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. ...
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 ...
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, ...
16
votes
Accepted
Positivity of certain Fourier transform
it is positive for $m=1$, but not for $m=2$, see this Mathematica output:
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 ...
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 ...
13
votes
Accepted
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.
...
13
votes
Accepted
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.
$\...
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 ...
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, $...
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 ...
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 ...
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 ...
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\...
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 ...
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 $n\to\infty$) ...
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]. ...
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 $X(t) = Y(t \wedge \tau)$ ...
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 ...
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. ...
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 ...
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 ...
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 $\alpha B_\cdot$, ...
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 ...
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,...
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 ...
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|^...
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 ...
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 ...
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 ...
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