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.
9,027 questions
2
votes
1
answer
122
views
change the sign of volatility
Assume the time inhomogeneous SDE
$dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$
has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE
$dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))...
- 23
1
vote
1
answer
113
views
What is the probability that all numbers in a set P are unique and each number in P is chosen randomly between 1 and n^3? [closed]
Hope someone can help me answer this question.
The problem is described as below.
I want to form a set (P) of n numbers. I randomly choose a number between 1 and n^3 and I choose n times.
My ...
- 11
8
votes
1
answer
1k
views
Topological necessary and sufficient condition for tightness
Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:
For each $\varepsilon>0$, we can find a compact subset $K$ of $X$...
- 3,993
1
vote
0
answers
217
views
Calculating or estimating a combinatorial multivariate sum
Dear all,
I'm currently looking at a problem in which the following combinatorial product emerges:
$c(m_1,\dots,m_\lambda;n_1,\dots,n_\lambda)=\frac{m_1 !}{(m_1-n_1)!}\frac{(m_1+m_2-n_1)!}{(m_1+m_2-...
- 41
1
vote
1
answer
231
views
Asymptotic behaviour of a mean
Fix $x>0$ and $c\in\mathbb{N}$. Let $f(x):=\frac{c}{4c-2+2x^2}$ and
$$m_N(x):=\frac{1}{N} \sum_{i=0}^{f(x)N} \log(\frac{c N}{2}-i(2c-1))$$
I'm pretty sure $m_N(x)\to\infty$ as $N\to\infty$.
I ...
- 293
10
votes
1
answer
210
views
Distribution of the maximum of the norm of k-averages of n i.i.d. d-dimensional random vectors
Suppose $X_1, ... X_n$ are i.i.d. random vectors in $d$-dimensional space (i.e., $R^d$) with continuous centrally symmetric density function $f(\cdot)$ (i.e., symmetric with respect to the origin). ...
- 121
11
votes
0
answers
536
views
Bounding the probability that a random variable is maximal
Question: Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$.
I am looking ...
- 119
1
vote
0
answers
177
views
Conditioning over Conditional probability? (also: $\phi$-mixing sequences)
For two sub $\sigma-$fields $\mathscr{F}$ and $\mathscr{G}$ of a probability space $(\Omega , \mathscr{A} , P)$ we define $\phi$ mixing as follows:
$$
\phi(\mathscr{F},\mathscr{G}) = \sup \{ |P(G|F) - ...
- 11
0
votes
1
answer
343
views
Path properties of Levy Processes
I would appreciate if someone helps me with introducing a reference explaining the path properties of Levy Processes. In other words, I want to know a good interpretation of the Levy - Khintchine ...
- 1
6
votes
4
answers
3k
views
Calculating the probability of an event defined by a condition on a Gaussian random process
Although the question itself can be expressed succinctly, I couldn't come up with a nice self-explanatory title - suggestions are welcome.
Motivation/Background
I was investigating whether it would ...
18
votes
1
answer
996
views
Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
1
vote
1
answer
166
views
Is the following statement true? $E[\xi U^{'}(\xi)] < +\infty$?
I encounter the following problem today. It seems a simple question.
Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions:
(1) $U$ is concave, continuous, ...
- 23
7
votes
0
answers
813
views
Wasserstein distance between two diffusion processes.
I would like to know if there exists a formula to compute the $L^2$-Wasserstein distance between the laws $P_1$ and $P_2$ in path space of two diffusion processes:
$$dx_t=f_1(x_t)dt + \sigma_1(x_t)dB^...
- 840
3
votes
1
answer
515
views
A probability question about removing stones from piles
I have run across a question that seems like it should have a well known answer, but I can't find one, so I thought I would ask this hive mind:
Suppose we start with t piles of s rocks each. In a ...
- 205
4
votes
0
answers
1k
views
The spectrum of a Markov Operator and Invariant Measures
Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
- 509
3
votes
2
answers
352
views
Continuity of hitting distributions
Hi everybody
Let $U$ be the domain (as shown in the picture) and $\bar{U}$ its closure, further more set $\partial_r U$ to be the reflecting boundary and $\partial_a U$ the absorbing one. The process ...
- 279
31
votes
4
answers
2k
views
Probability of zero in a random matrix
Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, ...
