All Questions
Tagged with pr.probability stochastic-processes
1,618 questions
3
votes
1
answer
84
views
What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?
Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having ...
- 73
3
votes
0
answers
80
views
Asymptotics of number of running maxima of iid random variables
Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities.
Given a realisation $\omega$ of the random variables, we say that $X_i (\...
- 6,155
5
votes
1
answer
375
views
Convergence of random functions
Suppose I have a sequence of random continuous functions, $f^{n} : [0, t] \to \mathbb{R}$. Suppose there also exists a random continuous function, $f: [0, t] \to \mathbb{R}$, defined on the same ...
- 85
0
votes
0
answers
31
views
Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales
Does anybody know a reference for the following theorem?
Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale.
Then, for any constant $c > 0$, the event $(\exists
> t)\, X_t \...
- 183
4
votes
0
answers
116
views
Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
- 41
2
votes
1
answer
65
views
On the stationarity of Gaussian processes
I am trying to understand and prove the statement:
The normal (or Gaussian) process is stationary in the wide sense if and only if it is strictly stationary.
I know the following:
A strictly ...
1
vote
0
answers
58
views
Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
0
votes
1
answer
51
views
Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, ...
- 5,405
4
votes
0
answers
62
views
Why optional stopping theorems require continuity conditions of martingales?
If we want to prove some form of optional stopping theorem (with a stopping time $T$) for continuous time martingales $M_t$, a typical strategy is to assume that $\mathbb E[M_{T\wedge n}] = \mathbb E[...
- 1,755
1
vote
0
answers
41
views
Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
- 1,677
2
votes
0
answers
67
views
The unique weak solution to some SDE yields the unique strong solution?
For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, ...
- 1,389
2
votes
1
answer
111
views
What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...
- 6,155
8
votes
1
answer
582
views
One flip coin game
Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which.
He has \$1 with which to bet with. On ...
- 6,155
0
votes
1
answer
57
views
Lower bounding an alternating series with signs from a martingale difference sequence
Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that
$$M_n := \sum_{i = 0}^n \epsilon_i$$
is a martingale.
We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
- 6,155
7
votes
2
answers
841
views
Why is $\mathbb R^{\mathbb N}$ not high-dimensional enough?
In this paper [1], the authors consider the limiting distribution of $$S_{n,p}:=\frac{1}{\sqrt n}\sum_{k=1}^nX_k$$ for $p\rightarrow\infty$ as $n\rightarrow\infty$, where $X_1, X_2,\dots, X_n$ are ...
- 181
4
votes
0
answers
132
views
Absolute Continuity of the Karhunen-Loeve expansion coefficients
The Karhunen-Loeve theorem (see these notes or the wikipedia page, for example) states the following:
Theorem: For a continuous, square-integrable, centered stochastic process $(X_t)_{t \in T}$ (with ...
- 73
3
votes
0
answers
144
views
Distribution of Brownian motion conditional on linear growth
Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.
Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event
$$ E_T := \{|B_s| \geq \lambda s\ \...
- 6,155
9
votes
1
answer
257
views
Higher or lower? (#2)
$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or ...
- 6,155
2
votes
2
answers
88
views
Can the solution to a controlled SDE with additive noise have non full support?
Let $W$ be a standard $d$-dimensional Brownian motion. Consider the following SDE
$$dX_t = b(X_t, u_t) \, dt + dW_t$$
with initial condition $X_0 = 0$ a.s., $b: \mathbb R^d \times \mathbb R^n \to \...
- 6,155
6
votes
1
answer
659
views
On the martingale betting scheme
For a fixed probability $0 < p < 1$, let $X^p$ be the martingale that goes up by $1$ with probability $p$, and goes down by $\frac{p}{q}$ with probability $q := 1-p$.
Write $X$ for the ...
- 6,155
1
vote
0
answers
40
views
Renewal Process with inter-arrival times having quadratic tails
Consider a sequence $(X_n)_{n \ge 1}$ of i.i.d. Pareto($2$) random variables, which means
$$
\mathbb{P}( X_1 > x) =
\begin{cases}
1/x \qquad &\text{for } x \ge 1
\\
1 \qquad & \text{else}....
