Sequences

A sequence is an ordered list of numbers indexed by the positive integers. In biology a sequence arises naturally as the number of cases in each successive generation of transmission, or as a population census taken year by year. Sequences model iterative estimators, sampling schemes, and the limiting behavior that underlies convergence results in statistics.

Definition and notation

A sequence $\{a_n\}$ assigns a value $a_n$ to each index $n = 1, 2, 3, \dots$ . A finite sequence has a last term; an infinite one continues forever. A sample of size $N$ is often written $\{x_i\}_{i=1}^{N}$ .

Bounded and monotonic sequences

Bounded: there exist numbers $m, M$ with $m \le a_n \le M$ for all $n$ .
Increasing: $a_{n+1} \ge a_n$ for all $n$ ; decreasing: $a_{n+1} \le a_n$ .
Strictly monotonic: the inequality is strict, $a_{n+1} > a_n$ (or $<$ ), so no two terms are equal.

Strict monotonicity matters because a strictly increasing function is invertible — this is exactly why a continuous cumulative distribution function (CDF), which is strictly increasing on its support, has a well-defined inverse (the quantile function).

Worked example

Let $a_n = \sqrt{n}$ : the terms $1, 1.414, 1.732, 2, \dots$ are strictly increasing and unbounded above.

Let $b_n = 1/n$ : the terms $1, 0.5, 0.333, 0.25, \dots$ are strictly decreasing, bounded in $(0, 1]$ , and approach $0$ as $n \to \infty$ .

An outbreak’s case counts often form a sequence that increases, then, being bounded by the finite pool of susceptibles, levels off — the monotone, bounded pattern of an epidemic that grows and then saturates.

Computing it

R

n <- 1:10
a <- sqrt(n)     # increasing
b <- 1 / n       # decreasing to 0
plot(n, a, type = "b")
all(diff(a) > 0)   # TRUE (strictly increasing)
all(diff(b) < 0)   # TRUE (strictly decreasing)

Python

import numpy as np
import matplotlib.pyplot as plt

n = np.arange(1, 11)
a = np.sqrt(n)          # increasing
b = 1 / n               # decreasing to 0
plt.plot(n, a, marker="o")
print(np.all(np.diff(a) > 0))   # True
print(np.all(np.diff(b) < 0))   # True

True
True

Julia

using Plots

n = 1:10
a = sqrt.(n)            # increasing
b = 1 ./ n              # decreasing to 0
plot(n, a, marker = :circle)
all(diff(a) .> 0)       # true
all(diff(b) .< 0)       # true

Why it matters for statistics

Estimators computed over growing samples form sequences, and questions like “does $\bar{X}_n$ settle down?” are questions about sequence convergence. Strict monotonicity of CDFs guarantees quantiles are uniquely defined, which underpins simulation, inverse-transform sampling, and confidence intervals.