- 37.7k
9
votes
3
answers
486
views
Representing a real number as the value of a countably infinite game
Is it true that for any real number $p$ between 0 and 1, there exist finite or infinite sequences $x_m$ and $y_n$ of positive real numbers, and a finite or infinite matrix of numbers $\varphi_{mn}$ ...
- 187
5
votes
2
answers
356
views
$L^\infty$ properties of an infinite-dimensional Gaussian semigroup
Let $W$ be a separable Banach space and $\mu$ a Gaussian Borel measure on $W$ which is centered and non-degenerate. For $F : W \to \mathbb{R}$ bounded Borel and $t \ge 0$, let
$$P_t F(x) = \int_W F(x+...
- 29.8k
1
vote
0
answers
223
views
Percolation on infinite percolation clusters
Let's consider an infinite percolation cluster $\mathcal{C}_p$ that takes shape in the supercritical phase ($p>p_c$) of a bond percolation in $\mathbb{Z}^d$. What could be said about a bond ...
user16782
14
votes
1
answer
956
views
Partitioning the vertices of an n-cube with random hyperplane cuts
An evolutionary biologist asked me a question which boils down, at least in part, to what seems to me an interesting question of combinatorial/probabilistic geometry.
It is an old chestnut of a ...
- 19.2k
8
votes
3
answers
2k
views
What is the optimal growth of the constant in BDG?
Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |X|^...
- 4,010
1
vote
2
answers
528
views
Conditional expectation and algebraic expressions
Let $\mathcal{A}$ and $\mathcal{B}$ be two sub-$\sigma$-algebras in a measure space. To each one, there is a conditional expectation associated, respectively $E^\mathcal{A}$ and $E^\mathcal{B}$. Given ...
- 157
0
votes
1
answer
101
views
multimodal circular model
Hi, can someone provide me with a list of probability models that is akin to Von Mises but consists multiple (potentially infinite) modes that takes into account attractors in the entire 2-D spatial ...
- 61
2
votes
2
answers
625
views
Measuring the independence between the components of a stochastic process
In a context of blind source separation (e.g. you want to extract the voice of a singer from a song), many approaches consist in maximizing the independence between the components of a certain ...
- 103
0
votes
2
answers
225
views
Estimating joint and conditional probabilities with incomplete information
I'm working on an application for which it would be great to have the following functionality:
Say that you have a collection $C$ of $n$ events, for now let's set $n = 3$ and call the events $a, b,$ ...
- 21
3
votes
2
answers
247
views
Exact simulation of a large sample histogram
Say I want to create a histogram of $N$ random points from some simple compactly supported distribution on $\mathbb{R}$, where $N$ is very large, say $N = 10^{30}$. The histogram has $K$ disjoint bins,...
- 223
0
votes
1
answer
329
views
Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]
I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
11
votes
0
answers
601
views
High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
- 1,666
4
votes
0
answers
216
views
How should one generate a random set of mappings?
My motivation for this question comes from the study of synchronizing automata. There is a general consensus that random automata are synchronizing and have short synchronizing words. I am hoping ...
- 38.6k
8
votes
2
answers
990
views
What is the tropical Robinson-Schensted-Knuth correspondence?
And what are it's applications? A conceptual explanation would be great! Is there an expository note about this somewhere?
Some references have already appeared in the answers and comments below. To ...
- 85.6k
1
vote
1
answer
191
views
Generalization of Gauss's inequality for not necessarily unimodal distributions?
Gauss's inequality is for unimodal distributions, concerning distance from the mode.
A similar result is Vysochanskiï–Petunin inequality, which is for the distance from the mean rather than the mode....
- 357
17
votes
3
answers
923
views
Random permutations from Brownian motion
Let $B(t)$ be a Brownian motion. The ordering of $(0, B(1), ..., B(n-1)) $ is a random permutation in $S_n$. This is not uniform for $n>2$ since the probabilities of the identity permutation $[123.....
- 28k
0
votes
0
answers
130
views
span of symmetrically truncated symmetric random variables
If $X_i$ are symmetric independent random variables, is $\vert \sum X_i I_{\vert X_i \vert < N_i}\vert $ stochastically smaller than $\vert \sum X_i \vert$ ? Is it comparable in any way which ...
- 90
4
votes
2
answers
420
views
Generating a group by randomly sampling generators
Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is full if it acts as a nonidentity element of $G$ in ...
- 2,649
4
votes
0
answers
753
views
Monte Carlo sampling high dimensions with the halton sequence?
