Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,021 questions
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Maximizing expectation of gaussian process over covariance matrix with fixed trace
Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded.
For each $\Sigma \...
- 4,653
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votes
0
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36
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Contribution of Fisher information near jump points in convolved probability distributions
I am trying to compute the contribution to the Fisher information from jump points $b_i(\theta)$ in the convolved function $f(x; \theta)$ with respect to the parameter $\theta$. I am unsure whether it ...
- 63.9k
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1
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65
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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 ...
- 128k
3
votes
1
answer
219
views
Interpretation of an asymptotic result in probability
A result in asymptotic theory says the following: Let $Y$ be a real random variable with full support and $E(|Y|)<\infty$. Assume that:
$$
(A)\quad \lim_{y\rightarrow \infty} \frac{\Pr(Y\geq y)}{\...
- 128k
4
votes
1
answer
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Expected value of a stochastic integral expression
I am wondering if the following expression can be processed a bit analytically,
$$
E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right],
$$
where $W_u$ is the normal Brownian motion (1D Wiener process), and $...
CommunityBot
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2
votes
1
answer
242
views
Modify a random variable to make its range Borel?
Let $X: \Omega\to{\mathbb R}$ be a random variable. Is it always possible to modify it (i.e. change the value of $X$ on a subset of $\Omega$ of zero measure) so that the range of $X$ is a Borel set?
...
49
votes
13
answers
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Why is it so cool to square numbers (in terms of finding the standard deviation)?
When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why ...
- 30.1k
4
votes
0
answers
219
views
Conditional distribution of steps of random walk given the sum
Set-up. Consider a random walk $S_n=\sum_{i=1}^n X_i$, where $\{ X_i, 1\leq i < \infty \} $
is a sequence of i.i.d. random variables with distribution $\mu$, $\mathbb{E}X_1 = 0$. Let $a > 0$.
...
- 724
3
votes
1
answer
405
views
Moments of a random variable related to uniform distribution on sphere
Let $u$ be taken uniformly from the unit sphere $\mathbb S^{n-1}$ and $D$ be a diagonal matrix. I'd like to find a general formula for
$$
\mathbb E[(u^\top D u)^m]
$$
for $m=1,2,3, \dots$, in terms of ...
- 188k
1
vote
1
answer
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Limit (convergence) of stopping times
Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
CommunityBot
- 1
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votes
0
answers
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Combinatorial/probabilistic interpretation of a quantity of union closed family
Let $\mathcal{F}\subseteq2^{[n]}$ be a union-closed family of sets. For a set $S\in[n]$ (not necessary belong to $\mathcal{F}$), define $w_{\mathcal{F}}(S)$ to be the number of subset of $S$ which ...
- 32.5k
3
votes
1
answer
158
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Upper and lower bounds for a Rademacher-type expectation
Suppose that $\varepsilon_i$
are independent Rademacher random variables
(that is,
$
\mathbb{P}(\varepsilon_i=-1)
=
\mathbb{P}(\varepsilon_i=1)
=1/2
$.
Fix an $a\in\mathbb{R}^n$
and define the random ...
- 258
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votes
1
answer
66
views
Does convergence in probability of iid samples imply convergence in measure of the sampled functions?
Let $g_i: [0, 1] \to \mathbb R$ be $L^1$ functions, equibounded in $L^1$ norm. Let $X_i$ a sequence of iid uniform random variables on $[0, 1]$. Suppose that
$$\frac{1}{n} \sum_{i = 1}^n g_i (X_i) \to ...
- 128k
8
votes
1
answer
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The cars problem, again
Consider the following simple problem: We are given $2n$ parking spots, labelled from 0 to $2n-1$. There are $n$ cars on the first $n$ spots, and the remaining $n$ spots are free. At every step, every ...
-3
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0
answers
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Approximation on Dirichlet's arithmetic progression by means of central limit theorem
In this video lecture on
Number theory over function fields taught by Will Sawin
is presented a 'conceptional' reason for error estimation
$\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \}
=\frac{1}...
- 39.9k
0
votes
1
answer
227
views
Constructing Markov chain
Let $(A_1,B_1)$ and $(A_2,B_2)$ be two random variables with the joint distributions $p_{A_1B_1}$ and $p_{A_2B_2}$, respectively. Moreover, we have
$$\mathbb{P}[(A_1,B_1)\neq (A_2,B_2)]=\alpha.$$
Then,...
