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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Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent?

Let $\varphi_k\in\mathbb{C}$ be a primitive $k$-th root of unity, and define the sets ...
7
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2answers
166 views

concentration inequality for entropy from sample

Consider a measure $\mu$ on a finite set, and let $x_1, \ldots, x_n$ be i.i.d samples from $\mu$. Then the expression $S_n = -\frac{1}{n} \sum_{i=1}^n \log \mu(x_i)$ converges by a.s. to the entropy ...
2
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1answer
117 views

Approximation of the cumulative normal distribution

As is well known, there is no explicit formula for $\int_{-\infty}^\infty step(t−x)\cdot e^{−t^2/2}dt=\int_x^\infty e^{−t^2/2} dt$ for generic $x,$ where $step(z)$ is the step function, $step(z)=1$ ...
2
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1answer
83 views

Shift Invariance of Backward Martingales for tail trivial probability measures

Consider the infinite cartesian product $\Omega=\{0,1\}^{\mathbb{N}}$ as a measurable space endowed with the $\sigma$-algebra $\mathscr{F}$ generated by the cylinder sets and $\sigma:\Omega\to\Omega$ ...
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2answers
195 views

Why are the vectors with this special structure linearly independent with high probability?

Given $a_i\in\mathbb{R}^m$ for $i=1,\ldots,2m$ with independent and identically random entries of some continuous distribution. Every choice of $m$ vectors from $\{a_1\ldots,a_{2m}\}$ is linearly ...
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1answer
79 views

Distribution of maximum unique number of several random numbers

Suppose discrete random variables $\{X_1, X_2, ..., X_n\}$ are i.i.d. described by the probability function: $f(x) \equiv \text{Pr}(X_i = x)$, and $X_i \in \{1,2,3, ..., m\}$. Let $Y$ be the ...
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100 views

Expectation of a specific random variable on the probability space of $n\times n$ matrices over $\{0,1\}$

Let $\mathcal{G}_{n,\frac{1}{2}}$ be the probability space of $n\times n$ matrices over $\{0,1\}$ and each entry of the matrix is independently equal to 1 with probability $\frac{1}{2}$ and equal to 0 ...
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89 views

Soft question: take complex analysis or cryptology? [migrated]

I am math major junior considering math grad school. I need to decide whether to take complex analysis or cryptology this semester. Complex analysis seems to be a recommended course for people ...
3
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1answer
149 views

A lottery on coins in a convex set

You play the following game. You get $4n$ gold coins and have to arrange them in the unit square in general position (no two coins have the same x or the same y coordinate). Call this set of coins ...
3
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2answers
155 views

Brownian motion in $\mathbb{R}^n$, probability of hitting a set

Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...
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37 views

Exploiting conditional independence for inference in Bayesian networks

How is conditional independence used for making probabilistic inference in Bayes networks easier or more efficient? For example, given the following Bayes network: Let's say I want to compute ...
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1answer
57 views

Local extrema of a posterior probability

Let $x$ be a binary random variable and $z$ be an arbitrary random variable. $x$ and $z$ are, in general, not independent. Let $y_1, \ldots y_n$ be $n$ identically distributed binary random variables ...
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0answers
50 views

Order statistic of Markov chain sample path and related probabilities

Consider a 1D sample path, denoted as $\{X(1), ..., X(t), ..., X(n)\}$, generated from a discrete time finite state (time homogeneous) Markov chain over states $\{1,...,m\}$, with transition ...
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1answer
162 views

Does walk on $Z^d$ with steps $(\pm 1,\pm1,\ldots,\pm 1)$ return to origin?

If the steps are iid uniform as in the title, is the return probability known? Is it positive? Answers, comments, references welcome. Clearly each of these steps is not equivalent to $d$ steps of type ...
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2answers
137 views

Multivariate CLT with varying dimension size

If $X_i$ is a sequence of $d$ dimensional i.i.d. integer valued random vectors with covariance matrix $\Sigma$ and $\mathbb{E}(X_i) = \mu$. Let each element of $X_i$ be chosen i.u.d. from $\{-1,1\}$. ...
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1answer
68 views

Question about the weak convergence of probability

Let $\mu$ be a probability measure on $\mathbb R$ and set $$c(K):=\int_{\mathbb R}(x-K)^+d\mu(x).$$ Assume that one has a sequence of probability measures $(\mu_n)_{n\ge 1}$ s.t. $$\int_{\mathbb ...
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51 views

Definiteness and infinite divisibility of kernels including heat semigroup

Let $P_{t}$ be the usual heat semigroup. Can one show (preferably) or disprove that for arbitrary $k \in \mathbb{R}_{>0}$ and $n \in \mathbb{N}$ we have \begin{equation} ...
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1answer
105 views

Question abouth Skorokhod representation of random variables (II)

This is a continuation of Question abouth Skorokhod representation of random variables Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that $$\int_{\mathbb R}|x|^pd\mu(x),~ ...
4
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1answer
104 views

Hellinger integral for the Student/Cauchy family

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$. Let now $p$ be ...
4
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1answer
96 views

Question abouth Prokhorov metric

Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that $$E\left[|X-Y|\right]<\varepsilon.$$ Set $Law(X)=\mu$ and $Law(Y)=\nu$, ...
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81 views

Maximal Correlation versus Correlation Coefficient When one RV is Gaussian

Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation ...
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0answers
66 views

Conditions which might imply a function is independent over its arguments

Suppose $f: [0,1]^{k+1} \to [0,1]$ and let $(a_0, \dots, a_k, b)$ be a tuple of $[0,1]$-valued random variables. Suppose for each $0 \leq i \leq k$ there is a collection of tuples $$ A_i = ...
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37 views

