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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46 views

How to verify a vector has a steady gradual increase mathematically? [on hold]

I rank the values inside my vector then look to verify the values have a gradual but steady increase. Ideally if I had 10 numbers the smallest would occur first and the second smallest would rank ...
3
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3answers
126 views
+50

Probability that a sum of intependent random variables hits a point

Let $X_1,\ldots,X_n$ be independent random elements of a normed space $X$. Suppose that $\sup_{x\in X}\mathbb{P}(X_i=x)=p_i$. What is the best known upper bound for $$\sup_{x\in X} ...
0
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1answer
57 views

Nagakami behavoir

Is the sum of square Nagakami random variables Erlang distributed? What is the distribution of euclidean norm of complex Nagakami? Cheers!
0
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1answer
54 views

Help in finding distribution of the following function of random variable

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
0
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0answers
34 views

Rate of convergence in narrow convergence

Does anyone help me in the following question? I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...
8
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2answers
341 views
+50

Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game

The game Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. Lucy picks $n$ different balls in any way Lucy likes, and paint them red. Lucy then giftwrap all the balls so that it is ...
3
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1answer
86 views

Ising model: probability of a long path of minus under plus boundary conditions

Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions. Low temperature, one minus spin. With a Peierls argument one can prove that, given a ...
0
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45 views

Green's function of the Ornstein-Uhlenbeck operator

Consider $\mathbb R^d$ with the Gaussian measure $d\gamma(x) = e^{\frac{1}{4}|x|^2}\,dx$. The Ornstein-Uhlenbeck operator $L$ is given by $$ Lu = \Delta u- \frac{1}{2}x\cdot \nabla u. $$ Is there a ...
2
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0answers
59 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
3
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1answer
121 views

Real points of zero-dimensional real algebraic varieties

There have been a number of discussions of zeros of random polynomials here (the most recent being: Why do roots of polynomials tend to have absolute value close to 1?). Here is a closely related ...
3
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1answer
155 views

Chances for a cosine polynomial to be positive at a point

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...
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27 views

How to define Product of Conditional Measures?

I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below. If $(X,\Sigma)$ is a measurable space, then the function $\mu : ...
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0answers
52 views

Lemma on Polish Spaces [on hold]

I asked the following question on StackExchange earlier, but received no replies yet. Let $(X_1,\Sigma_1,\mu_1)$ and $(X_2,\Sigma_2,\mu_2)$ be probability spaces. Suppose the following conditions ...
0
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0answers
26 views

Isotropic correlation function for a vector valued random field

I'm having trouble with some of the implications of the following theorem. Let $\mathbf{T} (\mathbf{x})$ be a mean-square continuous vector valued random field on $\mathbb{R}^3$ satisfying conditions ...
0
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0answers
95 views

A probability application question

Suppose there are two possible states $H$ and $L$, with prior probability $p$ and $1-p$ respectively. There are infinite rounds with a discount factor $ d$. In round 1, you could choose a value ...
0
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1answer
168 views

Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$ [on hold]

Let $X$ be the random variable obtained as the maximum of a throw of $m$ dice (each of which is $n$-sided). In other words, $X = \max\{l_1,\cdots, l_m\}$ where $l_i$ can take any value between $1$ and ...
0
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0answers
13 views

Dirichlet distribution: Normalization of alpha values [migrated]

I'm a programmer and currently trying to apply the Latent Dirichlet Allocation algorithm by Blei et al. on a text mining problem. I am using a library called gensim for this, which takes, among ...
2
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0answers
62 views

A 1-D random variable from a random distribution

I have a random variable $X$ that is drawn from the pdf $$ f(x; \mu, \sigma, \sigma_{\mu}, \sigma_{\sigma}) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{|\hat{\sigma}|\sqrt{2\pi }} ...
13
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2answers
598 views

Parity of $\lfloor 1/(x y) \rfloor$ not equally distributed

A curious puzzle for which I would appreciate an explanation. For $x$ and $y$ both uniformly and independently distributed in $[0,1]$, the value of $\lfloor 1/(x y) \rfloor$ has a bias toward odd ...
9
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1answer
253 views

Guessing the larger integer: A game-theoretic twist

The starting point for this question is the old chestnut, already discussed on MO, about a game show on which the host has chosen two distinct integers and the contestant gets to reveal one of them at ...
0
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2answers
88 views

Asymptotics for Hitting the sphere from the Outside

The problem is: consider A a solid ball centered at 0 and the exterior starting point $x\in A^{c}$, what is the behavior of $P_{x}(T_{B_{r}(0)}>t)$ for $d\geq 3$ as $t\to \infty$,where ...
4
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1answer
256 views

Strong Law of Large Numbers for arrays of partly dependent random variables

Suppose $X_1$, $X_2$ are two independent real-valued random variables. Let $F$ be a continuous (unbounded) function from $\mathbb{R^2}$ to $\mathbb{R}$. Assume that the necessary measurability and ...
2
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0answers
71 views

(Reference) Asymptotics of hitting probability by Brownian motion

The problem is: Given compact set A with positive finite volume (eg. ball,cube), what happens to $P_{x}(T_{A}>t)$ as $t\to \infty$, where $T_{A}=inf_{t>0}(B_{t}\in A)$ and x is in the "exterior" ...
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0answers
15 views

