Tail Bounds for Heavy-Tailed Sums

A comparison of tail-bounding methods for sums of i.i.d. random variables, with a focus on the Lomax (Pareto Type II) distribution. Implements a Riesz–Hölder / Chebyshev–Markov lower bound alongside classical competitors like Paley–Zygmund. We compare with upper bounds (Bonferroni, Chernoff, Bernstein, Bennett) and numerical approximatioms of the truth (CDF).

Structure

.
├── main.py               # entry point / experiment runner
├── config.py             # global parameters
├── requirements.txt
└── utils/
    ├── distributions.py  # tail / CDF functions
    ├── simulation.py     # Monte Carlo samplers
    ├── bounds.py         # upper and lower bounds
    ├── rh_cm_lb.py       # RH-CM target bound
    ├── scaling.py        # t_n = c, c*sqrt(n), c*n regimes
    └── plotting.py       # matplotlib figures
└── examples/
    ├── demo.py  # demonstration of the package

Quickstart

pip install -r requirements.txt
python main.py

The default run uses a Lomax(alpha=0.9) distribution with n=3 summands and a linear scaling regime. Four diagnostic plots are produced: raw tail probabilities, approximation/truth ratio, log-ratio, and log-probability.

Configuration

Edit config.py or pass a modified dict directly to run_experiment:

from config import get_config
from main import run_experiment

cfg = get_config()
cfg["n"]     = 10
cfg["alpha"] = 1.5
cfg["dist"]  = "lomax"
run_experiment(cfg)

Key parameters:

Parameter	Description
n	number of summands
alpha	Lomax tail index
lambda	Lomax scale
q	Hölder exponent (negative → conjugate pair)
M	MC sample size
scaling	"constant", "sqrt", or "linear"
dist	"lomax" or "gaussian"