- 116
6
votes
1
answer
133
views
Coupling/Ordering of Brownian bridges
Suppose I have two 1D Brownian bridges $(B^{(1)}_t,t\in [0,1]),(B^{(2)}_t,t\in [0,1])$, one from $0$ to $0$ and one from $x$ to $y$ where $x,y \geq 0$. Is there a neat way to show that there exists a ...
- 228
1
vote
1
answer
60
views
Reverse Doob’s maximal inequality for bounded martingales
Consider the set of discrete or continuous time $L^\infty$-bounded martingales $X$ with $X_0 = 0$ almost surely. Here $L^\infty$-bounded means $\|X\|_{\infty} := \sup_t \mathbb \|X_t\|_{L^\infty(\...
- 6,155
3
votes
1
answer
181
views
A nice terminal inequality for martingales
Let $X_t$ be a continuous time martingale taking with $\sup_t \mathbb E[X_t^-] < \infty$, and $X_0 = 0$ almost surely. Assume further that $X_1$ admits a probability density function.
Is it true ...
- 6,155
5
votes
2
answers
557
views
A race to the bottom
Nate has a biased coin that comes up heads $\frac{1}{2} + \delta$ proportion of the time, where $0 < \delta \leq \frac{1}{2}$. He is competing against a large number $N$ people who each have fair ...
- 6,155
0
votes
1
answer
64
views
Sharpening Doob’s upcrossing inequality for Brownian motion
Note: This question is heavily related to a series of posts ([1], [2]) by user GJC20.
Provided a martingale $X$ in continuous-time, Doob's upcrosssing inequality states:
If $U(a,b)$ denotes the number ...
- 6,155
0
votes
2
answers
60
views
Do continuous martingales satisfy this nice terminal inequality?
Let $X$ be a continuous, non negative martingale on $[0, 1]$ with $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. Assume further that $X_1$ admits a probability density function. Is it true that the ...
- 6,155
4
votes
1
answer
150
views
Convex order between Gamma distributions and Exponential distributions
Let $ (b_1, \dots, b_n) $ be a tuple of positive integers. Define independent random variables $ Y_i \sim \text{Gamma}(b_i, b_i) $ (shape and rate parameter both equal to $ b_i $) for $( i = 1, \dots, ...
- 55
2
votes
0
answers
71
views
Assumptions Wald's second equation?
Let $(X_n)_{n\in \mathbb{N}}$ be an i.i.d. sequence of random variables and $N$ an $\mathbb{N}_0$ valued random variable. Let $X_1 \in \mathcal{L}^2$ and $N \in \mathcal{L}^1$. Let $S_n := \sum_{i=1}^...
- 261
3
votes
1
answer
218
views
Pathwise linearization of diffusion processes
Let $W$ be a standard $n$-dimensional Brownian motion, and $X$ the diffusion process given by the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $\mu: \mathbb R^n \to \...
- 6,155
5
votes
0
answers
112
views
Discrete random walk in an expanding cage (i.e. in a growing domain)
In the book "A guide to First-Passage Processes" by Sidney Redner, a section is dedicated to the survival probability of a random walker in a growing domain.
For a fixed-length interval $[0,...
- 274
4
votes
1
answer
110
views
Scaling of stopped Hölder norm of Brownian motion
I'm interested in the behaviour of the stopped $\alpha$-Hölder norm of a one-dimensional real-valued Brownian motion $(B_t)_{t \geq 0}$ for $\alpha < 1/2$.
For fixed $T>0$, self similarity ...
- 153
3
votes
0
answers
145
views
What is an example of a non-tight probability measure?
Billingsley (Convergence of Probability Measures, 1968) and van der Vaart and Wellner (Weak Convergence and Empirical Processes, 2023) discuss the concept of tight probability measures and use the ...
- 277
4
votes
0
answers
113
views
SPDE Renormalization
some SPDE (in higher dimensions) can only be interpreted in a "renormalised" sense. For example considering $\Phi_2^4$ on $\mathbb{R}_+\times \mathbb{T}^d$ the solution is defined as the ...
