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.
9,028 questions
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Sum of digits iterated
Original version.
I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of ...
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Donsker Theorem Billingsley
Theorems $16.1$ and $16.3$ in Billingsley Convergence of measures.
$16.1$ reads : Random variables $u_1,\ldots$ on $(\Omega,\mathcal{B},\mathbb P)$
and are i.i.d. with $0$ mean and finite variance $\...
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Continuity on a measure one set versus measure one set of points of continuity
In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...
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Is there a statistical interpretation of Green's theorem, Stokes' theorem, or the divergence theorem?
This is cross-posted from math.stackexchange and stats.stackexchange. Probably there is no great answer to this question, but I thought I'd give it a shot here.
I'm teaching a class on integration ...
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Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
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Markov Chains based on sampled transition probabilities [closed]
If I have a process that transitions between states with some set, unknown probability, I can sample to find the transition probability. This probability is a sample average, with a well understood ...
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Probability of having many unique elements
If you sample $n$ integers from the range $1$ to $n$ inclusive it seems intuitive that you are likely to get a lot of numbers exactly once. Call $X_n$ the number of integers you get that occur ...
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what's the best way to characterise the distribution of prime elements in simple perfect squared squares
DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...
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Expected value of logarithm of a binomial random variable
Suppose $X\sim \mathrm{Binomial}(n,p),$ i.e. $X$ takes values in $\{0,1,2,\ldots, n\}$ and $P(X=i) = {n\choose i} p^i(1-p)^{n-i}.$
I am looking for a good estimate for $\mathbb{E}\log(X+\alpha).$ I ...
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Finding the probability that $n$ points which are randomly selected from $d-1$ dimensional spherical surface are 'semi-spherical'
Let $S^{d-1}=\{(x_1,\cdots,x_{d})\in {\mathbb R}^{d}|{x_1}^2+\cdots+{x_d}^2=1\}$, and let us call the intersection of any $d-1$ dimensional subset which passes through the origin and $S^{d-1}$ 'a ...
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Brownian motion inside a domain
Consider a regular domain $D \subset \mathbb{R}^d$ and a Brownian motion $B_t$ conditioned to stay inside $D$ for time $t \in [0,T]$. In the limit $T \to \infty$ the conditioned Brownian motion ...
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Characterising sets on which two probability measures coincide
Let $(\Omega,\mathcal{B})$ be a measurable space and let $\mu$ and $\nu$ be two probability measures on this space. It is well known that $\mathcal{A} = \{A\in \mathcal{B} \enspace | \enspace \mu (A) ...
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Is this a closed set?
Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
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Limit of pushforward measures of random variables is "represented" by a random variable
Suppose we have an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sequence of real random variables $X_n:\Omega\to\mathbb{R}$ such that the pushforward measures $(X_n)_*(\mathbb{P}...
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Symmetries of the standard probability space
The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In applications,...
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Variance of central limit distribution for $P(x) \sim 1/x^{1+\alpha}$ for finite but large $N$?
Is it known what the next-to-leading order term is in the variance of the central limit distribution for the average of $N$ variables each of which is distributed according to $P(x) \sim 1/x^{1+\alpha}...
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Singularity of sparse random matrices
The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...
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Linear combination of i.i.d. $Z_i$ distributed as $Z_1$
A classical property of the Gaussian distribution is that, if $\{Z_i\}_{1 \leq i \leq n}$ are i.i.d. standardised Gaussian distributions (i.e. $Z_i \sim N(0,1)$) and $S = \sum_{i=1}^n a_i Z_i$ where $...
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Central limit theorem for $P(x)\sim 1/x^3$ distribution
I have a random variable $x \in (0,\infty)$ with distribution $P(x)$ falling off slowly $P(x) \sim 1/x^3$ for large $x$. So the expectation value $\bar{x}$ is finite but the second moment $\bar{x^2}$ ...
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order of convergence of the conditional entropy (2)
Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g_n(X_n)$, then by Fano's inequality $$\...
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Is this inverted integral transform valid?
I have the following transform:
$$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$
with the following conditions:
$f(x)$ and $F(y)$ must be ...
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Cumulative distribution function of hypergeometric distribution
Does anyone know a closed form or a good approximation of the cumulative distribution function of hypergeometric distribution?
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Expectation of the time t standard brownian motion stopped at itself's square
I have a one dimensional standard brownian motion $W$ defined under a stochastic basis with probability $\mathbf{Q}$ and filtration $\left(\mathscr{F}\right)_{t\in{\mathbf{R}}_{+}}$, and I want to ...
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Continuity of caglad process
Consider a non increasing, caglad process $(X_t)_{t\geq0}$ such that, for each $t$, the distribution function $F_t(x):=P(X_t\leq x)$ is a continuous function of (real) $x$. Are there any sufficient ...
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bound the tail distribution
Suppose that Z_1, ... , Z_n are binomial distributions with E[Z_i]=z_i.
If (Z_i) are pairwise independent, then, It's well known that the Chebyshev inequality can bound the tail distributions.
If (...
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bilinear form tail bound
Let $A\in \mathbb{R}^{n\times n}$ and $u$ and $v$ be independent random $\{-1,1\}^n$-vectors. (i.e., each coordinate of $u$ is $1$ with prob. $1/2$ and $-1$ with prob. 1/2 and the coordinates of $u$ ...
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Bunimovich stadium bouncing ball
http://terrytao.wordpress.com/2008/07/07/hassells-proof-of-scarring-for-the-bunimovich-stadium/
I cannot relate how the eigenfunctions normalizaed correspond to the probabality distribution of the ...
