Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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15
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414 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
12
votes
0answers
151 views

Integrating a matrix function involving a determinant and exponential trace

I am trying to find the normalizing constant for a probability distribution and ran into a difficult integral. When $X$ is an $p \times k$ matrix, $a>0,$ and $g>0,$ how can I compute $$\int ...
11
votes
0answers
156 views

Upper and Lower Bounds on $Var(Var(X\mid Y))$

Are there any particular properties that \begin{align*} Var(Var(X\mid Y)) \end{align*} satisfies so that we can derive any upper and lower bounds on it. For example, if we replace $Var$ with ...
11
votes
0answers
816 views

Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
10
votes
0answers
155 views

Multiplicative version of Mcdiarmid's inequality?

Suppose you have $n$ i.i.d. random variables taking values in $\{0,1\}$, and $X$ represents their sum. Then you can use a Chernoff bound to control the deviation of $X$ from its expectation. The ...
10
votes
0answers
174 views

Calculating $\sum_{y=0}^x \Pr[Y= y] \Pr[Z\leq k-y]^2$ when Y,Z are binomially distributed?

Remark: I recently rewrote this post, hoping to get answers! I am analyzing the following experiment: Pick an $x \in \{0,\ldots,2k\}$ uniformly at random Pick $(2k+1)$-bit bitstring $b_1=(u,v_1)$ ...
9
votes
0answers
118 views

Probability on entering direction of a simple random walk

Let $X(n)$ be a simple random walk on $\Bbb{Z}^2$. Also we define $S_{R} = \inf\{n > 0 : X(n) \notin [-R, R]^2 \} $ : the exit time of the square $[-R, R]^2$, $T_{v} = \inf\{n > 0 : X(n) = ...
9
votes
0answers
273 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...
9
votes
0answers
947 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
9
votes
0answers
255 views

Do probability distributions form a comonad?

$\def\unit{{\rm unit}}\def\join{{\rm join}}$It's well known that (discrete) probability distributions form a monad. Specifically, if we let $PX$ be the set of discrete probability distributions on ...
9
votes
0answers
221 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin ...
9
votes
0answers
308 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
7
votes
0answers
104 views

Existence of Radon Nikodym derivative of Stieltjes measures

Let $X$ be a real valued random variable and let $Q_{X}$ be a right-continuous quantile function for $X$ (alternatively this is a right-continuous generalized inverse to the CDF of $X$ ). ...
7
votes
0answers
116 views

A generalization of simple random walk

Suppose $S_n, n\geq 0$ is a martingale on $\mathbb{R}$ such that $S_0=0$ and $|S_{n+1}-S_{n}|\in [\frac{1}{2}, 1]$. Prove that there exists $c,C>0$ s.t. $$ \frac{c}{\sqrt{n}} \leq P( S_1\geq ...
7
votes
0answers
207 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
7
votes
0answers
3k views

How to find probability distribution function given the Moment Generating Function

After searching, I found two questions like mine, but didn't see my answer to my question. Finding a probability distribution given the moment generating function Finding probability using ...
7
votes
0answers
240 views

Proving existence of limit by Martingale.

I'm thinking about a question: Suppose $y_n > −1$ for all $n$ and $\sum |y_n| < \infty$. Show that $\prod_{m=1}^\infty (1 + y_m)$ exists. Since $\sum |y_n| < \infty$, we must be able ...
7
votes
0answers
213 views

Closure in the Space of Probability Measures with the Prohorov metric

I have seen this result stated countless times: assume the metric space $(\theta,d)$ is separable; then $(\theta,d)$ is complete if and only if the space $(\mathcal{P}(\Theta),\rho)$ (the space of ...
7
votes
0answers
657 views

Idempotence and the Rao–Blackwell theorem

Original question: In the Wikipedia article on the Rao–Blackwell theorem, we read: In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased ...
6
votes
0answers
61 views

A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega ...
6
votes
0answers
94 views

Higher math and statistics/probability

So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ...
6
votes
0answers
230 views

Relation between Shannon Entropy and Total Variation distance

Let $p_1(\cdot), p_2(\cdot)$ be two discrete distributions on $\mathbb{Z}.$ Total variation distance is defined as $d_{TV}(p_1,p_2)= \frac{1}{2} \displaystyle \sum_{k \in \mathbb{Z}}|p_1(k)-p_2(k)|$ ...
6
votes
0answers
82 views

Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R $ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
6
votes
0answers
119 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
6
votes
0answers
2k views

A difficult integral

For $\gamma>0,\delta>0$, trying to evaluate this integral: $$ I=\int_0^H\frac{e^{i t x} \log\left(\frac{H}{H-x}\right) ^{\frac{1}{\gamma }-1} \left(\left(\frac{k}{H \log ...
6
votes
0answers
372 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ ...
6
votes
0answers
243 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
6
votes
0answers
260 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; ...
6
votes
0answers
115 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
6
votes
0answers
188 views

How to compute or simplify this integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
6
votes
0answers
115 views

Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: ...
6
votes
0answers
2k views

Different versions of functional central limit theorem (aka Donsker theorem)?

