Random walk

Example of eight random walks in one dimension starting at 0. The plot shows the current position on the line (vertical axis) versus the time steps (horizontal axis).

A random walk is a mathematical object which describes a path that consists of a succession of random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, superstring behavior, the price of a fluctuating stock and the financial status of a gambler can all be approximated by random walk models, even though they may not be truly random in reality. As illustrated by those examples, random walks have applications to many scientific fields including ecology, psychology, computer science, physics, chemistry, and biology, and also to economics. Random walks explain the observed behaviors of many processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. As a more mathematical application, the value of pi can be approximated by the usage of random walk in agent-based modelling environment.^[1]^[2] The term random walk was first introduced by Karl Pearson in 1905.^[3]

Various types of random walks are of interest, which can differ in several ways. The term itself most often refers to a special category of Markov chains or Markov processes, but many time-dependent processes are referred to as random walks, with a modifier indicating their specific properties. Random walks (Markov or not) can also take place on a variety of spaces: commonly studied ones include graphs, others on the integers or the real line, in the plane or in higher-dimensional vector spaces, on curved surfaces or higher-dimensional Riemannian manifolds, and also on groups finite, finitely generated or Lie. The time parameter can also be manipulated. In the simplest context the walk is in discrete time, that is a sequence of random variables (X
t) = (X
1, X
2,...) indexed by the natural numbers. However, it is also possible to define random walks which take their steps at random times, and in that case the position X
t has to be defined for all times t ∈ [0,+∞]. Specific cases or limits of random walks include the Lévy flight and diffusion models such as Brownian motion.

Random walks are a fundamental topic in discussions of Markov processes. Their mathematical study has been extensive. Several properties, including dispersal distributions, first-passage or hitting times, encounter rates, recurrence or transience, have been introduced to quantify their behaviour.

Lattice random walk

A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a simple random walk, the location can only jump to neighboring sites of the lattice, forming a lattice path. In simple symmetric random walk on a locally finite lattice, the probabilities of the location jumping to each one of its immediate neighbours are the same. The best studied example is of random walk on the d-dimensional integer lattice (sometimes called the hypercubic lattice) $\mathbb {Z} ^{d}$ .^[4]

If the state space is limited to finite dimensions, the random walk model is called bordered symmetric random walk and the transition probabilities depend on the location of the state, because on margin and corner states the movement is limited.^[5]

One-dimensional random walk

An elementary example of a random walk is the random walk on the integer number line, $\mathbb {Z}$ , 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 1, −1, 3, −3, 5, or −5. With five flips, three heads and two tails, in any order, 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.

All possible random walk outcomes after 5 flips of a fair coin

Random walk in two dimensions (animated version)

Random walk in two dimensions with 25 thousand steps (animated version)

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 $Z_{1},Z_{2},\dots$ , where each variable is either 1 or −1, with a 50% probability for either value, and set $S_{0}=0\,\!$ and $S_{n}=\sum _{j=1}^{n}Z_{j}.$ The series $\{S_{n}\}\,\!$ is called the simple random walk on $\mathbb {Z}$ . This series (the sum of the sequence of −1s and 1s) gives the distance walked, if each part of the walk is of length one. The expectation $E(S_{n})\,\!$ of $S_{n}\,\!$ 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:

E(S_{n})=\sum _{j=1}^{n}E(Z_{j})=0.

A similar calculation, using the independence of the random variables and the fact that $E(Z_{n}^{2})=1$ , shows that:

E(S_{n}^{2})=\sum _{i=1}^{n}\sum _{j=1}^{n}E(Z_{j}Z_{i})=n.

This hints that $E(|S_{n}|)\,\!$ , the expected translation distance after n steps, should be of the order of ${\sqrt {n}}$ . In fact,^[6]

\lim _{n\to \infty }{\frac {E(|S_{n}|)}{\sqrt {n}}}={\sqrt {\frac {2}{\pi }}}.

This result shows that diffusion is ineffective for mixing because of the way the square root behaves for large $N$ .

How many times will a random walk cross a boundary line if permitted to continue walking forever? A simple random walk on $\mathbb {Z}$ 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 $a/(a+b)$ , which can be derived from the fact that simple random walk is a martingale.

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 2ⁿ. For the simple random walk, each of these walks are equally likely. In order for S_n 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. The number of walks which satisfy $S_{n}=k$ is equally the number of ways of choosing (n - k)/2 with n is the number of allowed moves,^[7] denoted ${n \choose (n-k)/2}$ . For this to have meaning, it is necessary that n and k be even numbers. Therefore, the probability that $S_{n}=k$ is equal to $2^{-n}{n \choose (n-k)/2}$ . By representing entries of Pascal's triangle in terms of factorials and using Stirling's formula, one can obtain good estimates for these probabilities for large values of $n$ .

