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
0
answers
263
views
A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$
Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $...
CommunityBot
- 1
0
votes
0
answers
151
views
Inequality relating stationary probabilities and transition probabilities
Let $P$ be the transition probability matrix of a aperiodic irreducible DTMC and let $\pi$ be its stationary distribution. I would like to know if there is any literature on types of Markov chains ...
- 1,655
5
votes
1
answer
322
views
A colorful fractal structure on a graph provided by Wilson's algorithm: any explanation?
Consider a big finite rescaled piece of $\mathbb{Z}^2$, i.e. consider a unit square with a thick grid. Famous Wilson's method allows to generate a colored spanning tree of such a graph in a uniform ...
- 151k
4
votes
0
answers
173
views
On understanding Discrete-Valued Stochastic Processes( time series, panel data )
It seems to me that a significant proportion of work in probability theory, statistics and machine learning are on understanding continuous-valued, relatively weakly dependent, or linear dependent ...
- 41
2
votes
0
answers
112
views
How can we describe explicitly the "infinitely complex differentiable" complex-valued local martingales?
Let $\mathcal{F}_t$ be a continuous filtration on a probability space, and let $B$ be a standard $\mathbb{C}$-valued $\mathcal{F}_t$-Brownian motion. Let's call a complex-valued process $X$, possibly ...
- 4,010
4
votes
1
answer
400
views
Speed of random walks in groups
I've seen some estimates for the decay in $d$ of the probability a SRW makes a distance $d$ in time $n$, but is there any reference for the "speed" of a random walk in a group? I'm interested mostly ...
- 992
2
votes
2
answers
291
views
How many boxes so that there is $k$ of same of color from $n$ different colors?
Say you have $m$ boxes each of which is colored with one of $n$ colors. What should $m$ be so that the probability that there is atleast $k$ boxes with one same color is strictly greater than $\frac{1}...
- 5,432
2
votes
2
answers
2k
views
Change of measure Markov process
We begin with example. For the Poisson process with an intensity $\lambda_1$ there is an equivalent change of measure which makes it intensity to $\lambda_2$.
I would like to find the conditions ...
- 1,655
4
votes
1
answer
2k
views
Multivariate Central Limit Theorem For Non-Identical Distribution [closed]
Among the different generalizations of the CLT available on the web, I found these
CLT for the sum of non-identical (and independent) random variables
CLT for the sum of identical (and independent) ...
- 6,694
0
votes
2
answers
429
views
E[log(Z_t^2)], proof of convergence with Law of Large Numbers
Hi all,
question:
Let $Z_t$ be an iid sequence with $$\mathbb{E}\log(Z_t^2)<0 $$
Show that $$\sum_{j=0}^\infty Z_t^2 Z_{t-1}^2 ... Z_{t-j}^2 < \infty$$ almost surely
I am supposed to use LLN ...
- 5,721
6
votes
0
answers
183
views
Local structure in the stochastic sandpile model
Here's a question that came up at the recent AIM conference on chip-firing and generalizations.
The stochastic sandpile model, I think originally due to Manna, is a stochastic process that (in one ...
- 19.2k
8
votes
0
answers
130
views
Functions between Markov chains that preserve local harmonicity
Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is ...
- 19.7k
5
votes
1
answer
469
views
Universally measurable map coincides a.e. with a Borel map
Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set $A\...
3
votes
1
answer
143
views
Maps that are a.e. equal have almost the same graphs
Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two ...
- 60.6k
4
votes
0
answers
980
views
Inverse Fourier Transform involving a Bessel Function, Exponential, and Power
I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$:
$\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds $, where $J_{p/2-1}$ is a ...
- 41
5
votes
3
answers
1k
views
One can earn nothing on the Brownian motion, true ?
Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$).
Consider some "trading strategy" ...
- 3,069
10
votes
1
answer
831
views
Binomial distribution conjecture
Conjecture: Let $m$ and $n$ be fixed positive integers and let $f(k)$ be the probability that a Binomial($k(m+n)$, $p$) random variable is less than $kn$. Then for sufficiently small $p$, $f(k)$ is an ...
- 4,425
4
votes
2
answers
1k
views
Probability distribution over cluster size in Erdős–Rényi random graph.
My question is about the probability distribution over the possible size of the containing cluster of a randomly chosen node in an Erdős–Rényi random graph.
Let G(n,p) be an Erdős–Rényi random graph (...
