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.
9,021 questions
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Discrepancy between probability measures, tested against bounded functions of bounded variance
When studying some concentration inequalities, it became relevant to consider the following discrepancy between two probability measures $\pi$ and $\nu$ (treating $\sigma \in \left( 0, \frac{1}{2} \...
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156
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Nontrivial nonrandom properties of prime numbers
What are some nontrivial nonrandom properties of prime numbers. Consider the simple model where each number is prime with probability 1/log(n) by Montgomery and extensions of it. Once you add some ...
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Statistics of random Voronoi S-tessellations
Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to ...
3
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58
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Infinitesimal generators of random evolutions
Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
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158
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Conformally mapping between the upper half complex plane, and the plane with a tree on spatial points removed
A stochastic process such as SLE$_{\kappa}$ can be defined by taking the scaling limit of a curve in the upper half complex plane: put simply, one removes a line segment, then another, $n$ times, each ...
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134
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Asymptotics of a ratio on the unit sphere
Let $(a_n)_{n \geq 1}$ be a nonnegative, strictly decreasing sequence with $a_n \to 0$ as $n \to \infty$.
Consider the ratio (for $k \geq n$)
$$
R_{n, k} = \mathbb{E}_{u \sim \text{Unif}(\mathbb{S}^{k-...
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106
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When is there a Lipschitz Kantorovich Potential?
Let $c:\mathbb{R}^d\times \mathbb{R}^d\to [0,\infty)$ be a Lipschitz cost function and consider the optimal transport problem
$$
C(\mu,\nu):=\inf_{\pi}\, \int c(x,y)\,\pi(dxdy)
$$
where, as usual, the ...
5
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2
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How to calculate an integral over the complex unit sphere
We want to calculate the following integral over the complex unit sphere $S^{2n-1}$:
$$\int_{S^{2n-1}} \frac{1 }{|1 - \langle z, \zeta \rangle|^2} \, d\sigma(\zeta),$$
where $ z $ is a fixed point in ...
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114
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An urn model with weighted objects and replacement
Consider the following game:
In an urn, there are $K$ balls, $x_0$ of them are blue and light (mass $m_0$), $x_1$ are blue and heavy ($m_1$), $x_2$ are red and light ($m_2$), the rest $x_3$ are red ...
2
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Characterisation of Bessel process
Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that
For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
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45
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Inequality Involving Concave Monotonic Function
Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
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93
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Distance between binomial and normal distributions
I want to compare binomial distribution $Bin(n,p)$ with a constant $p$ when $n\rightarrow \infty$, to a normal distribution with $\mu=np,\sigma^2=np(1-p)$.
How close are they with the discrete ...
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370
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Convergence of iterated conditional expectations
Notation: We write $\mathbb E_{\mathcal F} X$ for the conditional expectation $\mathbb E[X|\mathcal F]$ of a random variable $X$ with respect to a $\sigma$-algebra $\mathcal F$.
Let $X$ be an ...
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1
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158
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Techniques for bounding the operator norm of the expectation of random matrix?
Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix
$$
A = I_n - uu^T.
$$
Question: What techniques are available to provide (reasonably ...
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139
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Dispersion of random walk with scaled step sizes
Let $Y_j$ be a sequence of independent Gaussian random variables with mean zero and unit variance ($\mathbb{E} Y_j = 0$ and $\mathbb{E} Y_j^2 = 1$) and let $\sigma:\mathbb{R}\to [1,2]$.
We define the ...
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69
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Simulating binomial distribution
$\DeclareMathOperator\Bin{Bin}\DeclareMathOperator\Pr{Pr}$I have a series of distributions $D_k=\Bin(3k,\frac{1+k^{-1/3}}{3})$, and a distribution $D_{k,\ell} = k +\Bin(k,\ell)$ parametrized by $\ell\...
