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Results tagged with pr.probability
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user 5371
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
14
votes
What is the probability that the range of a set of N randomly chosen real numbers in [0, 1] ...
The keyword is the order statistics. The distributions of the maximum and minimum values of a sample of $n$ independent uniformly distributed random variables are given respectively by the laws
$$U_{ …
- 22.3k
5
votes
Accepted
Two geometric probability questions (one answered, one more to go)
There is an asymptotic formula for the minimal spherical distance when $n$ is large (see e.g. the PhD thesis "Random Diameters and Other U-Max-Statistics" by M. Mayer, Corollary 3.37):
Theorem. As …
- 22.3k
4
votes
Accepted
Intuitive "proof" or explanation of a result in Friedman's urn
You might be interested in the article by David A. Freedman on Friedman's urn. He reports a simple and intuitive proof due to Ornstein, which only uses the strong law
of large numbers.
In his notati …
- 22.3k
10
votes
Accepted
Expected value as decision criterion in the context of rare events
The Kelly criterion is the optimal betting strategy for a player with limited resources (and if one had an infinite amount of capital they probably would not be interested in buying lottery tickets an …
- 22.3k
12
votes
A Markov process which is not a strong markov process?
Let $X(t) = f(W(t) + \pi)$, where $W(t)$ is a standard Wiener process and
$$f(x) = \begin{cases} (x,0), & x\leq 0 \\\ \\\ (\sin x,1-\cos x), & 0 < x < 2\pi \\\
\\\ (x-2\pi,0), & x\geq 2\pi
\end{cas …
- 22.3k
17
votes
Accepted
Estimate probability( 0 is in the convex hull of N random points ) ?
This is a classical and essentially geometric problem. In fact, the answer does not depend
on the distribution of the points (as long as the distribution is centrally symmetric).
The following resu …
- 22.3k
3
votes
Why do Littlewood-Paley projections behave like iid random variables
There is a quantitative way to express the somewhat vague notion of "almost independence of the Littlewood-Paley projections".
Let $\mathcal F_n$, $n\in\mathbb Z$, be the minimal $\sigma$-algebra gene …
- 22.3k
6
votes
Accepted
Föllmer: "Calcul d'Ito sans probabilités" in English or German?
Föllmer's approach was mainly adopted by specialists in Mathematical Finance.
Have a look at Introduction to Stochastic Calculus for Finance by D. Sondermann. This is an intro lecture course based o …
- 22.3k
10
votes
What are Central Limit Theorems and why are they called so?
From the introduction to History of the Central Limit Theorem: From Laplace to Donsker by Hans Fischer:
The term “central limit theorem” most likely traces back to Georg Pólya. As he
recapitulat …
- 22.3k
9
votes
Accepted
Quantitative bounds for multivariate central limit theorem
There is a bunch of such statements which can be obtained by Stein's method.
You might be interested in the paper "On the Rate of Convergence in the Multivariate CLT" by Gotze, which is specifically …
- 22.3k
2
votes
expected values over binomial distributions
This should have been a comment but I don't have enough reputation points to post comments.
The expression for $F(n)$ looks very similar to the Bernstein approximation (or Bernstein polynomial) to t …
- 22.3k
1
vote
Accepted
Random walks and Lyapunov exponents
Random dynamical systems by Ludwig Arnold contains a thorough discussion of various multiplicative ergodic theorems (including the Furstenberg-Kesten result), but not the central limit theorems. As fa …
- 22.3k
14
votes
Accepted
approximately linear functions
Let $E$ and $E'$ be Banach spaces. Mappings $f:E\to E'$, which satisfy the inequality
$$\|f(x + y) − f(x) − f(y)\| \leq\epsilon$$
for all $x, y \in E$, are called $\epsilon$-additive (or approximate …
- 22.3k