration inequality for empirical CDF (iid sum)
Consider the empirical and population CDF,
$$
F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad
F(t) = \mathbb{E} [F_n(t)],
$$
where above $X_1, \dots, X_n$ are iid, real-...
- 390
2
votes
1
answer
421
views
A question about Gaussian Processes suprema
Suppose $\{X_t; t \in \mathcal{X}\}$ is a centered Gaussian Process with covariance function $k(\cdot,\cdot)$, and let $d(x,y) = \mathbb{E}[(X_x-X_y)^2]$.
I am trying to find a tail bound for the ...
- 145
1
vote
1
answer
143
views
Comparison of hitting probability of two Markov chains both with only one absorbing state version 2 under stronger condition
Let $N_n:=\{1,2,\cdots,n\}$. Given two finite states Markov chains $\big(X^{(j)}_i\in N_n\}\big)_{i=0}^\infty$ for $j\in\{1,2\}$, both of which have two absorbing states at $1$ and $n$.
$\text{Pr}\...
- 2,239
1
vote
1
answer
335
views
Finding a connection between two types of convergence
Please, help me find connections between two types of convergence:
Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences:
1) $X_n \...
- 31
0
votes
0
answers
87
views
Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables
Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
0
votes
1
answer
92
views
Does point process ordering ever imply conditional intensity ordering?
Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
- 215