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73
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Tight tail bounds for sums of random variables
Let $X_1, X_2, \dots$ be iid uniformly on $[0,1]$. Define $Z_i^{(a)} = (X_i - a)^2$. Let $Y_n = \sum_{k=1}^n Z_k^{(1/k)}$. I am interested in matching tail bounds for $Y_n$ as $n \to \infty$. In ...
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148
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An inequality about binomial distribution
Statement
Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...
3
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Convergence rate of the sum of squares of inverse distances of random points which become dense in a region
$n$ points $\{X_i\}$ are drawn at random from a uniform distribution over a domain $\Omega\subset \mathbb{R}^m$ with a Lipschitz boundary. $D_n$ is defined as $$D_n = \sqrt{\frac{1}{\sum\limits_{1\le ...
2
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1
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105
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Inequality for Gaussian measures
Let $\mu$ denote a centered Gaussian measure on $\mathbb{R}^k$, $K=(-\infty, a] \times \mathbb{R}^{k-1}$ ($a\ge 0$) and $L=\mathbb{R}\times C$ where $C$ is a convex set in $\mathbb{R}^{k-1}$, ...
0
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0
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31
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What is the Fisher information matrix of the von Mises-Fisher distribution?
Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...
19
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2
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2k
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Higher or lower?
Consider the following game - I draw a number from $[0, 1]$ uniformly, and show it to you. I tell you I am going to draw another $1000$ numbers in sequence, independently and uniformly. Your task is ...
5
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1
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192
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Non-equivalent definitions of Markov process
As far as I know, there are three definitions of Markov processes (or of Markov chains).
DEFINITION 1 (WEAKER). A process $(X_t)_{t\in[0,\infty)}$ on $(\Omega,\mathcal{F},\mathbb{P})$ with values in ...
8
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1
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522
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One step forward, one step back
$N \geq 2$ players play a cooperative game on the integers $\mathbb Z$. All of them start from $0$. At each turn, they are simultanously given the same yes or no question to answer.
The questions ...
1
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0
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80
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Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
2
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56
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Sum of independent Wisharts
Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
2
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29
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Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
1
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0
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66
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Random lattice always has trivial automorphism group?
In example 2.5 of a paper [LS17] written by Lenstra and Silverberg, it is written that “Random” lattices have $Aut(L) = \{ \pm 1 \}$, I guess the 'Random' here refers to the distribution in Siegel's ...
1
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55
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Limit process of a sequence of Gaussian variables on mesh grid going to zero
Consider the interval $[0,1]$ and a partition $\mathscr{P}_n = \{ [t_i,t_{i+1}), \, i=1,\ldots,N_n \, : \, 0=t_0 < \ldots < t_{N_n} = 1\}$. Suppose that for all $i$ and $t \in [t_i,t_{i+1})$, we ...
3
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1
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194
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Dynamics of a random stretch map
Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric.
Let $\{\epsilon_n\}_{n \geq 1}$ be iid uniformly ...
34
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7
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3k
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A hat puzzle question—how to prove the standard solution is optimal?
I am currently writing an essay on hat puzzles, and for the warm-up section I introduce some of the standard finite hat puzzles. One of these proceeds as follows:
You and two friends are each given a ...
9
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1
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155
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How to sample exactly k indices given the inclusion probabilities of all indices?
Let $k<d$ two positive integers, and $\{p_i\}_{i=1}^d$ a series of probabilities, with $p_i \in (0,1)$ and $\sum_{i=1}^d p_i = k$.
We wish to sample exactly $k$ distinct indices $\mathcal{I}\...
2
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1
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276
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Probability of visible permutations
A visible permutation $\sigma$ of $[1,2, ...,n]$ has a permutation matrix such that all "1" locations are visible from the origin $(0,0)$.
Two "1" locations are visible if the two ...
0
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0
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30
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Why is the $\alpha$-divergence unique in positive measure space $\mathcal{M}$?