I have seen several versions of functional central limit theorem (see the end of this post). I am confused, and hope someone could help to clarify their relations and differences. For example, I am ...
6
votes
0answers
332 views

Convergence of Martingale.

The question is: 5.2.11. Let $X_n$ and $Y_n$ be positive integrable and adapted to Fn. Suppose $\mathbb E(X_{n+1}|\mathcal F_n) ≤ (1 + Y_n )X_n$ with $ \sum Y_n < \infty$ a.s. Prove that ...
6
votes
0answers
240 views

Regular Version of Conditional Gaussian Distribution

Let $Z_{1}$ and $Z_{2}$ be two independent normally distributed random variables with expectations $\mu_{1},\mu_{2}\in\mathbb{R}$ and variances $\sigma_{1}^2,\sigma_{2}^2\in (0,\infty)$ . I would ...
6
votes
0answers
379 views

Calculating probability of some event using geometric considerations

I want to estimate exponentially the following probability: Let $\bf{U}\in\mathbb{R}^n$ be a random vector uniformly distributed on the $n$-dimensional hypersphere, centered at the origin with radius ...
6
votes
0answers
646 views

Topological necessary and sufficient condition for tightness

Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$: For each $\varepsilon>0$, we can find a compact subset $K$ of ...
6
votes
0answers
213 views

Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
5
votes
0answers
63 views

Properties of characteristic functions under statistical dependence

Given random variables $X,Y,Z$,and $\phi(.)$ denoting the characteristic function, I can see that the following is true when $Z$ is independent of $X,Y$: $|\phi_{X+Z,Y} (t, s) − \phi_{X+Z}(t)f_{Y} ...
5
votes
0answers
42 views

Poisson Process: indepedent increment

Let $\{N(t): t\geq0\}$ be a Poisson process of rate $\lambda$, and let $S_n$ denote the time until the $n_{th}$ event occurs. compute $P(S_3>5|N(2)=1)$ Attempt: Notice that ...
5
votes
0answers
71 views

Nonzero solutions to $\mathbb E[e^{\theta X}] = 1$?

Suppose $X$ is a random variable with $\mu=\mathbb E[X]\ne0$ and that $X$ has a finite moment generating function on some open interval containing $0$. Then for what $\theta\ne0$ does the following ...
5
votes
0answers
53 views

There exists a real number so that $X_n$ is a martingale

I am working on the following problem: Let $Y_n$ be a sequence for which there exists constants $\alpha$ and $\beta$ with $$ E(Y_{n+1}\mid \mathcal{F}_n)=\alpha Y_n +\beta Y_{n-1} $$ for each ...
5
votes
0answers
37 views

Find a Markov chain transition kernel

Let $f_{X}$ be a density we would like to sample from. For some reasons, $f_{X}$ may be analytically intractable or expensive to evaluate. A solution consists in considering a density $(x,y) \in X ...
5
votes
0answers
45 views

Tail estimate for $L^1$ functions.

Suppose $f\in{L^1(\mu)}$ for some probability measure $\mu$. Pick $\epsilon>0$ and let $A_n=\{x:|f(x)|>\epsilon{n}\}$. I want to show that $$\mu(A_1)+\mu(A_2)+\dots<\infty$$ My first ...
5
votes
0answers
36 views

Almost surely, for all $s \ge 0$, there exist $t$, $u \ge s$ with $B_t < 0 < B_u$?

Let $(B_t)_{t \ge 0}$ be a Brownian motion starting from $0$. Then, do we have that, almost surely, for all $s \ge 0$, there exist $t$, $u \ge s$ with $B_t < 0 < B_u$?
5
votes
0answers
73 views

Intuition for almost sure convergence = fast enough convergence in probability

I know the meaning of convergence in probability and almost convergence. From zero-one law, we can derive that if a sequence of random variables converges in probability fast enough, then it converges ...
5
votes
0answers
76 views

Probability of another 3 integers with same sum and product as the first 3 integers

Let us suppose $3$ integers are selected at random from a large range, say $$-1000\leq x\leq y\leq z\leq 1000$$ Now, we define the sum and product: $$\begin{align*}s&=x+y+z ...
5
votes
0answers
60 views

Why row vectors in stochastic processes?

It seems reasonable to state that column vectors $\mathbf{x}$ are the most frequently seen standard notation, often using $\mathbf{x}^\intercal$ to denote a row vector (transposed column vector). ...
5
votes
0answers
47 views

Generalized Binomial Model independent in the limit

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ With $0\leq \theta_n+ ...
5
votes
0answers
128 views

How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
5
votes
0answers
54 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...