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.
8,667
questions
0
votes
1
answer
161
views
Finding the right σ-algebra. Question on uncertainty related to the secretary problem
Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.
In this setting it is relevant what is the distribution of the values of the ...
- 25
2
votes
1
answer
264
views
A generalization of negative binomial distribution
Assume we have a set of $n$ balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of ...
- 61
2
votes
1
answer
209
views
Ask for a special function related to the error function
I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...
- 1,619
1
vote
1
answer
64
views
Conditioned sum of n Poissons versus unconditioned Poissons
Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$:
$$\...
- 457
4
votes
1
answer
746
views
Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$
I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...
- 483
3
votes
1
answer
252
views
Zero-one law in binomial random graph model $G(n,p)$
Consider the binomial random graph model $G(n,p)$ with $0<p<1$. We say that $G(n,p)$ satisfies the Zero-One law if for every first order property $Q$ one has $\lim\limits_{n \rightarrow \infty} ...
- 39
7
votes
1
answer
489
views
Distributional equation X+Y=2X
Let $X$ be a positive real-valued random variable. Let $Y$ be an independent copy of $X$ and assume that the equality $X+Y=2X$ holds in distribution. Does this imply that $X$ is constant?
- 406
1
vote
0
answers
134
views
A probability question related to combinatoric problem
I am trying to solve a combinatoric problem. The problem is the following:
There are A,B,C three types of people. There are totally N people arriving sequentially and make a choice between two boxes X ...
- 191
0
votes
0
answers
56
views
Where can I find this article of Doléans-Dade?
I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade.
I could not find a pdf version online, and my university library does not have a printed version.
Thank ...
- 610
4
votes
1
answer
1k
views
Range of random walk
I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...
user92031
4
votes
2
answers
309
views
Convexity of truncated expectation
Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
- 41
0
votes
0
answers
242
views
Hadamard product (Schur product) in $L^2[0,1]$
Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
- 195
16
votes
3
answers
1k
views
Is there a rate of convergence for Donsker's theorem?
For the standard CLT, one can easily estimate a rate of convergence if you assume that the random variables have a little more than two moments.
Let $S_n$ be the centered-scaled sum of $n$ iid ...
- 921
1
vote
1
answer
705
views
Feller property for Ito diffusion with Lipschitz coefficients
Consider the following Ito diffusion $X_t$ satisfying
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$
with Lipschitz coefficients $b,\sigma$.
It can be shown that if $g$ is bounded ...
- 483
4
votes
3
answers
773
views
Why does the overhand shuffle converge to the uniform distribution on $S_n$?
Pemantle 1989 proves, among other things, that the Markov chain on $S_n$ induced by repeatedly and independently performing an overhand shuffle on a deck of $n$ cards is ergodic and has limiting ...
- 633
2
votes
0
answers
205
views
markov processes and ergodic theory
For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
- 169
3
votes
2
answers
247
views
Probability of no $k$ 1's in arithmetic progression in binary sequence of length $n$
It is well known [it's on Wolfram Mathworld, for example] that the probability of no runs of $k$ consecutive $1$'s will occur in a $\{0,1\}$-valued sequence of length $n$ is exactly equal to $$\frac{F^...
- 10.1k
0
votes
1
answer
191
views
What is the relationship between $E(X\mid\mathcal{A})$ and $E(X\mid A)$?
This question seems obvious, but not sure how to prove it.
Let $\mathcal{A}$ be a $\sigma$-algebra, and $X$ be a random variable.
Suppose $E(X\mid A)\le1$ for any $A\in\mathcal{A}$, can we conclude ...
- 87
1
vote
1
answer
184
views
Difficulty with a formula for a probability related to card shuffling
I've been reading this article on the overhand shuffle. In it the author uses a simplied mathematical model of the shuffle:
Pemantle’s model for the overhand shuffle is
parameterized by a ...
- 633
7
votes
1
answer
258
views
Is this simple-looking moment inequality true?
Let $p \ge 1$ be an integer. Does there exist a constant $C_p$ such that for every random variable $X \ge 0$,
$$
\mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C_p \...
- 562
2
votes
0
answers
345
views
formula for density of maximal Poisson disk sampling of radius 1?
Maximal Poisson disk sampling of radius r, applied to a finite planar region, is defined by successively choosing sample points uniformly randomly from the part of the region that is not within ...
- 147
4
votes
1
answer
144
views
Probability of existence of a base in the span of sparse vectors in GF(2)
For $i=1,2,\dots,l$, let $\mathbf{v}_i =(v_{i1},v_{i2},\dots,v_{in}) \in \mathbb{F}_2^n$ be a sparse vector in GF(2) such that all $v_{ij}$'s are independent for all $1 \le i \le l, 1 \le j \le n$ and ...
- 297
2
votes
0
answers
70
views
Existence of probability distribution satisfying upper/lower bounds on events
Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...
- 21
0
votes
0
answers
80
views
Prokhorov convergence of Gaussian measures
Consider a Hilbert space $\mathcal{H}$ and a sequence of centered Gaussian measures $\mu_n$ on it. The covariance operators of $\mu_n$ are defined via their eigenpair(eigenbasis and eigenvalue)) as ...
- 157
12
votes
1
answer
315
views
Convergence of an implicitly defined sequence of random variables
Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
- 171
18
votes
2
answers
633
views
What is the expected value of an N-dim vector of uniform randoms that sum to 1 which have been sorted into descending order?
What is the expected value of an N-dimensional vector of uniformly distributed random numbers which sum to 1 and have been sorted in descending order?
Here is the algorithm for drawing a sample from ...
