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Questions tagged [branching-process]

22 questions
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Why convergence in probability?

I'm reading the proof of Corollary 3.13 of this paper. Let $(X(t))_{t\ge0}$ be a Markov branching process associated with a replacement matrix $R$, with eigenvalues $\lambda_i$ associated with left eigenvectors $v_i$ and right eigenvectors $u_i$. We…
Dada
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A monotonic behaviour in Branching Brownian Motion

Consider the usual branching Brownian motion with dyadic branching, where a particle starts from origin and travels according to the law of a standard $1$ dimensional Brownian motion. After an $exp(1)$ time, it dies and gives birth to two identical…
L--
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Geometric Distribution in Simple Random Walk

My question is related to this question here. This says that if $S_{n}$ is a simple random walk (with steps $+1$ or $-1$ with probability $p$ and $q$ respectively) started at $S_{0}=1$, and if $T=\min\{k\geq 0:S_{k}=0\}$ then $$Z_k = \sum…
Blitzkrieg
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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 into either 2 tokens to the left, or 3 tokens to…
Good Boy
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Density of independent but not identically distributed exponential random variables

I am reading the paper "Limit distributions for multitype branching processes of m-ary search trees" by Chauvin, Liu and Pouyanne. In section 5, the following random variable is defined: $Z=X_1+\dots+X_n,$ where the $X_i$ are $\text{Exp}(i)$…
CampFire
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How to get the Corollary from the Theorem

I would like to understand how this Theorem implies the subsequent Corollary: Theorem. Let A be an irreducible matrix, related to a continuous time Markov branching process $X(t)$, with dominant eigenvalue $\lambda_1$ and associated eigenvector…
Dada
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How to prove a Branching Process is transient for all $k>0$ and calculate extinction probability

Consider a branching process $\{z_n\}$. $z_0=0$, $z_{n+1}=\sum_{j=1}^{z_n}\xi_{z_nj}$ and the extinction probability is $\rho$. $\xi_{ij}$ are i.i.d. (i). If $\mathbb{P}(\xi_{ij}=1)<1$, prove that all $k>0$ is transient. (Hint: consider two cases:…
Kaven Lin
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Probability number of vertices in large component of Erdös-Renyi graph is close to survival probability

I am currently take a course on Erdös-Renyi graphs where we have the probability that two vertices are connected is given by $p = \lambda/n$ where n is the number of vertices of the graph G. Then one observe different regimes. The one I am…
GG314
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Branching Process confusion

I'm having an hard time at understanding how to explicitly calculate the distribution law of a branching process. Let's call $(X_{i,j})_{i,j\,\geq\,1}$ a double set of indipendent random variables, all with the same discrete law $\nu$ such that…
Donson
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For a branching process $\left\{X_{n}\right\}$ with PGF $\varphi(s)=\frac{1-(b+c)}{1-c}+\frac{b s}{1-cs}$ find $ \lim \text{Pr}\{X_{n}=k|X_{n}>0\} $

Consider a discrete time branching process $\left\{X_{n}\right\}$ with probability generating function $$ \varphi(s)=\frac{1-(b+c)}{1-c}+\frac{b s}{1-c s}, \quad 01$. Assume $X_{0}=1$. Determine the conditional…
zaira
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Average Finite Component Sizes of Random Graph under Stochastic Dominance

Let $X$ and $Y$ be two $\mathbb{N}$-valued random variables. We say $Y$ stochastically dominates $X$ in the first order, written $X\le Y,$ if and only if $\mathbb{P}(Y>k) \ge \mathbb{P}(X>k)$ for all $k \in \mathbb{N}$. It is easy to show that if…
deej
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Proof involving pgfs of branching process

Let $\{X_n\}$ be a branching process, that is, $X_0=k\quad k\in \mathbb{N}$ $X_{n+1}=\sum_{r=1}^{X_n}\xi_r^{(n)}$ where $\xi_1^{(n)}, \xi_2^{(n)}, \cdots $ are non-negative i.i.d. random variables having distribution $P(\xi=i)=p_i$ Let $\phi_n(s)$…
Savitr
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Proving that a branching process survives with probability 1 at a fixed generation

I have the following problem about a sort of "reversed" time-inhomogeneous branching process that I'm trying to solve. Let $Z_0,Z_1,\ldots$ be a time-inhomogeneous branching process with $Z_0=1$. Every individual in my process is complete graph,…
matteo
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Branching Process and Random Walk

Consider this version of branching process: each individual has 2 offspring with probability p, and 0 offspring with probability 1-p. Let the 0th generation has 1 individual, then I managed to find the extinction probability q is \begin{equation} …
user1444522
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Intuitively understand this inequality

Let $b\in\mathbb{R}^d$ and consider a $d$-dimensional Markov branching process. Define the stopping time $$\tau_b(n)=\min\{t\ge0:b\cdot X(t)\ge n\}$$ for $n\ge0$, where the $\cdot$ denotes the inner product between the two vectors. I would like to…
Dada
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