Probability same football square wins all four quarters [migrated]

A local casino has a Super Bowl football squares promotion going on. All the 100 squares are guaranteed to be filled by the customers. The casino will pay a certain amount to the winner of each ...
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141 views

Groups with probability measures

Are there algebraic structures that integrate groups with probability measures? For instance, can the closure operation on a group be assigned a probability that says "how much" a member belongs to ...
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1answer
80 views

Negative population variable importance

I asked this question on stats.stackexchange and even elsewhere, but it never received an answer. I just state the probabilistic problem here. It is about the optimality of the conditional ...
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2answers
112 views

divisibility of uniform distribution [closed]

Let $X$ and $Y$ be independent and identically distributed random variables. Can $X+Y$ be a uniform distribution? (Please prove.) In other words, is a uniform distribution divisible? The meaning of ...
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68 views

Conditional version of martingale difference concentration inequality

Let $M_n$ be a $\mathscr{F}_n=\sigma(\eta_m,\theta_m, m\leq n)$ measurable martingale difference sequence. Then is it possible to find a exponential tail bound for the following $$P(|M_{n+1}| > ...
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89 views

Interplay between CLT and convergence in Total Variation

Given a random variable $X$ with bounded moments such that $E[X] = 0, E[X^2] = 1$, let $F_n$ denote the distribution $\sum\limits_{i=1}^d\frac{X_i}{\sqrt{n}}$ where each $X_i$ is an independent copy ...
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30 views

Comparison between the entrance measure and the harmonic measure

Consider the standard two-dimensional Brownian motion, and define $\tau(A)$ to be the hitting time of $A\subset \mathbb{R}^2$. Let $hm_A$ be the harmonic measure (from infinity) on $A$. Let $B(r)$ be ...
3
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1answer
103 views

Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...
3
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1answer
148 views

Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e. $$\rho(\mu,\nu)<\varepsilon,$$ then there exist two random ...
4
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1answer
160 views

Short papers in applied probability

Which journals publish short papers/technical note in applied probability, for example stochastic approximation ?
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79 views

standard auction model

I'm not familiar enough with the auction theory to know where to look, but this seems close to what seems to be known as the "standard auction model". Say an asset is up for auction.The true value of ...
6
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1answer
220 views

Finding discrete entropy via differential entropy

In a recent math.se question the following was asked which I have slightly edited. " Consider a fixed and given $n$ by $n$ matrix $M$ whose elements are chosen from $\{-1,1\}$. Consider also a random ...
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0answers
89 views

Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and ...
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67 views

$\sum_{n\in\mathbb{Z}^2}d\mu(x-2\pi n)=0\Rightarrow$ the summands are pairwise mutually singular

Let $\mu$ a finite measure supported by $\Gamma$ (smooth curve in $\mathbb{R}^2$) and absolutely continuos with respect to the arc length measure on $\Gamma$. Please why if ...
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4answers
853 views

Probability two products are equal

I am interested in the following simple looking problem on which I am stuck. Let $M$ be a fixed $m$ by $n$ matrix with $\pm1$ elements. Let $x$ and $y$ be two independently sampled random ...
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1answer
58 views

“Convergence speed” results for the Langevin process

The Langevin process is defined by the following stochastic differential equation: $$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$ Its equilibrium distribution is the following: $$ p_\infty (x) \propto ...
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465 views

Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...
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312 views

Adaptive version of the Azuma–Hoeffding inequality

The Azuma inequality states that if we have a martingale $X_1,\ldots,X_N$ that satisfies a bounded difference condition: $$|X_k - X_{k-1}| \leq c_k$$ Then: $$\Pr\left[X_N - X_0 \geq \sqrt{2\sum_kc_k^2 ...
6
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1answer
264 views

Law of unconsious statistician: application in characteristic function

Let $g(x)=(x-a)\mathbf 1_{x\ge a}$ for some $a>0$ and let $X$ be a non-negative random variable with cdf $F$ and $E[X]<+\infty$. I want to calculate $$\frac{d}{da}E[g(X)]$$ To do that I thought ...
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Full version of Soucaliuc's research announcement “Réflexion entre deux diffusions conjuguées”

Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes: [1] F. Soucaliuc, Réflexion entre deux diffusions ...
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180 views

Constructing an independent uniform random variable from two independent ones

Does there exist a continuous (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable ...
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1answer
59 views

Alternative formula of a Green's function for average density of eigenvalues of random matrix

A Green's function is defined as follows: $$G(\omega) = \frac{1}{N}\mathrm{E}\big[ \mathrm{Tr}\frac{1}{I\omega - J} \big]$$, where $I$ is the $N$-dimensional identity and $E$ means expectation value ...
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258 views

Proofs of main probability results from other fields

Making connections between different areas is very exciting and probability has already made connections with other fields (BM used in proving complex analysis and PDE results). To keep it short, I ...
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169 views

Is the limsup or liminf of n-wise independent events independent?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. ...
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64 views

Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
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82 views

Calculate correlation values of an ensemble of $N\times N$ real asymmetric random matrix from Gaussian measure

I am now reading a paper by Sommers, H. J., et al. "Spectrum of large random asymmetric matrices." Physical Review Letters 60.19 (1988): 1895-1898., it claims a mathematical statement (equation (2) in ...
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359 views

Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in ...
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163 views

Derandomization barriers in complexity theory applicable as barriers to constructive arguments replacing probabilistic method

The probabilistic method as first pioneered by Erdős (although others used this before) shows existence of a certain object while finding that object may take exponential time. (1) Is there any ...