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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.

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1answer
137 views
+50

Showing $o(1)$ convergence for ratio of successive binomial tail probabilities

For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that $$ \frac{P(X>c)}{P(X>c-1)}=1-o(1) $$ uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...
4
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1answer
51 views

The minimum of the reciprocals of some Poisson random variables

Let $X_1,\dots,X_k$ denote a collection of independent samples of a Poisson random variable whose mean also happens to be equal to $k$. Does the quantity $$k\boldsymbol{E}\min\left\{ \frac{1}{1+X_{1}}...
0
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1answer
72 views

Estimating expectation of a slightly strange sum

Let $X$ be a random variable with support on the positive integers (you can assume $\mathbb{E}[X^2] <\infty$ if needed, or even higher moments if needed), and let $S(i)=\mathbb{P}(X\geq I)$. ...
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0answers
37 views

Mathematics grade 12 probability [on hold]

Stella has 12 coins in her hands, six of these coins are Looney. She accidentally (at random) drops three of these points. Determine the probability that exactly two of the drop coins are loonies.
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1answer
47 views

What is the expectation/variance of the GOE (Airy-1) point process on a partition of the real line?

Let $\chi^{\mathrm{Ai}}(I)$ denote the GUE (Airy-2) point process on the interval $I \subset \mathbb{R}$. Soshnikov proved \begin{align} \mathbb{E}(\chi^{\mathrm{Ai}}(-T, +\infty)) &\sim \...
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0answers
36 views

Calculating the probability of obtaining exactly four distinct values when a die is rolled six times [on hold]

What is the probability of getting a) 4 distinct numbers (no order in the outcome e.g. 1,2,3,4 or 4,5,6,2 etc) b) 5 distinct from rolling a die 6 times So far I was able to calculate the ...
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0answers
69 views

Markovian Control in a stochastic control problem

I have a very simple control problem. \begin{align*} V(0,X) &= \sup_{(C_t)_t} E_0 \left[ |X_t| 1_{C_t = 0} + C_t \right] \\ \text{ s.t. } & d X_t = C_t d B_t, \quad X_t \in [-1,1], C_t \ge 0,...
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0answers
22 views

Asymptotics for a random set cover problem

Suppose you are given a positive integer $k$ and a probability distribution $f$ on the positive reals. I am interested in the limiting behavior of the following process as $n\to\infty$: Create an ...
0
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0answers
46 views

Ladder times of a Brownian motion with drift

Let $(B_s)_{s_\geq 0}$ be a standard Brownian motion and fix $t>0$. For $u>0$, set $T_u=\inf\{s>0, B_s+s t>u\}$. Now consider $x>0$ such that $\sup_{0 \leq s \leq x} (B_s+st)=B_x+xt$ ...
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2answers
342 views

Standard Brownian motion, Hölder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$

In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof. It is not hard to see that for every $k < \infty$, and every $\epsilon >...
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1answer
117 views
+50

Time discretization in the Feynman-Kac formula with boundary conditions

I am applying the Feynman-Kac theory for solving a PDE with boundary conditions. For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...
0
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1answer
45 views

Product of independent random variables and tail deconvolution

Suppose $X, Y$ are two independent non-negative random variables. The conditions (i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$ (ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...
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6answers
3k views

analog of principle of inclusion-exclusion

When I teach elementary probability to my finite math students, a common error is to mix up the concepts of disjointness and independence. At some point I thought that it might be helpful to some ...
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0answers
57 views

Birthday Calendar Gaps [on hold]

I work at a company that posts a birthday calendar. I noticed that there was a string of four consecutive days with no birthdays. What is the probability of that happening? Problem Statement Given n ...
3
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1answer
1k views

explicit expressions of the distribution of sums of i.i.d. logistic random variables

Where can I find the explicit expression of the distribution of the sum of n i.i.d. logistic random variables, for n=2,3,4... The expressions given in "On the convolution of logistic random variables,...
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1answer
95 views

Existence of certain event

Suppose that $X$ is an unbounded random variable such that $\operatorname EX=0$ and $\operatorname E|X|^q=1$ with some $q>2$. Only the distribution of $X$ matters, so the probability space can be ...
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2answers
809 views

Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that $$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$ ...
0
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1answer
42 views

How to calculate the probability of 2 events happening in time series under only cdf information?

In time domain $0\rightarrow T$, there are two independent events $A$ and $B$. $B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
5
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1answer
140 views

Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$

If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to ...
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0answers
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Is the effect of Tr on Y identifiable conditioning on G? [closed]

enter image description here I wonder if Tr and Y are independent conditioning on G? I am working on a causal inference problem, and I wonder whether and why the effect of Tr is identifiable if G is ...
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0answers
26 views

p-Variation distance defines semi-martingales

Question When, does the process $\tilde{X}_t$, defined path-wise by $$ \tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right), $$ define a ...
8
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2answers
503 views

Random Walks on high dimensional spaces

I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in arbitrary directions (uniformely on the unit sphere $S^1$, not left-right-up-down), ...
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1answer
38 views

Jump size of process in proof of martingale CLT bounded

I'm currently trying to understand the proof of the final theorem of (Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time) where I need the following: For $\{...
2
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0answers
110 views

Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
5
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1answer
165 views

Relation between the two possible KL divergences of two distributions

Given that I know $$D\left(P\parallel Q\right)<\alpha,$$ can I say anything about $D\left(Q\parallel P\right)$ in terms of an upper bound on it? Also, given this upper bound on $D\left(P\parallel ...
3
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1answer
67 views

