# Tagged Questions

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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### The (infinite) invariant measure of an SPDE

Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type: \left\{ \begin{aligned} &\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...
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### Let X be a Geometric (1/4) and Y be a Geometric (1/2) be two independent random variables [on hold]

Let X be a Geometric (1/4) and Y be a Geometric (1/2) be two independent random variables. Obtain the conditional distribution of Y , given that X - Y = 1. The Answer I got was one. However, I just ...
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### Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$. Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
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### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
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### Markov Chain: Number of communicating classes of a power of the irreducible transition matrix [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P_k$. In terms of $d$ and $k$, how many communicating classes does $P_k$ have, and what is the period ...
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### American put option pricing by “binomial trees”

Dear MO World, I'm teaching a financial mathematics course and have found a fascinating (to me) numerical phenomenon and wonder if anyone has studied it, or knows anything similar. I'll try and give ...
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### Looking for an exposition of a certain theorem of Talagrand

The following is a theorem by Talagrand (as stated here, http://arxiv.org/pdf/1511.08609v1.pdf), Let $(X, \mu)$ be a probability space. Let $F : X \rightarrow \{0,1\}$ be a family of functions ...
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### Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel

Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, $D \subseteq X \times A$ be an analytic set, and $D_x := \{a \in A : (x, a) \in D\}$ denote the $x$-section of $D$ at $x \in X$. ...
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### Do binary symmetric channels maximize mutual information?

Consider the following setup: $(X, Y)$ is a doubly symmetric binary source with parameter $0 < p < 1/2$, i.e., $X \sim \text{Bernoulli}(1/2)$, $Z \sim \text{Bernoulli}(p)$ and $Y = X \oplus Z$. ...
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### On the sum of uniform independent random variables

Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$? ...
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### resampling over Bowen balls

Hello MO World I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...
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### Concentration bounds on weighted sum of i.i.d. Bernoulli random variables

Let $X_1,\dots, X_n\sim\operatorname{Bern}(\frac{1}{2})$ be independent, identically distributed random variables, and $\alpha=(\alpha_1,\dots,\alpha_n)\in[0,1]^n$ a vector of non-negative weights ...
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### How much can the integrability at zero tell about the decay rate around zero? [migrated]

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
$B_t$ is a Brownian process, starting from the origin in $\mathbb{R}^n$. Let $\theta_X(T)$ denote the probability that the particle hits the set $X \subset\mathbb{R}^n$ within time $T$. Keeping $T$ ...