25 votes

Positivity of certain Fourier transform

A simple argument that the Fourier transform cannot be nonnegative for any $m>1$, integer or not, is given in my 1991 paper with Odlyzko and Rush: Noam D. Elkies, Andrew M. Odlyzko, and Jason A. ...
Noam D. Elkies's user avatar
18 votes
Accepted

Expected length of longest stick in a stick snapping process

It seems that the length of the longest stick is of order $n^{2\sqrt{2}-3} = n^{-0.171\ldots}$ as $n\to\infty$. This follows from a discrete-time analogue of the homogeneous fragmentation process, see ...
Timothy Budd's user avatar
  • 3,545
18 votes

What phenomena are better modelled by SDE instead of ODE?

This is a really broad question, but in general noise terms become important if there are few degrees of freedom; for example, chemical reaction kinetics can be accurately described by coupled ODE's, ...
Carlo Beenakker's user avatar
16 votes
Accepted

Positivity of certain Fourier transform

it is positive for $m=1$, but not for $m=2$, see this Mathematica output:
Carlo Beenakker's user avatar
16 votes

Positivity of certain Fourier transform

I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor. Another reference is Paul Lévy "Sur une application de la dérivée d’ordre ...
Abdelmalek Abdesselam's user avatar
13 votes

Naturally occuring counting process with a 1/log asymptotics?

No. One can create many sets of integers whose density decays like 1/log. An easy variant of the set of primes is the set of integers $n$ not divisible by any prime less than $n^\alpha$, for some ...
Greg Martin's user avatar
  • 12.7k
13 votes
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Does there exist an almost surely differentiable martingale?

The answer is no. Indeed, if a martingale is a.s. everywhere differentiable, then its quadratic variation is a.s $0$. So, by the Burkholder--Davis--Gundy inequality, the martingale is a.s. constant. ...
Iosif Pinelis's user avatar
13 votes
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A curious martingale

No. see Martingale Convergence Theorem 4 Let X be a continuous martingale. Then, almost surely, one of the following is satisfied $X_\infty=\lim_{t\rightarrow\infty}X_t$ exists and is finite. $\...
Thomas Kojar's user avatar
  • 4,414
11 votes

James-Stein phenomenon: What does it mean that a James-Stein estimator beats least squares estimator?

There is an excellent issue of Statistical Science that address the James-Stein phenomena from various aspects. https://www.jstor.org/stable/i23208816 Question What does it mean that a James-Stein ...
Henry.L's user avatar
  • 7,951
11 votes
Accepted

A theorem by Harald Cramér?

This is a special case of a general law of the iterated logarithm for non-iid random variables (r.v.'s), which states the following: Suppose that $Y_1,Y_2,\dots$ are independent zero-mean r.v.'s, $...
Iosif Pinelis's user avatar
11 votes
Accepted

Recursive random number generator based on irrational numbers

Of course the $X_k$ are not independent as random variables. So I assume you are referring to some notion of asymptotic independence, and it would help if you state your conjecture more precisely. One ...
Yuval Peres's user avatar
11 votes
Accepted

Mathematical construction of $\phi^4$ Euclidean field theory

When $d=2$, this works fine and this is precisely how Nelson originally constructed the $\Phi^4$ measure (in finite volume). Already for $d = 3$, the $\Phi^4$ measure is singular with respect to the ...
Martin Hairer's user avatar
11 votes

What phenomena are better modelled by SDE instead of ODE?

Brownian motion is an obvious example. Brownian motion described particles dispersed in a liquid that are large enough that the random jossling of the water molecules becomes important. Being one of ...
AccidentalTaylorExpansion's user avatar
10 votes
Accepted

Law of large numbers for martingales

The answer is yes, and it is based on an idea by Prokhorov (cf. e.g. Theorem 10 in Section 3 of Ch. IX in [Petrov, V. V., Sums of independent random variables, Springer-Verlag, 1975]). We have $EX_n^2\...
Iosif Pinelis's user avatar
10 votes
Accepted

Random walk to stay in an interval forever

Yes. Indeed, if $s = \sum_{i \geq 1} t_i^2 <1$, then $$ \mathbb{P}[ \ \ \forall n, \sum_{i=1}^n X_i \in [-1,1] \ \ ] \geq 1-s > 0. $$ To see this, note that $M_n = |\sum_{i=1}^n X_i|$ is a ...
js21's user avatar
  • 7,199
10 votes
Accepted

