Editing Random walk (section)

===One-dimensional random walk===
An elementary example of a random walk is the random walk on the [[integer]] number line, <math>\Z</math>, which starts at 0 and at each step moves +1 or −1 with equal probability.

This walk can be illustrated as follows. A marker is placed at zero on the number line, and a fair coin is flipped. If it lands on heads, the marker is moved one unit to the right. If it lands on tails, the marker is moved one unit to the left. After five flips, the marker could now be on -5, -3, -1, 1, 3, 5. With five flips, three heads and two tails, in any order, it will land on 1. There are 10 ways of landing on 1 (by flipping three heads and two tails), 10 ways of landing on −1 (by flipping three tails and two heads), 5 ways of landing on 3 (by flipping four heads and one tail), 5 ways of landing on −3 (by flipping four tails and one head), 1 way of landing on 5 (by flipping five heads), and 1 way of landing on −5 (by flipping five tails). See the figure below for an illustration of the possible outcomes of 5 flips.

[[File:Flips.svg|thumb|800px|center|All possible random walk outcomes after 5 flips of a fair coin]]
[[File:random walk 2500.svg|right|thumb|280px|Random walk in two dimensions ([http://upload.wikimedia.org/wikipedia/commons/f/f3/Random_walk_2500_animated.svg animated version])]]
[[File:random walk 25000 not animated.svg|right|thumb|280px|Random walk in two dimensions with 25 thousand steps ([http://upload.wikimedia.org/wikipedia/commons/c/cb/Random_walk_25000.svg animated version])]]
[[File:Random walk 2000000.png|right|thumb|280px|Random walk in two dimensions with two million even smaller steps. This image was generated in such a way that points that are more frequently traversed are darker. In the limit, for very small steps, one obtains [[Brownian motion]].]]

To define this walk formally, take independent random variables <math>Z_1, Z_2,\dots</math>, where each variable is either 1 or −1, with a 50% probability for either value, and set <math>S_0 = 0</math> and <math display="inline">S_n = \sum_{j=1}^n Z_j.</math> The [[Series (mathematics)|series]] <math>\{S_n\}</math> is called the ''simple random walk on <math>\Z</math>''.  This series (the sum of the sequence of −1s and 1s) gives the net distance walked, if each part of the walk is of length one.
The [[expected value|expectation]] <math>E(S_n)</math> of <math>S_n</math> is zero. That is, the mean of all coin flips approaches zero as the number of flips increases. This follows by the finite additivity property of expectation:
<math display="block">E(S_n)=\sum_{j=1}^n E(Z_j)=0.</math>

A similar calculation, using the independence of the random variables and the fact that <math>E(Z_n^2)=1</math>, shows that:
<math display="block">E(S_n^2)=\sum_{i=1}^n E(Z_i^2)+2\sum_{1 \le i < j \le n}E(Z_i Z_j)=n.</math>

This hints that <math>E(|S_n|)\,\!</math>, the [[expected value|expected]] translation distance after ''n'' steps, should be [[Big O notation|of the order of]] {{nowrap|<math>\sqrt n</math>.}} In fact,<ref>{{cite web|url=http://mathworld.wolfram.com/RandomWalk1-Dimensional.html |title=Random Walk-1-Dimensional – from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2000-04-26 |access-date=2016-11-02}}</ref>
<math display="block">\lim_{n\to\infty} \frac{E(|S_n|)}{\sqrt n}= \sqrt{\frac {2}{\pi}}.</math>

To answer the question of how many times will a random walk cross a boundary line if permitted to continue walking forever, a simple random walk on <math>\mathbb Z</math> will cross every point an infinite number of times. This result has many names: the ''level-crossing phenomenon'', ''recurrence'' or the ''[[gambler's ruin]]''. The reason for the last name is as follows: a gambler with a finite amount of money will eventually lose when playing ''a fair game'' against a bank with an infinite amount of money. The gambler's money will perform a random walk, and it will reach zero at some point, and the game will be over.

If ''a'' and ''b'' are positive integers, then the expected number of steps until a one-dimensional simple random walk starting at 0 first hits ''b'' or −''a'' is ''ab''.  The probability that this walk will hit ''b'' before −''a'' is <math>a/(a+b)</math>, which can be derived from the fact that simple random walk is a [[martingale (probability theory)|martingale]]. And these expectations and hitting probabilities can be computed in <math> O(a+b) </math> in the general one-dimensional random walk Markov chain.
<!-- Maybe a reference to the iterated log law should come here? -->

Some of the results mentioned above can be derived from properties of [[Pascal's triangle]]. The number of different walks of ''n'' steps where each step is +1 or −1 is 2<sup>''n''</sup>. For the simple random walk, each of these walks is equally likely. 
In order for ''S<sub>n</sub>'' to be equal to a number ''k'' it is necessary and sufficient that the number of +1 in the walk exceeds those of −1 by ''k''. It follows +1 must appear (''n''&nbsp;+&nbsp;''k'')/2 times among ''n'' steps of a walk, hence the number of walks which satisfy <math>S_n=k</math> equals the number of ways of choosing (''n''&nbsp;+&nbsp;''k'')/2 elements from an ''n'' element set,<ref>Edward A. Codling et al., Random walk models in biology, Journal of the Royal Society Interface, 2008</ref> denoted <math display="inline">n \choose (n+k)/2</math>. For this to have meaning, it is necessary that ''n''&nbsp;+&nbsp;''k'' be an even number, which implies ''n'' and ''k'' are either both even or both odd. Therefore, the probability that <math>S_n=k</math> is equal to <math display="inline">2^{-n}{n\choose (n+k)/2}</math>. By representing entries of Pascal's triangle in terms of [[factorial]]s and using [[Stirling formula|Stirling's formula]], one can obtain good estimates for these probabilities for large values of <math>n</math>.

If space is confined to <math>\mathbb Z</math>+ for brevity, the number of ways in which a random walk will land on any given number having five flips can be shown as {0,5,0,4,0,1}.

This relation with Pascal's triangle is demonstrated for small values of ''n''. At zero turns, the only possibility will be to remain at zero. However, at one turn, there is one chance of landing on −1 or one chance of landing on 1. At two turns, a marker at 1 could move to 2 or back to zero. A marker at −1, could move to −2 or back to zero. Therefore, there is one chance of landing on −2, two chances of landing on zero, and one chance of landing on 2.

<!--[[File:PascalTriangleRandomWalk.JPG|thumb|center|600px|Pascal's triangle in a random walk]] Commenting out previous table from pic-->
{| class="wikitable" style="text-align:center"
|-
! k
! style="width:2em" | −5
! style="width:2em" | −4
! style="width:2em" | −3
! style="width:2em" | −2
! style="width:2em" | −1
! style="width:2em" | 0
! style="width:2em" | 1
! style="width:2em" | 2
! style="width:2em" | 3
! style="width:2em" | 4
! style="width:2em" | 5
|-
| <math>P[S_0=k]</math>
|
|
|
|
|
| 1
|
|
|
|
|
|-
| <math>2P[S_1=k]</math>
|
|
|
|
| 1
|
| 1
|
|
|
|
|-
| <math>2^2P[S_2=k]</math>
|
|
|
| 1
|
| 2
|
| 1
|
|
|
|-
| <math>2^3P[S_3=k]</math>
|
|
| 1
|
| 3
|
| 3
|
| 1
|
|
|-
| <math>2^4P[S_4=k]</math>
|
| 1
|
| 4
|
| 6
|
| 4
|
| 1
|
|-
| <math>2^5P[S_5=k]</math>
| 1
|
| 5
|
| 10
|
| 10
|
| 5
|
| 1
|}

The [[central limit theorem]] and the [[law of the iterated logarithm]] describe important aspects of the behavior of simple random walks on <math>\mathbb Z</math>. In particular, the former entails that as ''n'' increases, the probabilities (proportional to the numbers in each row) approach a [[normal distribution]].

To be precise, knowing that <math display="inline"> \mathbb{P}(X_n= k )= 2^{-n}\binom{n}{(n+k)/2} </math>,
and using [[Stirling formula|Stirling's formula]] one has

<math display="block">{\log \mathbb{P}(X_n= k )} = n\left[\left({1+\frac{k}{n}+\frac{1}{2n}}\right)\log\left(1+\frac{k}{n}\right)+\left({1-\frac{k}{n}+\frac{1}{2n}}\right)\log\left(1-\frac{k}{n}\right)\right] +\log {\frac{\sqrt{2}}{\sqrt{\pi}}} +o(1).</math>

Fixing the scaling <math display="inline">k=\lfloor \sqrt{n}x\rfloor</math>,  for <math display="inline">x</math> fixed, and using the expansion <math display="inline"> \log(1+{k}/{n})=k/n-k^2/2n^2+ \dots </math> when <math display="inline">k/n</math> vanishes, it follows

<math display="block">{\mathbb{P}\left(\frac{X_n}{n}= \frac{\lfloor \sqrt{n}x\rfloor}{\sqrt{n}} \right)} = \frac{1}{\sqrt{n}} \frac{1}{2\sqrt{\pi}}e^{-{x^2}}(1+o(1)). </math>

taking the limit (and observing that <math display="inline">{1}/{\sqrt{n}}</math> corresponds to the spacing of the scaling grid) one finds the gaussian density <math display="inline"> f(x) = \frac{1}{2\sqrt{\pi}}e^{-{x^2}} </math>. Indeed, for a absolutely continuous random variable <math display="inline">X</math> with density <math display="inline">f_X</math> it holds <math display="inline">\mathbb{P}\left(X \in [x,x+dx)\right)=f_X(x)dx</math>, with <math display="inline">dx</math> corresponding to an infinitesimal spacing.

As a direct generalization, one can consider random walks on crystal lattices (infinite-fold abelian covering graphs over finite graphs). Actually it is possible to establish the central limit theorem and large deviation theorem in this setting.<ref>{{cite book |author=Kotani, M. |author2=[[Toshikazu Sunada|Sunada, T.]] |year= 2003 |title= Spectral geometry of crystal lattices  |volume= 338 |pages= 271–305 |doi=10.1090/conm/338/06077|series= Contemporary Mathematics |isbn= 978-0-8218-3383-4 |doi-access= free }}</ref><ref>{{cite journal |author=Kotani, M. |author2=[[Toshikazu Sunada|Sunada, T.]] |year= 2006 |title= Large deviation and the tangent cone at infinity of a crystal lattice |journal= Math. Z. |volume= 254 |issue= 4 |pages= 837–870 |doi=10.1007/s00209-006-0951-9|s2cid= 122531716 }}</ref>

====As a Markov chain====
A one-dimensional ''random walk'' can also be looked at as a [[Markov chain]] whose state space is given by the integers <math>i=0,\pm 1,\pm 2,\dots .</math> For some number ''p'' satisfying <math>\,0 < p < 1</math>, the transition probabilities (the probability ''P<sub>i,j</sub>'' of moving from state ''i'' to state ''j'') are given by
<math display="block">\,P_{i,i+1}=p=1-P_{i,i-1}.</math>

==== Heterogeneous generalization ====
{{Main|Heterogeneous random walk in one dimension}}
The heterogeneous random walk draws in each time step a random number that determines the local jumping probabilities and then a random number that determines the actual jump direction. The main question is the probability of staying in each of the various sites after <math>t</math> jumps, and in the limit of this probability when <math>t</math> is very large.