Stochastic convergence

Stochastic convergence is a mathematical concept intended to formalize the idea that a sequence of essentially random or unpredictable events sometimes is expected to settle into a pattern.

The pattern may for instance be

• convergence in the classical sense to a fixed value, perhaps itself coming from a random event.
• an increasing similarity of outcomes to what a purely deterministic function would produce
• an increasing preference towards a certain outcome
• an increasing "aversion" against straying far away from a certain outcome
• an increasing adherence to one particular probability distribution
• the series formed by calculating the expected value of the outcome's distance from a particular value may converge to 0
• the variability of the results may grow smaller and smaller, i.e. the variance converging to 0

Four different varieties of stochastic convergence are noted:

• Almost sure convergence
• Convergence in probability
• Convergence in distribution
• Convergence in rth order mean

Almost sure convergence

Example 1

Consider an animal of some short-lived species. We note the exact amount of food that this animal consumes day by day. This sequence of numbers will be unpredictable in advance, but we may be quite certain that one day the number will become zero, and will stay zero forever after.

Example 2

Consider a man who starts tomorrow to toss seven coins once every morning. Each afternoon, he donates a random amount of money to a certain charity. The first time the result is all tails, however, he will stop permanently.

Let $X_{1},X_{2},...$ be the day by day amounts the charity receives from him.

We may be almost sure that one day this amount will be zero, and stay zero forever after that.

However, when we consider any finite number of days, there is a nonzero probability the terminating condition will not occur.

Example 3

A business owner has two sources of income: His business, and interest from a large bank deposit with fixed interest and no withdrawal or deposits.

The business income varies unpredictably from month to month, while income from interest is predictable and given by a simple function f.

The income for month i can thus be modeled by a stochastic variable $U_{i}=X_{i}+f(i)$, where $X_{i}$ is the income from the business.

Now assume $X_{i}$ converges almost surely to 0 (history bears out that all businesses sooner or later fold up).

Then the total monthly income $U_{i}$ has almost sure convergence to the function f(i).

Formal definition

Let $\scriptstyle X_{0},X_{1},...$ be an infinite sequence of stochastic variables defined over a subset of R.

Then the actual outcomes will be an ordinary sequence of real numbers.

If the probability that this sequence will converge to a given real number a equals 1, then we say the original sequence of stochastic variables has almost sure convergence to a.

In more compact notation:

If $P(\lim _{i\to \infty }X_{i}=a)=1$ for some a, then the sequence has almost sure convergence to a.

Note that we may replace the real number a above by a real-valued function $f(i)$ of i, and obtain almost sure convergence to a function rather than a fixed number.

Convergence in probability

Example 1

An absent-minded professor gets a job in an unfamiliar part of town.

The first time he walks to work, he has difficulty finding his way, and ends up several hours late.

The next few dozen times, things generally improve, although sometimes he manages to get hopelessly lost again.

As the months and years go by, he gets to know the area very well, and falls into a routine that make him more and more punctual, although it may still happen occasionally that he is very late.

Example 2

We may keep tossing a die an infinite number of times and at every toss note the average outcome so far. The exact number thus obtained after each toss will be unpredictable, but for a fair die, it will tend to get closer and closer to the arithmetic average of 1,2,3,4,5 and 6, i.e. 3.5.

Formal definition

Let $\scriptstyle X_{0},X_{1},...$ be an infinite sequence of stochastic variables defined over a subset of R.

If there exists a real number a such that $\lim _{i\to \infty }P(|X_{i}-a|>\varepsilon )=0$ for all $\varepsilon >0$, then the sequence has convergence in probability to a.

Convergence in distribution

Example 1

The outcome from tossing a non-biased dice follows the uniform discrete distribution.

Assume a new dice factory has just been built.

The first few dices come out quite biased, due to imperfections in the production process. The outcome from tossing any of them will follow a distribution markedly different from the desired uniform discrete distribution.

As the factory is improved, the dices will be less and less loaded, and the outcomes from tossing a newly produced dice will follow the desired distribution more and more closely.

Example 2

Let $\scriptstyle X_{n}$ be the result of flipping n unbiased coins, and noting the fraction of heads.

$\scriptstyle X_{1}$ will then follow the uniform discrete probability distribution with expected value $\mu =0.5$ and variance $\sigma ^{2}=0.25$, but as n grows larger, $\scriptstyle X_{n}$ will follow a distribution that gradually takes on more and more similarity to the gaussian distribution .

Forming the stochastic sequence $\scriptstyle Z_{n}={\frac {(X_{n}-\mu )}{\frac {\sigma }{\sqrt {n}}}}$, we find the variables $Z_{n}$ becoming distributed more and more like the standard normal distribution as n increases.

We then say the sequence $\scriptstyle Z_{n}$ converges in distribution to the standard normal distribution.

(This convergence follows from the famous central limit theorem).

Convergence in rth order mean

Formal definition

If $\scriptstyle \lim _{n\to \infty }E(|X_{n}-a|^{r})=0$ for some real number a, then {$X_{n}$} converges in rth order mean to a.

Relations between the different modes of convergence

• If a stochastic sequence has almost sure convergence, then it also has convergence in probability.
• If a stochastic sequence has convergence in probability, then it also has convergence in distribution.
• If a stochastic sequence has convergence in (r+1)th order mean, then it also has convergence in rth order mean (r>0).
• If a stochastic sequence has convergence in rth order mean, then it also has convergence in probability.

See also

• Stochastic differential equations

Related topics

• Probability
• Probability theory
• Stochastic variable
• Differential equations
• Stochastic modeling