2
votes
1answer
51 views

Why does differencing create wide-sense stationary time series?

In time series analysis, a common assumption made is that the series is wide-sense stationary, ex. that it has time invariant mean and covariance. However, as this is often not the case in real life, ...
-1
votes
1answer
185 views

Generating independent random variable from two correlated random variables

Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
5
votes
1answer
95 views

Deviation bound for the maximum of the norm of Wiener process

Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof: $$ ...
0
votes
0answers
62 views

Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
1
vote
0answers
51 views

Stochastic process inference from partial observations

Consider a set $U$. My signal is a piece-wise constant "function" $Sig: t \mapsto s$, i.e. the signal at time $t$ equals to some subset $s \subset U$. One can see $Sig(t)$ as a stochastic process. ...
4
votes
1answer
133 views

diffusions corresponding to estimators

I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...
2
votes
1answer
141 views

Empirical estimator fot the total variation distance on a finite space

I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$): ...
4
votes
0answers
260 views

Inverse Fourier Transform involving a Bessel Function, Exponential, and Power

I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$: $\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds $, where $J_{p/2-1}$ is a ...
1
vote
1answer
68 views

How to simulate random paths of a non-homogeneous continuous-time Markov process with discrete state space for a given infinitesimal generator matrix?

Let $X=(X_{t},t \in T)$ be a non-homogeneous, continuous time Markov process with a finite state space S={1,...,K}. Let $\alpha_{i,j}(t)$ be the hazard rates of some $\varGamma$-distributed random ...
3
votes
0answers
58 views

Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
3
votes
2answers
270 views

Probability distribution for two-state system that depends on residence time

I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...
2
votes
0answers
86 views

A simplified MCMC / MH algorithm. Are there known convergence results?

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
0
votes
0answers
141 views

Spectral densities and their corresponding covariance functions.

Hey guys, I'm currently doing a course in stochastic processes and have come across something that has been wrecking my mind for a while. So, let's say that I have some even, symmetric function ...
5
votes
1answer
128 views

Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?

Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic? If so, what are necessary and sufficient conditions ...
0
votes
0answers
78 views

Markov renewal process with failure?

I hope this question is not too elementary for this site, and that it contains a sufficient degree of detail. I have a problem where I want to model sequences of variable length $\boldsymbol{e}_i = ...
1
vote
1answer
181 views

Kalman Filter…Denoising measurement data to track objects

Hi Everyone, I am about to implement a Kalman Filter in a software. I found this very helpful article here: http://bilgin.esme.org/BitsBytes/KalmanFilterforDummies.aspx The example helps a lot, ...
1
vote
1answer
196 views

Extending Wald's equation to two classes of i.d. random variables?

I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
1
vote
0answers
148 views

Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis. Correlation of the series changes over time and across ...
1
vote
0answers
99 views

time derivative of the median of a stochastic process

Suppose you have a cumulative distribution that is changing with time, namely $ P_t(x) $. Assume $ P_t $ is monotone increasing and smooth enough so that we can define $ x_t(P) = P_t^{-1} $. We want ...
7
votes
4answers
1k views

Recent impressive combinatorial developments in probability theory

In the preface to the second edition of Daniel Stroock's book "Probability Theory: An Analytic View", there is this striking claim (on p. xv) ... I suspect that, for at least a decade, the most ...
2
votes
1answer
341 views

MCMC with progressive demollification of delta distributions

Edit: I simplified the example to a canonical case for clarity. Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space ...
4
votes
1answer
185 views

Hyperplane arrangements and covering numbers

Let $H$ be a set of $(d-1)$-dimensional hyperplanes in $\mathbb{R}^d$. For each hyperplane $h \in H$ let $D(h)$ and $\bar{D}(h)$ be the corresponding half spaces of $\mathbb{R}^d$. For a point $x ...
0
votes
1answer
76 views

multimodal circular model

Hi, can someone provide me with a list of probability models that is akin to Von Mises but consists multiple (potentially infinite) modes that takes into account attractors in the entire 2-D spatial ...
1
vote
2answers
219 views

Gibbs sampling step size

I have some data generated using MCMC methods and in particular Gibbs sampling. I computed the autocorrelation but I'm unsure how to determine how many samples to skip. I'd like to determine that ...
4
votes
2answers
535 views

Are Gaussian Processes more important than other stochastic processes?

I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason ...
2
votes
0answers
280 views

Estimating Wiener process parameters

Consider a Wiener process with zero drift, infintesimal variance $\sigma^2$, and an unknown starting value $\nu$. That is, \begin{align} Y_t \sim \mathcal{N}(\nu, t\sigma^2). \end{align} Now, ...
0
votes
0answers
168 views

The spectral representation and isotropic covariance functions

Caveat: My apologies if this question is poorly phrased. I am an engineer/computer scientist teaching myself mathematics. The spectral representation of the covariance function of a second order ...
4
votes
3answers
665 views

Kolmogorov probability axioms without non-negativity condition

What is a minimal consistent modification of probability axioms to include negative values? Is it enough to use a minimal modification of axioms obtained by formal exclusion of non-negativity ...
9
votes
1answer
2k views

Coin Pusher Game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins). Essentially, one has a distribution of ...
1
vote
0answers
635 views

Moments of function of Poisson process

(I'm new to Poisson processes, so please edit if my terminology is incorrect.) Edit: per comments, here is a (more) general version of the originally posted problem (which is now at the bottom, below ...
21
votes
2answers
870 views

Drawing natural numbers without replacement.

Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all ...
4
votes
1answer
913 views

Squared residuals versus just residuals?

In statistics (in particular time series analysis, like ARCH/GARCH models) I've noticed that residual diagnostics usually look at autocorrelation of residuals and squared residuals. Why both? What ...
4
votes
3answers
385 views

Averaging over random walk on binary lattice

I have a function $f$ defined over a bit vector of length $n$. Equivalently, this is a function defined on the set of integers $[0,\ldots,2^n-1]$. I would like to compute the mean or variance or some ...
1
vote
1answer
722 views

Derivative of a differentiable stationary Gaussian process

Thanks for your help in advance. I'm interested in understanding the properties of derivatives of a differentiable stationary Gaussian process. Specifically, is the derivative also a Gaussian ...
1
vote
1answer
617 views

Autocorrelation of a ±1-valued random process with certain statistics

Suppose $f(t)$ is a continuous-valued, zero-mean stochastic signal with Gaussian autocorrelation (with variance $\sigma^2$). Suppose I then pass this signal through a step function, producing a new ...
3
votes
1answer
371 views

Stationary non-isotropic spatial stochastic processes

I asked this question in math.stackexchange but got no response; Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not ...
3
votes
2answers
407 views

An Upper Bound for the Average of Top Order Statistics

The following problem arises when we try to bound the expected offline optimal value of a simple online assignment problem with random values and unit weights, by its deterministic approximation. The ...
1
vote
2answers
118 views

is there an interpretation to the inverse of $I-M$ in multitype branching process, where $M$ is the mean matrix?

Assume we have a multitype branching process, i.e., we have a mean matrix $M_{ij}$ and $M_{ij}$ is the expected count of generating $j$ from $i$ in one time step, i.e.: $M_{ij} = \sum_{r} n(r,j)P(r | ...
56
votes
9answers
9k views

Is there a natural random process that is rigorously known to produce Zipf's law?

Zipf's law is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ ...