Sequences (Statistics)

A series of terms of a sequence are often denoted with upper case letter with a numberical subscript like so;

$$X_1, X_2, ... , X_n$$

It should be noted that this is distinct from a series of random variables, which is also notated like $$X_1, X_2, ... , X_n$$

because typically the assumption of independently distributed variates, the order does not matter, however for a sequence the order is of most importance.

= Specifying sequences =

closed form
when the sequence can be described in terms of some number n indicating the element in the sequence like so;

$s_n = n^2$  {n=1,2,...n}

then the sequence is said to be in closed form, a closed form is not always available for a sequence.

$a_n = (-1)^{n+1}$ is a useful trick to cause a sequence to alternative in sign.

recurrence system form
$a_{n+1} = a_n + 7 (n=1,2,...)$

first-order recurrence relation because only the n-1 previous term is involved.

$x_1 = 2, x_{n+1} = x_n^2 + 7 (n=1,2,...)$

is called a recurrence system

$x_1 = 2, x_{n} = x_{n-1}^2 + 7 (n=2,2,...)$

you can vary the starting index, or the relationship between the successive terms in the recurrence relation expression

= Sequence Types =

sequences can be composed of combinations of terms in additive or multiplicative structure, there are some basic types that you should recognize

Arithmetic
recurrence system for arithmetic sequence is formed like so;

$x_1 = a$,     $x_{n+1} = x_n + b $,  for      $(n=1,2,3....,k)$

the closed form of the same sequence would be

$b_n = a + (n-1)b$, $(n=1,2,3....,k)$

Constant sequence
sometimes nothing goes on, and in sequences like that are called constant sequences;

${1,1,1,1,1,1}$

$x_n = a$, $(n=1,2,3....)$

Geometric sequence
$x_1 = a$, $x_{n+1} = rx_n $, $(n=1,2,3....)$

has a closed form

$x_n = ar^{n-1} $,      $(n=1,2,3...)$

linear recurrence sequence
$x_1 = a $, $x_{n+1}=rx+n+d$, (n=1,2,3...)

second order recurrence systems
The fibonacci recurrence sequence is an example of a second order recurrence sequence

$u_{n+1} = pu_{n+1} + qu_{n}$

inhomogeneous recurrence sequence
this type of sequence includes terms other than terms of the sequence for example;

$u_{n+1} = u_{n+1} + u_{n} + 2n$

= Convergence =

Convergence to zero
for an example sequence like $$x_n = 100/n$$, for n:1,2,...n, the limit of $$x_n$$ as n tends to $$\infty$$ is equal to zero.

Convergence in probability
if the variance quantity has n in the denominator, then its variance decreases with sample size, n, hence the variances of Y tends to zero for large values of $$Y_n$$ sequences of random variables X_n are said to converge in probability if the probabilty that X_n differs from X by more than some positive value $$\varepsilon$$ tends towards zero as n increases.

Convergence of mean square
a sequence is said to converge in mean square to a random variable X if the expection of $$(X_n - X)^2$$ tends to zero as n increases

Convergence in distribution
suppose X_n

"convergence in distribution was introduced: as it turns out, this is weaker than the other two, and in fact convergence in probability implies convergence in distribution."

relationships in convergence
$$\{X_n \xrightarrow{ms} X\} \Rightarrow \{X_n \xrightarrow{p} X\} \Rightarrow \{X_n \xrightarrow{D} X\}$$