CommunityBot
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vote
0
answers
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views
How to optimize parametric information-theoretic bounds?
I am faced with an information-theoretic upper bound, such as
\begin{align}
\sqrt{\alpha'}2^{I_\alpha(X;Y)},
\end{align}
where $I_\alpha(X;Y)$ is the Rényi mutual information with parameter $\alpha>...
- 287
3
votes
0
answers
130
views
A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)
As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
- 141
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votes
1
answer
65
views
Strict positive definite function gradient tuple
I have a (Gaussian) random function (aka "stochastic process" or "random field") $(f(t))_{t\in \mathbb{R}^d}$. I now want to consider the vector valued random function $g=(f, \...
- 385
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votes
2
answers
126
views
Unique coupling
Let $X$ be a Polish metric space, and let $\mu,\nu$ be two Borel probability measures on $X$, when is the product measure the only coupling of $\mu$ and $\nu$. More formally, let $$\Gamma(\mu,\nu):=\{\...
- 1,724
7
votes
1
answer
763
views
Reference request: discretisation of probability measures on $\mathbb R^d$
Given a probability measures $\mu$ on $\mathbb R^d$ with finite first movement, i.e.
$$\int_{\mathbb R^d}|x|\mu(dx)~~<~~+\infty.$$
My concern is to approximate $\mu$ some $\mu_n$ that is ...
CommunityBot
- 1
10
votes
0
answers
338
views
Simultaneous strong law of large number classes?
Say that $C$ is a SSLLN class of subsets of some Polish space $V$ provided that for every sequence of Borel i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\...
- 2,555
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votes
0
answers
343
views
Can KL divergence go to 0, but $E[\log(p/q)^2]$ diverge in certain cases?
Let $p(x)$ be a fixed distribution over a discrete space.
Let $A, C > 0$ be constants.
Let $\epsilon > 0$. Can we find an example of a distribution
$q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\...
- 3,065
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votes
0
answers
52
views
Density of squared bessel process
I was trying to find a transition density function for a squared Bessel process. In the book "Continuous martingale and Brownian motion" by Revuz and Yor, I find a Corollary on page 441 that ...
- 3,065
-1
votes
0
answers
27
views
Number variance of random points (and deviations for empirical processes)
Let $X_1, X_2, \dots$ be i.i.d. random variables having uniform distribution on $[0,1]$. Write $I_{t,x}$ for the indicator function of an interval of length $x$ with center $t$. Consider
$$
V(N,x) = \...
- 2,207
1
vote
1
answer
197
views
Probability distribution on Python-dictionary-like objects?
I would like to examine information-theoretical properties of random variables that take as values objects which are akin to dictionaries in the Python programing language.
That is, each sample of the ...
CommunityBot
- 1
0
votes
1
answer
72
views
Lower Bound on the Probability for the Sum of IID Random Variables
Let $X_1,\ldots,X_n$ be $n$ iid normalized random variables (with finite variance, possibly sub-Gaussian).
Suppose further that $\mathbb{P}(X_1 > 0 ) > 1/2$, implying a positive skew in the ...
- 128k
10
votes
1
answer
1k
views
Joint law of the time integral of Brownian motion and its maximum
Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
$$I_t=\int\limits_0^tW_s\,\...
- 5,405
1
vote
2
answers
278
views
Is integral of adapted separable process adapted?
Assume $f(t,\omega)$ is (i)separable, (ii) measurable as function from $((0,T)×\Omega)$ into $R$ and (iii) is adapted to the filtration $F_t, 0<t<T$
Also $\int_0^Tf^2(s)ds<\infty$ almost sure....
CommunityBot
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4
votes
1
answer
530
views
On stochastic integration
This questions has been asked on math.stackexchange
I have two questions on stochastic integration.
(1) Constructing the Ito integral, there is the following remark in Jacod/Shiryaev (page 46, 2nd ...
CommunityBot
- 1
7
votes
0
answers
264
views
$\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]$ where $W$ and $V$ are independent
Let $V$ and $W$ be independent random variables. Assume that $V$ is standard normal.