Referring to the Halton Sequence, Swiler et al 2006 state that
In cases where a large number of input variables are sampled,
Robinson and Atcitty recommend using a leaped sequence, where the
...
- 225
3
votes
1
answer
555
views
Cover time and intersection time of random walks
Consider a simple lazy random walk on an $n$-vertex undirected, connected graph: this is the Markov chain which transitions from $i$ to $j$ with probability $p_{ij}=1/(2d(i))$ where $d(i)$ is the ...
- 571
3
votes
1
answer
335
views
Stochastic processes having Markov kernels
Let $(\Omega_1, \mathcal{F}_1, P_1)$ and $(\Omega_2, \mathcal{F}_2, P_2)$ be probability spaces and suppose $(X_t)$ and $(Y_t)$ are real-valued stochastic processes defined on the respective spaces. ...
- 33
8
votes
1
answer
2k
views
Van Den Berg-Kesten-Reimer inequality
Van Den Berg-Kesten-Reimer inequality
Let $n$ be a positive integer. For $i\in[n]$, let $\Omega_i$ be a finite set and $\mu_i$ a probability measure on it. Set $\Omega=\Omega_1\!\times\!\ldots\!\...
- 148
5
votes
0
answers
397
views
Concentration of functions of random unitary matrices
Suppose $U$ and $V$ are $n \times n$ random unitary matrices, chosen independently from the Haar measure. Is there any kind of concentration inequality which would be applicable to polynomials $p(U,V)$...
- 2,449
2
votes
0
answers
763
views
Brownian motion & time shift
Hi,
I am just starting to study the theory of Brownian motion and I was wondering whether the following was true.
We consider a one-dimensional, one sided, Brownian motion process.
For $A$ an ...
- 678
1
vote
0
answers
130
views
Divisible Random Variables
Suppose I can write a positive, real valued random variable
$$ X = m_1 X_1 + m_2 X_2,$$
where $m_1$ and $m_2$ are i.i.d, $X_1$ and $X_2$ are i.i.d and moreover, the $X_i$ are distributed like $X$. ...
- 195
4
votes
1
answer
2k
views
Bounding Entropy in terms of KL-Divergence
Let $h(X)$ be the differential entropy of a continuous random variable $X$ with density $f$, and let $Y$ be another continuous random variable with density $g$. If $KL(X\mid\mid Y)$ is the Kullback-...
- 49
12
votes
5
answers
3k
views
Properties preserved under passage to augmented filtration
Dear all,
generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration.
Recall that if $(\mathcal{F}_t)_{...
- 121
4
votes
1
answer
1k
views
Probabilty of two permutations having common elements?
What is the probability of two permutations on set X of size m (i.e. |X|=m) having at least n points of intersection? By this I mean that if two permutations, which I'll call g(x) and h(x), map a ...
- 55
0
votes
0
answers
321
views
Expected value of a logarithm of a Levy process
I have a strictly positive Levy process $(L_t)$ with no Brownian part, drift $\gamma$ and jump measure $\nu$. Is it possible to calculate the expected value of the logarithm of this process, so $\...
- 667
3
votes
1
answer
1k
views
"Generalized" monotonicity of the expected value
Let $X$ and $Y$ be random variables
with cumulative distribution functions $F_X(t)$ and $F_Y(t)$ respectively.
Suppose that
$\forall t \in \mathbb{R} \ F_X(t) \geq F_Y(t)$.
Does it imply that $E(X) \...
- 401
11
votes
2
answers
2k
views
Expected values of traces of products of random matrices
Suppose I want to compute a quantity of the type:
$\mathbb{E}\mathrm{tr}(AUBU^{\ast})$
where averaging is over Haar measure on the unitary group $\mathcal{U}(n)$ (one can of course consider higher ...
- 3,323
0
votes
2
answers
2k
views
Rank $k$ of a sequence of random variables
Suppose one has $n$ real random variables $X_1, X_2, \dots, X_n$ from a certain distribution. Sort these random variables to get a sequence $Y_1, Y_2, \dots, Y_n$. What is known about the distribution,...
- 3
2
votes
0
answers
105
views
Modelling a GI/G/1 queue with exceptional first vacation and multiple vacations
I am currently working on a stochastic modelling problem in networks. I have a G/G/1 queue where
the interarrival and inter-service times are iid
the arrival and service time distributions are ...
- 519