In this article https://bsi-ni.brain.riken.jp/database/file/298/303.pdf (S. Amari 2009), it is said that a $f$-divergence (eq. 17) which can be written by a decomposable Bregman divergence (eq. 53) ...
4
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127
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A "resampling identity" for the Bessel(3) process
I've come across the following resampling identity and was wondering if this is known since it seems rather natural. Take $X$ a two-sided Brownian motion conditioned to always stay below $1$. (So if ...
3
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1
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232
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Bounds on relative entropy for MLE in Bernoulli coin tosses
In the context of estimating the parameter $p$ from a dataset of $n$ i.i.d Bernoulli coin tosses, we often use the relative entropy $D(p \parallel \hat{p})$ to measure the performance of an estimator $...
4
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2
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312
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What is the expected size of the smallest hitting set?
Suppose we pick $n$ subsets of size $j$ of an $N$-element set $S$ uniformly at random. A hitting set is a subset of $S$ that intersects all our subsets. I am interested in the smallest size of an ...
1
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0
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43
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Moments on the Stiefel manifold
Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
2
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1
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202
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Strong Liouville property of virtually abelian groups
Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...
7
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5
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514
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Probability of $\operatorname{Bin}(n,p)=\operatorname{Bin}(n,q)$ is decreasing when $n$ increases
$\newcommand{\Bin}{\operatorname{Bin}}$I would like to show that $\mathbb P(\operatorname{Binomial}(n,p) = \operatorname{Binomial}(n,q))$ decreases when $n$ increases for a fixed pair $(p,q)$. This ...
3
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1
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99
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Intersection of IID fractal sets
Let $A, B \subset \mathbb R$ be IID random closed subset. Suppose that there exists $d \in (1/2, 1]$ such that the Hausdorff dimension of $A$ is equal to $d$ almost surely. Is it true that $\mathbf P\...
3
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0
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131
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Matrix-Gaussian distributions
The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
3
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0
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353
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Moments of normalized multivariate Gaussians (and Wick's/Isserlis theorems)
Suppose $x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$ is distributed according to the real two-dimensional Gaussian with mean-$0$ and covariance matrix $\Sigma$. I am interested in a closed form for ...
-1
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1
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103
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Convergence in $\mathbb{L}_1$ implies convergence "perturbed" conditional expectations
Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following ...
3
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1
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70
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Multiplicative approximation for a negative moment of the binomial distribution
Let $X$ be a binomial random variable with parameters $n,p$.
Define the function
$f(n, p, t) = E\frac{1}{1 + t X},
$
where $t > 0$.
Question: Can we find an elementary function $F(n, p, t)$ such ...
-2
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1
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43
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$E(\mathbf{y}|\mathbf{x}+\mathbf{z})=g(\mathbf{x})$ almost surely, if $\mathbf{z}\perp \!\!\! \perp \{\mathbf{y},\mathbf{x}\}$ jointly?
Let $\mathbf{y},\mathbf{x}$ and $\mathbf{z}$ be real-valued random vectors with possibly different dimensions.
If $\mathbf{z} \perp\!\!\!\perp \{\mathbf{y},\mathbf{x}\}$ (i.e., $\mathbf{z}$ is ...
2
votes
1
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526
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What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?
It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from ...
3
votes
1
answer
279
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Bounds on hitting time of sum of i.i.d. random variables
I have a sequence $(X_i)_{i\geq 1}$ of i.i.d. random variables taking values in $\mathbb Z$. I know that each $X_i$ has mean $0$ and finite variance $\sigma^2$. Let $S_n=X_1+\cdots+X_n$. Then I can ...
0
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0
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149
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Reference book for a probability course
In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete ...
2
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0
answers
70
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Poisson process subordinated by a gamma process
I am working on a problem and I encountered the following situation:
$(N(t): t \ge 0)$ is a Poisson process with parameter $\lambda t $. If $T_{n} = \sum_{i=1}^n W_i$ represents the $n^\text{th}$ ...