Population Variance PDF given Sample Variance [closed]

I require the pdf of the population variance (v2) conditioned on the sample variance (s2). I know that (n-1)s2/p2 follows a chi-squared distribution but can it be applied in the other direction? ...
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0answers
19 views

Perturbing moments of multivariable distributions

Let $P$ be a multivariate probability distribution on $\mathbb R^n$ which is moment-determinate and let $\{m_k : k \in \mathbb N_0^n\}$ be the sequence of moments $P$. Fix an order $p$ and consider ...
0
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1answer
100 views

Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
3
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0answers
123 views

Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...
2
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0answers
68 views

hitting time of the first quadrant for a 2-d random walk

Suppose $(S_n)_{n=1}^{\infty}$ is a simple random walk in $\mathbb{R}^2$. Let $\tau_{(a,b)}$ be the hitting of the first quadrant when $S_0 = (a,b)$. Is there a way to compute or estimate the ...
0
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1answer
84 views

Sum of n independent F distribution random variables [closed]

I need a help: What will be the distribution of sum of $n$ independent F distributed random variables with parameters 1 and 1 (i.e., $F(x;1,1)$? Formally, say $x_1,\ldots,x_n$ are i.i.d. as F(1,1), ...
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30 views

Stability of simple conditions on functions under convolution and/or mixture

We consider families of smooth probability densities defined on $\mathbb{R}^+$, $p=(x\in \mathbb{R}^+ \mapsto p_n(x))_{n\in\mathbb{N}}=(p_n)_{n\in\mathbb{N}}$ satisfying (i) $\int_{\mathbb{R}^+} ...
13
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2answers
354 views

What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$

Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x_1,x_2,\ldots ,x_n)=\sum _{S\subseteq [n]} \hat{f}(S)\prod_{i\in S} x_i $$ The degree of the ...
3
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1answer
338 views

Publishing an elementary proof of a less-general and less-useful version of a classic result?

Background Let $X_t$ be the continuous time Markov process on the state space {Working, Broken}. Let $U$ be the cumulative sojourn Working during an interval $[0,\tau]$ (the process's uptime). It is ...
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0answers
37 views

Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular, Have the numerical implications, if any, of the results in ...
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55 views

Why we mistaken coin toss to be an example of classical probability? [migrated]

It is now well known that a random coin toss has 1/6000 probability of landing on its edge. So the out-dated model that a coin toss always land on either heads or tails with probability 1/2 is wrong. ...
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0answers
32 views

Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit: $$ dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC $$ where $W$ is a standard Brownian motion ...
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1answer
75 views

Diffusion processes with different diffusion coefficients and absolute continuity

I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level. My initial situation is the following. Consider two SDEs: \begin{align} ...
0
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1answer
38 views

Lower bound for median of independent outcomes

Consider a stochastic variable $X$ taking positive real values and the events $P(X\geq a)\leq\frac{1}{3}$ and $P(X \leq b) \leq \frac{1}{2.9}$. We define $X_m$ as the median of $k$ independent ...
3
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1answer
125 views

Random non-intersecting circles in the plane

If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...
4
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1answer
351 views

Why are the angular differences of these random complex polynomial coefficients almost constant?

This is based on მამუკა ჯიბლაძე's (not-)answer here. I guess it is better to make up a new thread for it. Let me repeat the setup here: We consider polynomials whose complex roots are randomly ...
2
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2answers
83 views

Probability of a contiguous sub-sequence with different elements

Let $a$ and $b$ be two positive integers, and say $b\gg a$. Let $S$ be a random sequence with $ab$ elements, whose entries are all integers from $1$ to $a$, such that each number from $1$ to $a$ ...
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75 views

Reference: Bochner Integral`

What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
7
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3answers
333 views

Learning roadmap: 'combinatorial' probability

I am about to finish working through Williams's Probability With Martingales. I have studied analysis up to the first five chapters of Folland's text but have not studied any combinatorics yet. It ...
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1answer
50 views

Could somebody recomends a good book or article about numerical methods for Stochastic Partial Differential Equations

Could somebody recomend a good book or article about numerical methods for Stochastic Partial Differential Equations. I'm looking for a good introductory material thanks.
2
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1answer
82 views

Interpretation of riemannian geodesics in probability

Good morning everybody. My question is, as maybe already hinted in the title, rather philosopic. We know that geometric properties of a riemannian manifold can be interpreted in terms of certain ...
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0answers
43 views

Is there an efficient algorithm for sampling from the negative hypergeometric distribution? [closed]

I'm writing a small statistics library currently. One of the algorithms I'm implementing has two variants: one that samples the hypergeometric distribution and one that samples the negative ...
0
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0answers
33 views

Numerical method for self-consistency of one-dimensional probability density function

I have an integral equation for self-consistency of one-dimensional probability density function, like this $$\rho_x(x) = \frac{1}{|a|}\int \int \rho_x\left(\frac{s-b}{a}\right) \rho_P(p) ...
9
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3answers
227 views

Central limit theorem via maximal entropy

Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that: $$\int_\mathbb{R} \rho(x)\, dx = 1$$ and $$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$ ...
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3answers
313 views

The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small? More ...
5
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161 views

Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?

Recent questions showed that roots of a random polynomial tend to lie on the unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
2
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1answer
59 views

Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...