- 248
6
votes
2
answers
836
views
Probability of the maximum (Levy Stable) random variable in a list being greater than the sum of the rest?
Given a list of identical and independently distributed Levy Stable random variables, $(X_0, X_1, \dots, X_{n-1})$, what is the is the probability that the maximum exceeds the sum of the rest? i.e.:
...
CommunityBot
- 1
4
votes
3
answers
3k
views
What is the name for a non-normalized distribution?
For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...
CommunityBot
- 1
-2
votes
1
answer
292
views
Probability distribution needed [closed]
Let me clarify my needs. The PDF must comply to:
1. The mean is always in the shorter tail
2. Should have an inverse function
3. Be defined in the interval [0, 1]
4. Should have a shape parameter that ...
CommunityBot
- 1
6
votes
1
answer
644
views
Random path in a graph
Consider a finite graph $G$. I would like to define a random path between two vertices $s$ and $t$ of the graph $G$ by looking at a measure $\mu$ on all spanning trees. Then the probability of a given ...
- 4,432
6
votes
2
answers
720
views
Local concentration of measure on Erdos-Rényi graph
Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...
- 7,559
12
votes
4
answers
4k
views
Tail bound for Poisson random variable
Is the following fact about Poisson random variables true?
For any $\lambda \in (0,1)$ and integer $k > 0$, if $X$ is a Poisson random variable with mean $k \lambda$, then $\Pr(X < k) \geq e^{...
CommunityBot
- 1
4
votes
1
answer
495
views
A probability exercise related to Central Limit Thm
This exercise appears in K.L.Chung's A Course in Probability Theory, Chapter 7.
Ex.7.1-4
Let ${X_j}$ be independent r.v.'s such that $\max_{1\leqslant j\leqslant n} \frac{|X_j|}{b_n} \to 0$ in
pr. ...
- 3,993
1
vote
1
answer
16k
views
Calculating $E[X^2Y^2]$ given $E[X^2]$, $E[Y^2]$, $E[X]$, $E[Y]$, and that $X$, $Y$ are Gaussian. [closed]
Suppose $E[X]=E[Y]=0$, and $E[X^2]=E[Y^2]=1$. Can you show that $E[X^2Y^2] = 1 + 2\operatorname{cov}(X,Y)^2$? I am not even sure if this expression is correct, I found it in a geostatistics paper, ...
- 3,993
0
votes
0
answers
1k
views
Measure induced by function
It is known that an n-increasing, left-continuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3-increasing on $[0,\infty)^3$ but also 2-...
- 8,271
2
votes
1
answer
151
views
Large Deviations for $\nu_\epsilon = Z_\epsilon\exp\left(-\frac{1}{\epsilon}\Phi(x)\right)d\mu$
Given a probability measure of the form
$$\nu_\epsilon=Z_\epsilon\exp\left(-\frac{1}{\epsilon}\Phi(x)\right)d\mu$$
with $Z$ being the normalizing constant.
Under which conditions on $\mu$ and $\Phi$...
- 3,993
1
vote
1
answer
228
views
Is anything known about Large Deviation Principle for non additive functionals on Markov chains?
Let $\Sigma$ be a finite set of cardinality $|\Sigma |$ and
$$\Pi = \{ \pi(i,j)\}_{i,j = 1}^{|\Sigma|}$$
a stochastic matrix (ie a matrix whose elements are non negative and such that
each row sum ...
- 7,559
6
votes
0
answers
298
views
Is there an Arctic Circle phenomenon for Amman-Beenker tilings?
I found some slides on tilings and one of them pertained to Amman-Beenker tilings. It looks like there is an Arctic Circle phenomenon similar to that for dominos or lozenges. Is there any ...
CommunityBot
- 1
1
vote
1
answer
510
views
When is a coarsening of a Markov chain Markov?
Let $(X_t)_{t \in \mathbb{N}}$ be a discrete-time Markov process on the finite state space $\mathcal{X}$, with transition matrix $T$. Suppose $f: \mathcal{X} \to \mathcal{Y}$ is a (deterministic) ...
- 1,769
8
votes
2
answers
1k
views
Sufficient Condition for Exponential Decay in Chernoff Bound (Large Deviations)
Let $X_i$ ($i=1,...,n$) be a sequence of independent and identically distributed random variables. Denote $\mu=\mathbb{E}[X_i]$ and $S_n=\frac{1}{n}\sum_{i=1}^nX_i$. This question concerns the tail ...