- 183
4
votes
1
answer
195
views
Can samples be compressed?
The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^...
- 21.9k
5
votes
2
answers
514
views
Concentration of U-statistics for exchangable distributions (and the unbounded case)
Consider the following so-called $U$-statistic of order 2: $$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j)$$ where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If $|h(w_1,...
- 1,659
1
vote
0
answers
85
views
The role of absolute continuity in stochastic ordering defined over sets of probability distributions
This question is about a claim given in this paper (page 261, the remark), but without any proof.
It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ (...
- 359
4
votes
1
answer
1k
views
What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?
Let $X$ be a random variable with $|X|\le1$, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?
Existing results:
It has been known that $E|E(X|\mathcal{A}...
- 87
3
votes
1
answer
246
views
Solving algebraic recurrence relations on a cyclic graph
I have a set of $n$ variables $p_1, \ldots p_n$ with $0 \leq p_i \leq 1$ and a defining equation for each of one of the forms:
$p_i = 0$.
$p_i = 1$
$p_i = p_j p_k$ for some $j, k$ with $i, j, k$ all ...
- 1,331
2
votes
0
answers
95
views
Smoothness of Value function for SDE with discontinuous coefficients
Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function $v:\...
- 21
3
votes
2
answers
206
views
Moving a result from the unconditional to the conditional
I'm generally wary when lifting a result stated unconditionally to a situation where I'm conditioning on a random variable. Consider the following classical result in weak convergence:
Theorem. Let $...
- 321
6
votes
1
answer
827
views
Average minimum number of random k-sparse vectors in GF(2) to span the whole space?
What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...
- 297
2
votes
1
answer
2k
views
Expected value and variance of a stochastic process
I would like to ask if there is a way to find the expected value and the variance of the following process
$$
dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0
$$
where $a\in (-\infty,+\infty), b&...
- 323
2
votes
0
answers
123
views
Quadratic characteristic and constancy
Consider a change of measure on $\mathcal{F}_{t}$ defined by the restriction of two probability measures of the form
\begin{align}
\frac{dQ_{t}(\theta)}{dP_{t}}=\exp^{ \theta A_{t}-\kappa(\theta) S_{t}...
- 247
7
votes
1
answer
321
views
Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution
Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where
\begin{equation}
Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}.
\end{equation}
To ...
- 183
2
votes
0
answers
146
views
Question about continuity in the "complete Skorohod Topology"?
I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process"
https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf
In one of the exercises, exercise 8.9 ...
1
vote
1
answer
95
views
Reference for a local density theorem for binary vectors
I have the following theorem written on my whiteboard, but have misplaced the reference. I believe the probabilistic method may be involved in the proof. Any pointers appreciated.
Theorem Let $v\in\{...
- 10.1k
0
votes
1
answer
1k
views
Predictable quadratic Variation <.> has same intervals of constancy as the process
From
Revuz and Yor - Continuous Martingales and Brownian Motion 1999
Chapter IV Proposition 1.13
it is proven, that for a continuous local martingale $M_t$ the intervals of constancy ...
- 247
2
votes
0
answers
126
views
Construction of a random variable
I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the statement:...
- 701
2
votes
1
answer
181
views
Median of a uniform multinomial variable
Let $k\in\mathbb N^+$ be a positive integer.
Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$.
For $i\in \{1,2,\...
- 608
3
votes
0
answers
145
views
“Local” functional central limit theorem for the empirical distribution function
This question is a repost from Mathematics Stack Exchange, where it did not receive any answer.
Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[...
- 233
1
vote
0
answers
87
views
Convergence of an rcll process along a random subsequence
I have a process $X_s$, for $s \ge 0$, taking values in a Polish space $T$ with an rcll version where I have shown, for every nonrandom increasing sequence $s_n$, that $X_{s_n} \to c$ in probability, ...
- 321
9
votes
5
answers
385
views
Probability theory without deductive closure
Human knowledge is not deductively closed. Uncertainty can arise from that just as much as from lack of brute facts. (When a Harvard graduate was reported to have thought that the earth is farther ...
- 11.9k
1
vote
0
answers
85
views
Majorizing inequality on spectral norm of product of a random and a deterministic low-rank projection
Let $P$ be a rank $k$ uniformly randomly oriented projection matrix in ${\mathbb R}^d$ -- this is constructed as $R^T(RR^T)^{-1}R$ where $R$ is a $k\times d, k<d$ random matrix with i.i.d. 0-mean ...
- 517
9
votes
1
answer
2k
views
Law of large numbers for martingales
I apologize in advance if this question is too basic, but I've received no response on Math Stack Exchange, so perhaps it is more appropriate here:
Let $X_n$ be a square-integrable martingale with $\...
- 93
2
votes
0
answers
189
views
Infinitesimal generator and stationarity
The following question is bothering me. I think it is probably known but I cannot find any reference...
Let $(X_t)$, $(Y_t)$, $(Z_t)$ denote 3 Feller processes with respective infinitesimal ...
- 21
5
votes
1
answer
203
views
Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel
Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, $D \subseteq X \times A$ be an analytic set, and $D_x := \{a \in A : (x, a) \in D\}$ denote the $x$-section of $D$ at $x \in X$.
...
- 275
6
votes
3
answers
584
views
What is the order of the constant $K$ in the multidimensional Dvoretzky-Kiefer-Wolfowitz inequality($Ke^{-c z}$)?
Let $F_n$ be the empirical distribution obtained from an i.i.d. sample
of the distribution $F:R ^d \to [0, 1]$.
Kiefer (1961) shows that the convergence of the empirical distribution is like
$$
P\left(...
user24451