Approximate the expectation value of maximum of correlated random variables

Question

I am trying to find the expectation value of the maximal of some correlated random variables generated by Poisson point process (PPP).

Assume homogeneous PPP are on the 2-D plane with density $\lambda$. There are three overlapping hexagons demonstrated in the figure.

What is the expectation value of maximal number of points in one of the three hexagons?

Mathematically, the problem is described as follows:

Denote $X_i$ as the number of the points in the area $A_i$. Then, $Y_1=X_1+X_4+X_5+X_7$, $Y_2=X_2+X_4+X_6+X_7$ and $Y_3=X_3+X_5+X_6+X_7$ are the number of the points in the three hexagons, respectively. Calculate or approximate $E[\max(Y_1,Y_2,Y_3)]$.

Answer 1

Not an answer, but here is a python script to help estimate it:

I got approx. $6\frac{2}{3}\lambda A_7$ (assuming all the small triangles have the same area, and the larger ones are just three times the area.

from typing import List
from scipy.stats import poisson

# ppp definition
_lambda = 7
AREAS = [3, 3, 3, 1, 1, 1, 1]
ppp = [poisson(a*_lambda) for a in AREAS]

# map partition
HEXAGONS = {1: [0, 3, 4, 6], 2: [1, 3, 5, 6], 3: [2, 4, 5, 6]}

def ppp_realization() -> List[int]:
    """
    Return a realization of the poisson point process for each partition.
    """
    return [i.rvs(1)[0] for i in ppp]

def hex_count(hex_id: int, partition_vars:List[int]) -> int:
    """
    Calculate the counts inside the hexagon with hex_id given a realization of the 
    random variables associated with the partitioned area.
    """
    return sum(partition_vars[i] for i in HEXAGONS[hex_id])

def run_exp(n_runs:int = 1000) -> float:
    maxima_sum = 0
    for r in range(n_runs):
        area_counts = ppp_realization()
        hex_counts = [hex_count(i,area_counts) for i in HEXAGONS]
        maxima_sum += max(hex_counts)
    return maxima_sum/n_runs