- 7,559
6
votes
1
answer
798
views
Prohorov's theorem for random elements of Hilbert space: weak convergence
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(E,\mathcal{E})$ be a separable Hilbert space ($E$) with Borel $\sigma$-algebra $\mathcal{E}$. For concreteness let us set $E=L^{2}[a,b]$ ...
- 23.2k
-1
votes
1
answer
1k
views
Rank of covariance matrix whose diagonal elements are same [closed]
Suppose A is a covariance matrix whose diagonal elements are same, i.e. $A_{1,1}=A_{2,2}=\cdots=A_{N,N}$, can we conclude that A is full rank?
Suppose the absolute values of the off-diagonal elements ...
- 2,361
40
votes
1
answer
6k
views
The human body's random number generator
I remember learning in microbiology that the human body generates antibodies using a random process so that an enormous variety of antibodies can be produced with a simple genetic code.
Now that I'm ...
- 3,359
4
votes
1
answer
243
views
On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument
Theorem 3.3.4 in Davies' Heat Kernels and Spectral Theory begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator ...
- 7,197
19
votes
9
answers
3k
views
How can I generate random permutations of [n] with k cycles, where k is much larger than log n?
I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both ...
CommunityBot
- 1
1
vote
1
answer
353
views
Agreement of two topologies on a linear space
I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
- 16.5k
3
votes
1
answer
138
views
Is $R^n$ stochastically complete for the heat kernel of a Schrödinger operator?
Suppose $V:\mathbb{R}^{n} \to \mathbb{R}$ is just a positive polynomial and $K_{t}(x,y)$ is the heat kernel of $H = -\Delta + V$. Then does it follow
$$\int_{\mathbb{R}^{n}} K_{t}(x, \cdot)\,dy = 1?$$...
- 2,715
2
votes
0
answers
240
views
$n$-th return of a random walk on $\Bbb Z^d$
Lets define $f_n = P(X_n =0 , X_k \ne 0, k< n)$ the first return distribution of the random walk $X_n$ on $\mathbb{Z}^d$, and lets go ahead and assume that $f_n \approx n^{-(1+\alpha)}$ for some $\...
- 3,993
2
votes
0
answers
101
views
How to prove convexity for a complex integral including the variable in both limits and integrand?
During my research [on inventory management policies, i.e., something really applied ;-) ] I stumbled on integrals of the following type and I'm curious under which circumstances there are convex.
...
- 60.6k
36
votes
3
answers
4k
views
the following inequality is true，but I can't prove it
The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...
- 79.9k
1
vote
0
answers
100
views
Conditions on a measure to satisfy certain relation on moments.
Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$.
I'd like to impose some conditions on $\mu$ so the function
$$f:p\to \frac{\int_0^\infty t^...
- 173
8
votes
1
answer
1k
views
Conditional law as a random measure and convergence of random measures
I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider ...
0
votes
1
answer
145
views
Limiting probability question [closed]
Let $X$ denote an $m\times n$ matrix and suppose that each value $x_{ij}$ is an integer that is selected uniformly at random from ${1,\dots,n}$, independently of all other values. If we fix $m$ and ...
- 24.8k
1
vote
1
answer
405
views
Convergence to a k-dimensional Gaussian vector
Suppose I have a sequence of stochastic processes $X_{N}(t)$, $N=1,2,3,\ldots$ with mean zero and that I know for every fixed $t$, the random variable $X_{N}(t)$ converges in law to a Gaussian random ...
- 149k
5
votes
2
answers
321
views
Keeping time by randomly drifting a $q$-ary string
Imagine I have a string $s$ of length $L$ encoded over an alphabet of size $q$, e.g. $s = 000101$, where $L = 6$ & $q = 2$. For each of $T$ time intervals, $(t_1, ..., t_N) \in T$, I select a bit ...
- 851
7
votes
1
answer
12k
views
inner product of two gaussian random vectors?
Suppose that $x, y\sim N(0,I_n)$ are independent. Consider the inner product $\langle x, y\rangle$. Intuitively, $y$ behaves like a random vector of length $\sqrt n$, so $\langle x, y\rangle$ is close ...
- 802
10
votes
3
answers
1k
views
Rapid evaluation of multivariate normal integral
I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form
$$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, dz,$...
- 101
3
votes
2
answers
222
views
Random metrics on compact orientable surfaces
Hello everyone,
Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of ...
- 15.4k