Scaling in Mehta's integral

Yes, this follows by the de la Vallée-Poussin necessary and sufficient condition for the uniform integrability. Indeed, suppose that \begin{equation} \gamma n^2\to a \end{equation} (as $n\to\infty$) ...
Iosif Pinelis's user avatar
10 votes

The Wiener measure of an open set

This is known as the support theorem for Brownian motion. Besides the proof in the answer of Iosif Pinelis and the proof in Exercise 1.8 of [1], there is also a proof on page 59 of [2]. ...
Yuval Peres's user avatar
10 votes
Accepted

On martingale convergence

You can construct a continuous local martingale $X$ such that $X(n) = n$ almost surely. Indeed, for $t < 1$, set $Y(t) = B(t/(1-t))$ for $B$ a Brownian motion and then $X(t) = Y(t \wedge \tau)$ ...
Martin Hairer's user avatar
10 votes
Accepted

How to optimally bet on a biased coin?

The strategy suggested in Geoffry Irving's answer of betting the maximum amount each time is correct, but the argument given is incomplete. The expected final amount, conditional on the outcomes of ...
Will Sawin's user avatar
  • 135k
9 votes
Accepted

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

Your intutive reasoning is leading you astray because you are thinking of Brownian motion as behaving like a smooth curve, for which there is a well-defined "direction" in which it is heading. ...
Nate Eldredge's user avatar
9 votes
Accepted

Can all local martingales be represented using only Brownian motion and finite variation processes?

First, a martingale is always only specified with respect to a filtration, and so is thus a local martingale. You do not specify any filtration in your problem, so I assume you mean the natural ...
Stephan Sturm's user avatar
9 votes
Accepted

Kolmogorov continuity theorem and Holder norm

One can apply a deterministic result, called Garsia--Rodemich--Rumsey inequality, to estimate $\mathrm{E}[||X||^\alpha_{\gamma;[0,T]}]$. Here is a particular form of this result, which is most ...
zhoraster's user avatar
  • 1,493
9 votes
Accepted

Maxima of Brownian motion

A good way to measure the set of maxima is the Hausdorff dimension of the set of records, which for BM is a.s. 1/2. Because of time/scale invariance, the dimension is the same for $\alpha B_\cdot$, ...
ofer zeitouni's user avatar
9 votes
Accepted

Largeness of the set of zeroes of a Brownian motion

Yes, the local time (at zero) maps the zero set of Brownian motion to an interval. See e.g. Lemma 6.9 page 159 in [1] for continuity. [1] Brownian motion, by Peter Mörters and Yuval Peres. Cambridge ...
Yuval Peres's user avatar
9 votes
Accepted

The Wiener measure of an open set

$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\om}{\omega}$Let $g:=f_0$. There is some real $\de>0$ such that \begin{equation*} \om(g,\de):=\max\{|g(y)-g(x)|\colon x,y\in[0,...
Iosif Pinelis's user avatar
9 votes

Regularity of translations for Brownian motion

If $f$ is absolutely continuous with $\int_0^1 f'(t)^2\,dt<\infty$, then, by Girsanov's formula (see e.g. Theorem 5.1. in [1]), the process $(B_t+f(t))_{t\in[0,1]}$ is a standard Brownian motion ...
Iosif Pinelis's user avatar
9 votes
Accepted

How is the Gronwall lemma used in this paper?

$\newcommand\al\alpha\newcommand\be\beta\newcommand\la\lambda$The reasoning in the paper is probably as follows: For real $t\ge0$, letting $$u(t):=2\la(E|X_t|^2-|EX_0|^2)-1,$$ $$\al(t):=-1+2\la(E|X_0|^...
Iosif Pinelis's user avatar
9 votes

"Practical" use of time-continuous stochastic processes like Wiener process or Poisson (point) process?

You answered your own question, I think. Many physics/economics models involve partial differential equations which often are studied using Feynman-Kac (among many other methods), which involves ...
Thomas Kojar's user avatar
  • 4,414
9 votes
Accepted

Can independent Brownian motions hit zero at the same time?

Your question is asking whether two Brownian motions can both first hit zero simultaneously. In fact we can say something stronger; for $N$ independent Brownian motions, the set of times where each ...
Julius's user avatar
  • 301
9 votes
Accepted

Paper request : “A random integral and Orlicz spaces” from K. Urbanick

Here it is, not the best quality scan, but it should serve the purpose. Urbanik and Woyczynski (1967) (the URL is also archived on the Wayback Machine, so it should last) I notice that some older ...
Carlo Beenakker's user avatar

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