Powers of Frobenius norm of sum of random matrices

For $i= 1, \ldots, n$, let $A_i \in \mathbb{R}^{d \times d}$ be random i.i.d. matrices with $E [A_i] =0$. Can we relate (upper bound) $E[\|\sum_{i=1}^n A_i \|_F^4]$ to $E[\|A_i\|^4_F]$ ?
3
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0answers
72 views

Upper bounding the start of a distribution's CDF, given bounds on first moments

Take nonnegative random variables $X$ whose first $K$ moments have bounds: $\mu^k\leq E[X^k]\leq c\mu^k$ for each $k=1,\dots,K$. In this case what is an upper bound for $P(X\leq O(\mu))$? I am ...
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1answer
171 views

Approximating $\mathbb{E}[1/X]$

I am well aware (as for instance discussed here https://math.stackexchange.com/questions/910846/is-it-true-in-general-that-e1-x-1-ex) that for an arbitrary random variable $X$ it does not hold that $\...
4
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1answer
105 views

Is there a counterexample to the Thin Shell Conjecture for sub-exponential distributions?

The thin shell conjecture states that there exist universal constants $C,c>0$ such that every logconcave isotropic random vector $X$ in every Euclidean space $\mathbb{R}^n$ satisfies $$\mathbb{P}\...
3
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2answers
175 views

Example of measure for some algebra over N

$\mathcal F$ is set of events. Can you give an example of some algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive ...
14
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1answer
385 views

Inequalities for marginals of distribution on hyperplane

Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...
3
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1answer
109 views

Expected size of the smallest preimage set

Let $f$ a function from $\{0, 1 \}^{2n}$ to $\{0, 1 \}^{n}$ uniformly picked at random. I would like to have an estimation of the expected size of the smallest premiage set of $f$, more formally $\...
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4answers
690 views

Good introduction to statistics from a algebraic point of view?

There are already lots of questions on this subject like Is there an introduction to probability theory from a structuralist/categorical perspective? Is there a combinatorial/topological treatment ...
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0answers
79 views

On a much weaker version of the Normal conjecture

I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...
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0answers
104 views

Probability of degree $0$ gcd between every pair of random homogeneous polynomials shifted by random primes?

Take $n,d,B\in\mathbb Z_{>0}$ with $d<n$ and denote $\mathcal M_{n,d}$ to be set of all total degree $d$ monomials in $n$ variables $x_1,\dots,x_n$ with degree $\leq1$ in each variable (...
0
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1answer
43 views

Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$

Let $\mathbb S_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z_1,\ldots,z_n$ be a i.i.d sample from $\mathcal N(0, 1)$. Question Given $\epsilon > 0$ (may be assumed to be very small), what is ...
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1answer
455 views

Is the ito integral $\int_0^t \operatorname{sign}(W_s)\mathrm{d}W_s$ a Brownian motion?

Consider the ito integral of the sign of the Brownian motion $W_s$ from $0$ to $t$: $$\int_0^t \operatorname{sign}(W_s)\,dW_s$$ This appears for instance in the Tanaka formula. I think this is a ...
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0answers
79 views

The Kleisli Category of the Monad of Measures of Finite Support and its composition formula

In this post, I was introduced to the monad of finitely supported measures. $HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad. Let's call this ...
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2answers
167 views

Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
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2answers
104 views

Expectation of the exitpoint distance for the symmetric random walk

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$. Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...
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0answers
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A limit for two correlated variables [on hold]

Suppose we have two correlated Normal variables $X_A$ and $X_B$, with respective standard deviation $A$ and $B$, and correlation $\rho$. The variable $X_A + X_B$ has a standard deviation in excess (...
5
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1answer
495 views

What is the six positive real number for a dice producing a highest chance?

Say there is a dice with six faces, each face has a positive real number different from others. There is a chessman on the origin of the number axis. In each trial, the dice rolls infinite times. ...
2
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1answer
100 views

bp continuity of Markov operators / semigroups

Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to ...
2
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1answer
135 views

Fourier transform of a simple random walk

Consider the usual simple random walk on $\mathbb{Z}$, taking steps of +1 or -1 with equal probability. Of course, each trajectory corresponds uniquely to an element of $\{-1,1\}^\infty$. Now, there ...
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1answer
1k views

Random walk in a circle

Suppose we have a circle of radius $R$ centered in the origin of a $x,y$ cartesian reference frame. A particle starting from the center of the circle is moving with a speed given by: $$\overrightarrow{...
2
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1answer
99 views

How to uniformly sample a square (0,1)-matrix whose trace is 0 and whose row sums and column sums are the same?

Happy New Year! Suppose I would like to sample a $n \times n$ (0,1)-matrix whose trace is 0, and whose row sums and column sums are all $m$ with $1 \le m \le n-1.$ How can I sample this matrix ...
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0answers
66 views

mean distance between subspaces

Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their ...
4
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2answers
2k views

Eigenvalues of random Hamiltonian matrices

A real $2n\times 2n$ Hamiltonian matrix has the general form $$H=\begin{pmatrix} A & B \cr C & -A^T \end{pmatrix} $$ where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are ...
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0answers
61 views

Is there solution to a backward stochastic differential equation with $yz$ in the generator?

Please consider the following backward stochastic differential equation: $$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$ Here $a(s)$, $b(s)$ are square-integrable stochastic ...
4
votes
1answer
66 views

Sampling uniformly from the vertices of a polytope

I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that enumerating vertices is hard. I also note the MO ...