# Questions tagged [probability-limit-theorems]

For question about limit theorems of probability theory, like the law of large numbers, central limit theorem or the law of iterated logarithm.

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### Gambler's ruin in the limit (only stopping rule ruin)

Imagine a classical Gambler's ruin with winning probability p and losing probability q = 1-p. You start at 1\$and lose once you reach 0\$. The only stopping rule is that the game is over when the ...
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### Does this simple branching random walk on $\mathbb Z$ satisfy a central limit theorem?

Consider the following simple branching random walk on $\mathbb Z$ in discrete time: At stage 0, we start with one token at the site 0. At each time step, we randomly, independently split each token ...
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### Is the random limit in the Kesten–Stigum theorem random in this simple case?

Fix $p \in (0,1)$ and suppose we have a sequence of random variables defined as follows: let $X_0 = 1$, and given $X_n$, we have the binomial distribution $$X_{n+1} \sim \text{Bin}(X_n,p) + X_n.$$ I.e....
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### Almost sure convergence of sum of independent random variables in the book by Hall and Heyde

In Section 1.3 of the book "Martingale Limit Theory and Its Applications" by P. Hall and C. C. Heyde, the authors give a martingale proof for the following result: If $S_n = \sum_{i=1}^n X_i$...
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### Support of Bohr-Jessen random variable at 1

I am reading the book by E. Kowalski on Probabilistic Number Theory. Link to the book: https://people.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf I am stuck on Exercise $3.2.4$. It seems ...
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### What does Borel's large number theorem really mean?

According to this page in "Encyclopedia in Mathematics", the Borel's large number theorem can be stated as below. "Consider independent random variables $X_1,\dots,X_n,\dots$ which are ...
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Let $X_n$ be a sequence of random variables which converge in probability to $X$. Suppose also that for all $\varepsilon > 0$ there exists $\delta_\varepsilon > 0$ such that $P(|X| \leq \delta_\... 1 vote 1 answer 76 views ### Can a sequence$X_n$of positive random variables converge to infinity almost surely with expectations remaining bounded? Is there some probability space$ (Ω, \mathcal F, \mathbb P)$and a sequence$(X_n)$of positive random variables on$Ω$for which$X_n → ∞ $almost surely, but the sequence$\mathbb E[X_n]$of ... 1 vote 1 answer 90 views ### From almost surely to expectation Suppose you have two sequences of positive random variables,$(X_n)$and$(Y_n)$, such that a.s.$Y_n = o(X_n)$. Do you see any sufficient condition to obtain$\mathbb E(Y_n)=o(\mathbb E(X_n))$? -1 votes 1 answer 74 views ###$X_n\to X$and$Y_n\to Y$in distribution. What we can say about the convergence of$X_n+Y_n$? Suppose we have two sequence of random variables$X_n$and$Y_n$such that they converge in distribution respectively to$X\sim\Gamma(\frac{1}{2},1)$and$Y\sim\Gamma(\frac{1}{2},1)$.$X_n$and$Y_n$... 0 votes 0 answers 26 views ### What can we say about the sum of independent random variables with finite expectation? Currently reading up on some measure-theoretic probability, and got stuck on the following problem from an earlier exam in a probability theory course. Problem 3 Let$X_1, X_2, \ldots$be a sequence ... 3 votes 2 answers 176 views ### Central limit theorem for uncorrelated and identically distributed random variables I'm trying to determine whether a sum of identically distributed but only uncorrelated continous random variables could possibly converge to a normal distribution. First of all we define the i.i.d. ... 5 votes 1 answer 192 views ### Random walk on a k-dimensional grid Consider a$k$-dimensional grid with integer points and let's begin our random walk at the origin$(0,0,\dots,0)$. Every step you have to move on each cartesian axis in the following way:$x_1$axis: ... 7 votes 1 answer 72 views ### Bootstrap Weak Convergence Let$X_1, X_2, \dots, X_n$be an iid sample from an unknown distribution finite mean$\mu$and finite variance$\sigma^2$. Furthermore, let$R_1,R_2,\dots,R_n$. denote iid Rademacher random variables. ... 2 votes 0 answers 39 views ### What's the limiting distribution of the eigenvalue and eigenvectors of the sample covariance matrix? The question is from the book 'Statistical Models and Methods for Financial Markets', Page 43. Suppose$\boldsymbol{x_1,..., x_n}$are$n$independent observations from a multivariate population with ... 0 votes 1 answer 32 views ###$X_n$is the number of heads in the sequence of tosses before the tails appear for the$n$-th time. Check if$\frac{X_n}{n}$converges in probability. We consider an infinite sequence of tosses of a symmetrical coin. For a natural number$n$, let$X_n$denote the number of all the heads that appear in the sequence before the tails appear for the$n$-... 0 votes 0 answers 27 views ### Are the set of finitely supported probability measure in Euclidean space compact? Suppose X is a compact subset of Euclidean Space. Let P(X) be the set of all Borel probability measures on X. Now let's consider the set of all Borel probability measures on X with finite support. My ... 0 votes 0 answers 17 views ### (Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes Let$X_t^n$and$X_t$be stochastic processes (with finite moments), and assume that for every$t>0$,$\lambda>0$and bounded continuous function$\varphi$, \begin{equation} \int_0^te^{-\lambda ... 0 votes 0 answers 43 views ### We have {$X_n$}, n$\ge$1 - a sequence of independent and equally distributed random variables with finite E|$X_1$|. We need to prove that... The problem: We have {$X_n$}, n$\ge$1 - a sequence of independent and equally distributed random variables with finite E|$X_1$|. We need to prove that$E|\frac{S_n}{n} - E$X_1$|\rightarrow 0$, where$... 29 views

### Contradiction of key renewal equation applied to inter-arrival CDF

I'm looking for where I've made a mistake, since I'm reaching a contradiction after applying key renewal theorem and differentiating through the convolution. Let $(X_t)$ be a renewal process with ...
Let $X_1,X_2,...$ be positive integer-valued random variables with the same distribution but they can be dependent on each other. We want to prove that $\lim_{n\rightarrow \infty} E[\max_{i<n} X_i/... 1 vote 1 answer 50 views ### Converge in probability I'm trying to solve the following Probability problem: Consider$ X(\omega) : [0,1] \to [0,1] $, where$f(x) = 2x \mathbb{I}_{[0,1/2]}(x) + 0.75 \mathbb{I}_{\{1\}}(x) $and$X(\omega) = \omega = x$. ... 1 vote 2 answers 91 views ### Expectation of inverse of the sum of positive random variables Suppose we have a sequence of independent identically distributed positive random variables$X_1,X_2,\cdots\stackrel{i.i.d.}{\sim}\xi$, and I am puzzled with the existence of expectation $$\mathbb{E}\... 2 votes 0 answers 37 views ### Distribution of convergent sum of random variables [duplicate] Let \{X_n\}_{n=1}^\infty be an i.i.d. sequence of random variables that are distributed uniformly on \{\pm 1\}. Define$$Y=\sum_{n=1}^\infty \frac{X_n}{n}$$Can anything meaningful be said about ... 4 votes 1 answer 109 views ### Convergence of a "simple" stochastic algorithm In this question, we will consider the convergence of an algorithm as follows. Let$f(x) = \frac{1}{2}x^2$be our objective function. We define a so-called forcing function$\rho(\alpha) = \frac{1}{2}\...
Each day, a very volatile stock rises 70% or drops 50% in price, with equal probabilities and different days independent. Suppose a hedge fund manager always invests a fraction $\alpha$ of her current ...