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The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then $$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, …

Latent Dirichlet allocation - is quite a popular topic in data-mining. Wikepedia mentions thousands citations in few years. Question 0 Can one give some digest for a math minded person of the key i …

Consider general Brownian bridge W(0)=0; W(T) = a. (Here "general" means: $W(T)\ne 0$). What is the probability W(t) >= b, for all $ t \in [0, T] $ ? Is there close simple formula in terms of a …

Setup. Let casino generate a color: black or red with equal probability. Let client try to guess the color. If guess is correct - he earns 1 coin from casino, if not - he gives one to casino. If he lo …

Consider some manifold $M$ say compact smooth. Let $b_i$ be its Betti numbers (non-zero), i.e. its cohomology dimensions. Assume $M$ can be subsequently fibered by many manifolds, i.e. there is $ M_ …

Consider $S^k \subset R^{k+1} $. Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ). Consider Voronoi cell around each point. How many neighbours would a cell have …

Quote Wikipedia: Applications of graphical models include ... gene finding and diagnosis of diseases... Unfortunately there is no comment what are these applications... Can one comment on this ? Bac …

It is probably some basic statistics question, but... Informally 1: How to choose "criteria", such that it will guarantee that error decision probability is less than "epsilon", and maximize probabi …

Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$). Consider some "trading strategy" whic …

Context Working with some biological datasets it was puzzling to see the patterns like Figure 2 (right) below. The first feeling was, that it corresponds to some biological effects like correlations b …

Just out of curiosity. The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound. Saying that we cannot measure something with certain accuracy. However looking closer …

Have the following stochastic process $f(t)$ been studied in mathematics ? It is stationary, Gaussian, $f(t)-$complex independent Gaussians $N(0,1)$. The autocorrelation is given by the zero-order Be …

Let's start from a little bit far. Basic probability theory - chain rule reads: $$ P(AB) = P(A)P(B|A)$$ Example: consider n+m balls, where n - white balls, m - black balls, consider A - first cho …

Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes. What is known about their mathematical properties ? I know the …

Consider some compact Riemannian manifold $M$. Fix some point $p$. Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$. Consider growth function $g(r)$ to be volume of …