- 573
1
vote
1
answer
185
views
Sum of $X_k$ with $\mathbb{P}(X_k=\pm 1) = 1/2\pm 1/(2\sqrt{k})$
Let $\{X_k\}$ be a sequence of mutually independent random variables with
\begin{align}
\mathbb{P}(X_k = 1) & = \frac{1}{2} + \frac{1}{2\sqrt{k}},
\\
\mathbb{P}(X_k = -1) & = \frac{1}{2} - \...
- 269
2
votes
1
answer
59
views
The ranked mass process associated with a Lambda-coalescent
I am reading a paper by Pitman (1999), and I am confused by his Corollary 7. First some notation so that I can explain my confusion:
$\mathcal{P}_\infty$ is the space of partitions of $\mathbb{N}$, $\...
- 203
1
vote
1
answer
131
views
Breiman's first exit times from a square root boundary generalization
The paper "First exit times from a square root boundary" by Breiman, generalizes an observation made by Blackwell and Freedman. In summary: given a zero-mean random walk $S_n$ with i.i.d. ...
- 63
9
votes
3
answers
448
views
All stationary martingales are constant?
Suppose $(X_{n})_{n\geq{1}}$ is a stationary process that is a martingale with respect to some filtration. Suppose also that $\mathbb{E}X_{0}^{2}<\infty$ so that $\mathbb{E}X_{n}^{2}<\infty$ for ...
- 662
2
votes
1
answer
89
views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
- 177
3
votes
1
answer
139
views
On the permutation consistency condition in the Kolmogorov extension theorem
I am trying to understand the Kolmogorov extension theorem (KET). Wikipedia states the following two necessary consistency conditions for the probability measures $\nu_{t_1,\cdots,t_k}$:
$$\nu_{t_{\pi(...
- 133
0
votes
0
answers
85
views
Does a 2d random walk hit 0 for increasing distances AND time spans?
Question: For a simple symmetric random walk $(Z_t)_{t\geq 0}$ in $\mathbb{Z}^2$, does
$$\lim_{\beta\rightarrow 0}\mathbb{P}^{x_\beta}(Z_t=0\text{ for some }t\leq h(\beta)T)=0\quad (2.8)$$
where $|x_\...
- 31
4
votes
1
answer
196
views
(Lattice approximation) Does UV stability lead to continuum limit of a subsequence?
In the context of lattice approximation, the term "UV stability" seems to be used frequently. To me, it seems like
Uniform boundedness of the partition function in the limit where lattice ...
- 3,477
0
votes
0
answers
39
views
Random subsets of measure spaces
Related to generalizing reliability polynomials from graph theory to other spaces I ran into the following question.
To start, take a finite set $M$ and build a subset $X$ of $M$ at random by ...
- 646
4
votes
0
answers
87
views
Statistics of random Voronoi S-tessellations
Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to ...
- 41
2
votes
0
answers
157
views
Conformally mapping between the upper half complex plane, and the plane with a tree on spatial points removed
A stochastic process such as SLE$_{\kappa}$ can be defined by taking the scaling limit of a curve in the upper half complex plane: put simply, one removes a line segment, then another, $n$ times, each ...
- 640
1
vote
0
answers
114
views
An urn model with weighted objects and replacement
Consider the following game:
In an urn, there are $K$ balls, $x_0$ of them are blue and light (mass $m_0$), $x_1$ are blue and heavy ($m_1$), $x_2$ are red and light ($m_2$), the rest $x_3$ are red ...
- 31
2
votes
0
answers
61
views
Characterisation of Bessel process
Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that
For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
- 177
19
votes
2
answers
2k
views
Higher or lower?
Consider the following game - I draw a number from $[0, 1]$ uniformly, and show it to you. I tell you I am going to draw another $1000$ numbers in sequence, independently and uniformly. Your task is ...
- 6,155
5
votes
1
answer
192
views
Non-equivalent definitions of Markov process
As far as I know, there are three definitions of Markov processes (or of Markov chains).
DEFINITION 1 (WEAKER). A process $(X_t)_{t\in[0,\infty)}$ on $(\Omega,\mathcal{F},\mathbb{P})$ with values in ...
- 1,149
8
votes
1
answer
522
views
One step forward, one step back
$N \geq 2$ players play a cooperative game on the integers $\mathbb Z$. All of them start from $0$. At each turn, they are simultanously given the same yes or no question to answer.
The questions ...
- 6,155