Critical Galton--Watson processes: The maximum of total progenies within a large window
- Fleischmann, Klaus
- Vatutin, Vladimir A.
- Wachtel, Vitali
2010 Mathematics Subject Classification
- 60J80 60F17
- Branching of index one plus alpha, limit theorem, conditional invariance principle, tail asymptotics, moving window, maximal total progeny, lower deviation probabilities
Consider a critical Galton-Watson process Z=Z_n: n=0,1,... of index 1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in the window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m satisfies that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_infinity(j).