We are interested in the following limit
\begin{align}
\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]
\end{align}
...
- 671
1
vote
0
answers
58
views
Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
- 109k
2
votes
1
answer
415
views
High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)
Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
CommunityBot
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1
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Concentration of the norm of subGaussian random vectors
I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin.
I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I ...
CommunityBot
- 1
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votes
1
answer
95
views
On the behaviour of individual random walks of a Markov Chain
My current research (on Probabilistic Automaton) brought me to the following question regarding Markov Chains. I state the definitions for the sake of clarity.
Let $M$ be a discrete-time finite Markov ...
- 232
3
votes
1
answer
1k
views
Borel-Cantelli lemma for general measure spaces (those with infinite measure)
The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure.
But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
CommunityBot
- 1
8
votes
1
answer
585
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 ...
CommunityBot
- 1
3
votes
0
answers
92
views
Tighter Freedman's inequality for a special martingale difference sequence
Let $X_{1}, \ldots, X_{T} \in \{0, 1\}$ be a sequence of Boolean random variables with
$$
\mathbb{E}[X_{t} | X_{1}, \dots, X_{t - 1}] = p_{t}.
$$
Consider the sequence $Y_{t} := X_{t} - p_{t}$ (which ...
- 63.9k
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 $\...
- 3,669
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, ...
- 128k
2
votes
1
answer
119
views
Deriving the distribution of standardized variables with empirical mean and standard deviation
I'm working with a set of independent and identically distributed random variables $\{ x_i \}_{i=1}^N$, where each $x_i$ follows a Gaussian distribution $P_X(x) = \mathcal{N}(x; \mu, \sigma^2)$. This ...
CommunityBot
- 1
4
votes
1
answer
227
views
Problem in Probability Theory and Functional Analysis
Let's consider the vector space V of bounded scalar functions, which includes the constant function 1. We assume that any uniform limit of a bounded monotonic sequence of functions from V also ...
1
vote
1
answer
329
views
Hölder continuity of Radon transform of smooth function
Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
CommunityBot
- 1
3
votes
1
answer
604
views
Weighted sum of standard Brownian bridges
Let $\{B_j\}_{j=1}^k$ be a sequence of Brownian bridges.
Let us consider $$X(t)=\sum_{j=1}^m w_j(t)B_j(t),$$ where $w_j$ are positive weight functions.
Then what can we say about (distribution or may ...
CommunityBot
- 1
5
votes
2
answers
424
views
Existence of an invariant measure on an infinite dimensional space via Lyapunov functional
Set-up.
Assume that we have a complete separable metric space $\mathcal{X}$ that is not locally compact. Let $V: \mathcal{x} \to [0; +\infty]$ be a functional such that $K_r :=\{x \in \mathcal {X} : V ...
3
votes
2
answers
282
views
Nash equilibria of a "minority game"
An odd number $N \geq 3$ of players are playing a game - they bet on the outcome of a biased coin that comes up heads $p > \frac{1}{2}$ of the time, where $p$ is known to all of the players in ...
- 745
1
vote
0
answers
42
views
Sub-Gaussian analysis via bounded decomposition?
Let $\psi_\alpha(x) := \exp(x^\alpha)-1$.
The Sub-Gaussian Norm $\lVert X \rVert_{\psi_2}$ of a random variable $X$ is defined as
$$
\lVert X\rVert_{\psi_2} = \inf\{c>0\mid \mathbb{E}[\varphi_2(|X|/...
- 2,183
1
vote
1
answer
51
views
How do the total variation distances of the marginals relate to the total variation distance of the joint under independence?
Suppose there are two sets of random variables $X_1,...,X_n$ and $Y_1,...,Y_n$ with all the variables being defined over the same sample space, but not necessarily being identically distributed. Is ...
- 2,183
0
votes
2
answers
116
views
Upper bounds on quotients of binomial coefficients
Let $\gamma>1$ be a real number and let $n\in \mathbb{N}$.
Define $f\colon\mathbb{N}\to[0,1]$
$$
f(n_0) = \frac{\binom{n-n_0}{m}}{\binom{n}{m}},
$$
where
$$
m = \Big\lfloor{\frac{n}{\lceil\gamma ...
- 128k