Questions tagged [probability-theory]

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 [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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38
votes
0answers
3k views

A difficult integral

For $\gamma>0,\delta>0$, How do I 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 \left(\frac{H}{H-x}\right)...
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434 views

Does this Condition on Exit Times imply $X_t$ is a Local Supermartingale?

Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $$\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}.$$ We can interpret $\tau_{s,a,b}$ as the ...
24
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1answer
2k 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 ...
21
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0answers
458 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 ...
21
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2answers
646 views

What happens to a random walk when we increase the probabilities of going right?

Consider a random walk on the integers where the probability of transitioning from $n$ to $n+1$ is $p_n$ (and of course, the probability of transitioning from $n$ to $n-1$ is $1-p_n$); we assume all $...
20
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0answers
454 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 ...
19
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0answers
5k 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; i\...
18
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0answers
1k views

Difficult integral for a marginal distribution

I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral: $$ \operatorname{p}\left(y\right) = \int_0^{1/\sqrt{\,2\pi\,}}\!...
17
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203 views

Measurability with respect to equivalent sub-$\sigma$-algebras for mappings into countably generated spaces

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $\mathcal{G}_1$ and $\mathcal{G}_2$ be sub-$\sigma$-algebras of $\mathcal{F}$ that are equivalent to each other in the sense that $...
15
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0answers
669 views

Random sum in coupon collection

I have a problem which involves the standard coupon collector's problem to find a probability density from the generating convolution. I start by defining the problem and a few basic statistics. Let ...
13
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1answer
216 views

Limiting behavior of coins on a finite graph with flips activated by a full neighborhood

Let $G = (V,E)$ be a finite graph, and as particular concrete examples, consider the case of a path or cycle graph. Say a configuration on $G$ is a function $f: V \to \{0, 1\}$, which we can think of ...
13
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316 views

If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

Let $f\in C^3(\mathbb R)$ be positive $g:=\ln f$ $d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^...
13
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0answers
365 views

Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 32.13

This is a verification request for a counterexample that I think I have found for Problem 32.13 on page 427 in Patrick Billingsley’s Probability and Measure textbook (third edition, but the problem ...
13
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0answers
2k views

probability measures vs. probability distributions vs. measure of probability density

I am learning probability theory right now and am confused about some basic concepts. I have a few questions and am wondering if you can also check if the following is correct: Suppose we have a ...
13
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0answers
799 views

Distribution of the sum of absolutes values of T-distributed random variables

Where $X$ is a r.v. following a symmetric $T$ distribution with $0 $mean and tail parameter $\alpha$. I am looking for the distribution of the $n$-summed independent variables $ \sum_{1 \leq i \leq n}|...
11
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0answers
396 views

Discretization formula for system of differential equations. “Solution to one of these is the initial condition of the other”. In which sense?

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
11
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0answers
572 views

Interpreting the Lindeberg's condition

I know the Lindeberg's CLT but I don't have a good grasp of the intuition behind the Lindeberg's condition. Could you please give some intuition behind said condition via an example (or, perhaps, via ...
11
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0answers
298 views

Limit distributions for Markov chains $X\to\sqrt{U+X}$

This question spawned from a recent, very interesting problem. Let $\varphi=\frac{1+\sqrt{5}}{2}$ and $T$ denote the map on the space of continuous probability density functions supported over $\...
11
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0answers
820 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ $$\...
11
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1answer
7k 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 moment-...
11
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0answers
273 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 $b_1=(u,v_1)$ ...
11
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0answers
1k 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 ...
10
votes
1answer
276 views

Law of large numbers for dependent random variables with fixed covariance

Given identically distributed random variables $X_1,...,X_n$, where $ \mathbb E X_i = 0$, $\mathbb E [X_i^2] = 1$, and $\mathbb E [X_i X_j] = c<1$, define $S_n = X_1 + ... + X_n$. Will $$\frac{...
10
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0answers
463 views

What is the origin of the term “Crystal Ball Condition”?