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Is there monotonicity of measure concentration?
Suppose $X$ and $Y$ are nonnegative random variables such that $\mathrm{Pr}(X\geq t)\leq\mathrm{Pr}(Y\geq t)$ for all $t\geq0$. Now take $X_1,\ldots,X_n$ to be independent with the same distribution ...
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"Fractional sampling" from a probability distribution
My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to ...
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Distribute Monte Carlo samples among dimensions
Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
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Probability of disc-disc overlap for discs placed with uniform probability on a surface until a density $\rho$ is achieved
Imagine I place discs of radius $r$ on a two-dimensional plane, selecting their positions with uniform probability across the surface of the plane, and stop when I reach a disc density $\rho$. As a ...
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Maximum of a sequence of $n$ positive random variables where variance is an increasing function of $n$
Suppose I have a sequence of $n$ i.i.d. random variables $X_1,X_2,\ldots,X_n$. Each $X_i$ is positive and has variance $\sigma(n)$ that is an increasing function of the number of variables in the ...
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Binomial Expectation of Convex Function
Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate
$$\frac{\partial}{\partial \alpha} \...
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Distribution of last time Brownian motion crosses a line
Is the distribution of the last time Brownian motion crosses a line y=a*x known? (Equivalently, the distribution of the last time a Brownian motion with downwards drift hits 0.) It's not hard to give ...
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Finding the Levy triplet of a Levy process
I know the levy triplet of a Poisson process
$N_t$- $(0,0,\lambda\delta_{1}(y))$ and its characteristic function is
$\phi_N=exp[-t\Bigl(\intop_{0}^{\infty}(1-e^{iuy}+iuy1_{\{\mathbf{|}\mathbf{y}|&...
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Joint law for number of visits in transient simple random walk
Consider a simple $1$-dimensional random walk $X_n$.
Let $p$ and $q$ (with $p+q=1$) be the probability of transitions from $n$ to $n+1$ and from $n$ to $n-1$ respectively, and suppose that $p>q$ ...
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Expected size of $k$-th layer of a POSET
Is this known?
What is the expected width of the $k$-th layer (anti-chain layer) of a $d$-dimensional partially ordered set of $n$ elements formed by product of $d$ random linear orders chosen from ...
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$L^2$ convergence of a tight sequence [closed]
Let $(X_n,n\geqslant 1)$ be a tight sequence of stochastic processes defined on the same probability. Suppose $\lVert X_n\rVert_{L^2}$ converges to $\lVert X\rVert_{L^2}$. Under what conditions do we ...
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Reference request: stationary measures as convex combinations of ergodic measures
Does anyone know a good reference for the fact that a stationary probability measure is a convex combination of the stationary and ergodic probability measures?
I have found some references for the ...
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Rotation-invariant strict-inclusion-preserving preorderings on subsets of the circle
Say that a preordering $\le$ on a set of subsets of some space preserves strict inclusion provided that $A\lt B$ whenever $A\subset B$ (where $A\lt B$ iff $A\le B$ and $B \not\le A$).
Let the space ...
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Determining the asymptotic behavior of some function of random matrix
Consider a series of random matrices $X_n\in\mathbb{R}^{n\times m}$ consisting of i.i.d. entries, each with zero mean and variance $1/m$, and let $a_n,b_n\in\mathbb{R}^{n\times1}$ be two deterministic ...
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Number rank-k 0-1 matrices (characteristic 0)
What is the number of $n\times n$ 0/1-matrices with rank $k$?
(The rank is taken over the rationals.)
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Geometric Expectation Inequality
Suppose $f(z)$ is a discrete probability distribution with space $S$. Suppose $g(z),h(z)>0$ for all $z \in S$. Is it true that
$$\prod_{z \in S}{g(z)^{f(z)}}+\prod_{z \in S}{h(z)^{f(z)}} \leq \...
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Number of times a Gaussian process crosses zero in an interval
Using a probabilistic method for number theoretic purposes, I have encountered the following question (it may be very standard):
Let $X_t$ be a Gaussian process $(t>0)$ such that $X_0=0$. What ...
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Best and worst centrally symmetric convex covering shapes
Suppose you have a centrally symmetric convex 2D shape $C$ of area $A$, and you randomly throw
down copies of $C$ on the plane so that each $C$-center lies within a given unit square $S$,
until $S$ is ...
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Is "small" dependence enough for central limit theorem?
Writing down a paper about some estimation of some combinatorial quantities, i realized that i would have much more precise results if these two questions have positive answer:
1) Suppose you have a ...
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Does the strong law of Large Number hold for an infinite dimensional Brownian motion?
For finite-dimensional Brownian motion $W_t$, it is well known that
\begin{equation}
\lim_{t\to \infty}\frac{W_t}{t}=0,\text{ a.s. }\ \ \ \ \hspace{1cm} \langle 1\rangle
\end{equation}
Now suppose we ...
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Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)
Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, i....
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Argmax of random walk vs of Brownian motion
Consider a random walk on $\mathbb{Z}$ with triangular drift and jumps that are standard normals. That is,
$$
\begin{cases}
RW_{t+1} = RW_t - d + \epsilon_t, \quad t \geq 0,\\
RW_{t-1} = RW_t - d + \...
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Expected number of distinct elements out of random selection [closed]
What is the expected number of distinct elements when choosing N random elements of a set of size N?
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