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I have a time-inhomogeneous Galton-Watson binary branching process over a finite number of generations $n$. In each generation $i$, there is a probability $p_i$ of a child surviving; so each node has 2 children with probability $p_i^2$, 1 child with probability $2 p_i (1-p_i)$, and zero children with probability $(1-p_i)^2$.

I want to show that the process survives to time $T$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this?

The process is inhomogeneous. The number of expected survivors at level $i$ is $\mu_i = 2^i p_1 \dots p_i$, which I can compute. Is there some criterion of the form $\forall i, \mu_i \geq \text{poly}(n)$ that suffices to show survival?

I am sure that this is well-known, but I have not been able to find any simple description.

Thanks!

I have a time-inhomogeneous Galton-Watson binary branching process over a finite number of generations $n$. In each generation $i$, there is a probability $p_i$ of a child surviving; so each node has 2 children with probability $p_i^2$, 1 child with probability $2 p_i (1-p_i)$, and zero children with probability $(1-p_i)^2$. Furthermore $p_i$ is a decreasing function of $i$. So the process starts out as a super-critical branching and ends as a sub-critical branching.

I want to show that the process survives to time $n$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this?

The process is inhomogeneous. The number of expected survivors at level $i$ is $\mu_i = 2^i p_1 \dots p_i$, which is a unimodular function of $i$. It seems that a sufficient criterion should be $\mu_n \geq 1$ or maybe $\mu_n = \Omega(\text{poly}(n))$.

Thanks!

I have a time-inhomogeneous Galton-Watson branching process over a finite number of generations $n$. Each node may have 0, 1, or 2 offspring. I want to show that the process survives to time $T$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this?

The process is inhomogeneous, but I can compute the expected number of survivors $\mu_i$ at each level. Is there some criterion of the form $\forall i, \mu_i \geq \text{poly}(n)$ that suffices to show survival?

I am sure that this is well-known, but I have not been able to find any simple description.

Thanks!

I have a time-inhomogeneous Galton-Watson binary branching process over a finite number of generations $n$. In each generation $i$, there is a probability $p_i$ of a child surviving; so each node has 2 children with probability $p_i^2$, 1 child with probability $2 p_i (1-p_i)$, and zero children with probability $(1-p_i)^2$. 

I want to show that the process survives to time $T$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this?

The process is inhomogeneous. The number of expected survivors at level $i$ is $\mu_i = 2^i p_1 \dots p_i$, which I can compute. Is there some criterion of the form $\forall i, \mu_i \geq \text{poly}(n)$ that suffices to show survival?

I am sure that this is well-known, but I have not been able to find any simple description.

Thanks!

I have a time-inhomogeneous Galton-Watson branching process over a finite number of generations $n$. I want to show that the process survives to time $T$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this?

The process is inhomogeneous, but I can compute the expected number of survivors $\mu_i$ at each level. Is there some criterion of the form $\forall i, \mu_i \geq \text{poly}(n)$ that suffices to show survival?

I am sure that this is well-known, but I have not been able to find any simple description.

Thanks!

I have a time-inhomogeneous Galton-Watson branching process over a finite number of generations $n$. Each node may have 0, 1, or 2 offspring. I want to show that the process survives to time $T$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this?

The process is inhomogeneous, but I can compute the expected number of survivors $\mu_i$ at each level. Is there some criterion of the form $\forall i, \mu_i \geq \text{poly}(n)$ that suffices to show survival?

I am sure that this is well-known, but I have not been able to find any simple description.

Thanks!