I am reading the paper “Critical Threshols and the Limit Distribution in the Bak Sneppen Model” and I have difficulty proving the following statement:
The probability that at time $n$ we are in a block (b-avalanche) of range $k$ converges, as $n \rightarrow \infty$, to $$\frac{D_{N}(b|k)P_{N}(b,k)}{\sum_{l=3}^{N}D_{N}(b|l)P_{N}(b,l)} = \frac{D_{N}(b|k)P_{N}(b,k)}{D_{N}(b)}$$
where
$b$ is a threshold (that is fixed) and range is a block property
$P_{N}(b,k)$ is the probability that a block has range $k$
$D_{N}(b|k)$ denote the mean length of a block, given a block has range $k$
$D_{N}(b)$ denote the mean length of a block
My doubt is how he came in LHS. I thought so far:
Let $A = \{$I'm in block $x$ at time $n\}$ and $C = \{$range of block $x\}$
$P(\{A=T\} \cap \{C=k\}) = P(A=T | C=k)P(C=k) = P(A=T | C=k)P_{N}(b,k) = P_{N}(b,k)\sum_{x=1}^{T}P(A=x|C=k)$
If anyone can help me I would be very thankful.
Best regards