Lecture

Martingales and Brownian Movement

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In course

MATH-330: Martingales and Brownian motion

Introduction to the theory of discrete-time martingales (optional stopping and convergence theorems) and its applications (gambler's ruin, branching processes, Pólya urns...). Introduction to Brownian motion. ...

Description

This lecture covers the concepts of martingales and Brownian movement, including demonstrations and reminders related to martingales, mean numbers, and hypotheses about the generation of children. The instructor explains the independence hypothesis and the function of the number of children sharing a common ancestor.

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Related concepts (16)

Generation

A generation refers to all of the people born and living at about the same time, regarded collectively. It can also be described as, "the average period, generally considered to be about 20–⁠30 years, during which children are born and grow up, become adults, and begin to have children." In kinship terminology, it is a structural term designating the parent-child relationship. It is known as biogenesis, reproduction, or procreation in the biological sciences.

Mean

There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the arithmetic mean, also known as "arithmetic average", is a measure of central tendency of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead bar, .

Geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers a1, a2, ..., an, the geometric mean is defined as or, equivalently, as the arithmetic mean in logscale: Most commonly the numbers are restricted to being non-negative, to avoid complications related to negative numbers not having real roots, and frequently they are restricted to being positive, to enable the use of logarithms.

Greatest Generation

The Greatest Generation, also known as the G.I. Generation and the World War II generation, is the Western demographic cohort following the Lost Generation and preceding the Silent Generation. The generation is generally defined as people born from 1901 to 1927. They were shaped by the Great Depression and were the primary generation composing the enlisted forces in World War II. Most people of the Greatest Generation are the parents of the Silent Generation and Baby Boomers, and, in turn, were the children of the Lost Generation.

Generalized mean

In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). If p is a non-zero real number, and are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is (See p-norm).

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