Given $p>0$ and a family of random variables $\{X_{n}\}$, the uniform integrability of $\{|X_{n}|^{p}\}$ follows from the existence of a $\delta>0$ such that $$ \sup_{n} E(|X_{n}|^{p+\delta})&...
10
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0answers
521 views

Intuition behind the renewal equation

So. I've acquired the unenviable task of having to learn renewal theory on my own. I'm finding most of it to be pretty intuitive, except for one thing. The intuition behind the renewal equation has ...
10
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0answers
294 views

Random walks in $\mathbb{Z}^2$

Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 ...
10
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0answers
1k 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 \...
10
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0answers
281 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 0,\...
10
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823 views

Bound variance proxy of a subGaussian random variable by its variance

If I know $X$ is a sub-Gaussian random variable, and I know it has finite variance $\sigma^2$. Can I assert that $\sigma^2$ is a valid variance proxy for $X$? Definition (sub-Gaussian Random Variable)...
10
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0answers
615 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. ...
10
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0answers
803 views

the parametrization of a Gumbel in terms of a Gaussian

Extreme Value Distribution From a Gaussian. I was wondering how the parametrization of $\alpha$ and $\beta$ of a Gumbel $e^{-e^{-\frac{x-\alpha }{\beta }}}$ was done in terms of a cumulative Gaussian $...
9
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0answers
93 views

there is a positive constant $c$ such that for any $n$ real numbers $a_i$ with $\sum a_i^2=1$, $\mathbb P[|\sum \epsilon_i a_i|\le1]\ge c$

Question (4.8.2 from the Probabilistic Method 4th edition by Alon and Spencer): Show that there is a positive constant $c$ such that for any $n$ real numbers $a_1,\dots,a_n$ satisfying $\sum\limits_{i=...
9
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0answers
135 views

Is there a well-defined `uniform' distribution on $C([0, 1])$?

I'm wondering whether we can define a uniform distribution on the space of continuous functions over a compact set, e.g. $C([0, 1])$. If so, then how should I rigorously describe it? And how can I ...
9
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0answers
95 views

Optimal rate of growth of i.i.d. Gaussians?

Suppose I have a countable collection $\{N_k\}_{k=1}^\infty$ of independent $\mathcal{N}(0.1)$ random variables and I want to estimate their rate of growth in the sense that I want to find a function $...
9
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0answers
372 views

Practical applications of non-standard probability

Recently I read a paper by Benci et al. describing an alternative to Kolmogorov's construction of probability where the probability measure $P$ takes values in a non-Archimedian field and we have $P(A)...
9
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0answers
509 views

Conditions for the existence of a density with respect to Lebesgue measure

Let $X:\Omega \to \mathbb{R}$ be a random variable on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and denote by $$\chi(\xi) := \mathbb{E}e^{i \xi \cdot X}, \xi \in \mathbb{R}^d,$$ its ...
9
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0answers
340 views

Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
9
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0answers
132 views

If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...
9
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0answers
194 views

Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
9
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0answers
1k views

Why is a predictable stochastic process called *predictable*?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I$ be an index set $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be a stochastic ...
9
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0answers
283 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 $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
9
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0answers
181 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: \...
9
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0answers
434 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 ...
8
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0answers
392 views

New numeration system, mapping to binary numeration system

Let us consider $$Z = X_1 + X_1 X_2 + X_1 X_2 X_3 + \cdots.$$ Here the $X_i$'s can only take on two different values: either $X_i=a$ or $X_i=b$, with $0 < a < b < 1$ and $a+b = 1$. The $n$-...
8
votes
1answer
270 views

If $X^{(n)},X$ are càdlàg and $X^{(n)}\to X$ in distribution, do the corresponding transition semigroups strongly converge?

Let $\left(\kappa^{(n)}_t\right)_{t\ge0}$ and $(\kappa_t)_{t\ge0}$ be Markov semigroups on $(\mathbb R,\mathcal B(\mathbb R))$ for $n\in\mathbb N$ $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly ...
8
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0answers
2k views

Rigorous Proof of Slutsky's Theorem

I was hoping to type up my proof of Slutsky's Theorem and get confirmation on the excruciating details being all correct... Statement of Slutsky's Theorem: $$\text{Let }X_n, \ X,\ Y_n,\ Y,\text{ ...
8
votes
0answers
432 views

What is the intuition behind conditional expectation in a measure-theoretic treatment of probability?

What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment? You may assume I know: what a probability space $(\Omega, \mathcal{...
8
votes
0answers
145 views

Getting a bound for Gibbs distribution mean

Suppose $F$ is a strictly convex and increasing function, $U$ a random variable with support $[0,1]$ and density $$ f_U(u)= \frac{e^{-\frac{1}{T}F(u)}}{\int_{0}^{1} e^{-\frac{1}{T} F(x)} dx}.$$ Do we ...
8
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0answers
449 views

How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
8
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0answers
883 views

Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...

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