If the space is confined to $\mathbb {Z}$ + 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.

k	−5	−4	−3	−2	−1	0	1	2	3	4	5
$P[S_{0}=k]$						1
$2P[S_{1}=k]$					1		1
$2^{2}P[S_{2}=k]$				1		2		1
$2^{3}P[S_{3}=k]$			1		3		3		1
$2^{4}P[S_{4}=k]$		1		4		6		4		1
$2^{5}P[S_{5}=k]$	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 $\mathbb {Z}$ . In particular, the former entails that as n increases, the probabilities (proportional to the numbers in each row) approach a normal distribution.

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.^[8]^[9]

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 $i=0,\pm 1,\pm 2,\dots .$ For some number p satisfying $\,0<p<1$ , the transition probabilities (the probability P_i,j of moving from state i to state j) are given by

\,P_{i,i+1}=p=1-P_{i,i-1}.

Higher dimensions

Three random walks in three dimensions

In higher dimensions, the set of randomly walked points has interesting geometric properties. In fact, one gets a discrete fractal, that is, a set which exhibits stochastic self-similarity on large scales. On small scales, one can observe "jaggedness" resulting from the grid on which the walk is performed. Two books of Lawler referenced below are a good source on this topic. The trajectory of a random walk is the collection of points visited, considered as a set with disregard to when the walk arrived at the point. In one dimension, the trajectory is simply all points between the minimum height and the maximum height the walk achieved (both are, on average, on the order of √n).

To visualize the two dimensional case, one can imagine a person walking randomly around a city. The city is effectively infinite and arranged in a square grid of sidewalks. At every intersection, the person randomly chooses one of the four possible routes (including the one originally traveled from). Formally, this is a random walk on the set of all points in the plane with integer coordinates.

Will the person ever get back to the original starting point of the walk? This is the 2-dimensional equivalent of the level crossing problem discussed above. It turns out that the person almost surely will in a 2-dimensional random walk, but for 3 dimensions or higher, the probability of returning to the origin decreases as the number of dimensions increases. In 3 dimensions, the probability decreases to roughly 34%.^[10]

The asymptotic function for a two dimensional random walk as the number of steps increases is given by a Rayleigh distribution. The probability distribution is a function of the radius from the origin and the step length is constant for each step.

P(r)={\frac {2r}{N}}e^{-r^{2}/N}

Relation to Wiener process

Simulated steps approximating a Wiener process in two dimensions

A Wiener process is a stochastic process with similar behaviour to Brownian motion, the physical phenomenon of a minute particle diffusing in a fluid. (Sometimes the Wiener process is called "Brownian motion", although this is strictly speaking a confusion of a model with the phenomenon being modeled.)

A Wiener process is the scaling limit of random walk in dimension 1. This means that if you take a random walk with very small steps you get an approximation to a Wiener process (and, less accurately, to Brownian motion). To be more precise, if the step size is ε, one needs to take a walk of length L/ε² to approximate a Wiener length of L. As the step size tends to 0 (and the number of steps increases proportionally) random walk converges to a Wiener process in an appropriate sense. Formally, if B is the space of all paths of length L with the maximum topology, and if M is the space of measure over B with the norm topology, then the convergence is in the space M. Similarly, a Wiener process in several dimensions is the scaling limit of random walk in the same number of dimensions.

A random walk is a discrete fractal (a function with integer dimensions; 1, 2, ...), but a Wiener process trajectory is a true fractal, and there is a connection between the two. For example, take a random walk until it hits a circle of radius r times the step length. The average number of steps it performs is r². This fact is the discrete version of the fact that a Wiener process walk is a fractal of Hausdorff dimension 2.

In two dimensions, the average number of points the same random walk has on the boundary of its trajectory is r^4/3. This corresponds to the fact that the boundary of the trajectory of a Wiener process is a fractal of dimension 4/3, a fact predicted by Mandelbrot using simulations but proved only in 2000 by Lawler, Schramm and Werner.^[11]

A Wiener process enjoys many symmetries random walk does not. For example, a Wiener process walk is invariant to rotations, but random walk is not, since the underlying grid is not (random walk is invariant to rotations by 90 degrees, but Wiener processes are invariant to rotations by, for example, 17 degrees too). This means that in many cases, problems on random walk are easier to solve by translating them to a Wiener process, solving the problem there, and then translating back. On the other hand, some problems are easier to solve with random walks due to its discrete nature.

Random walk and Wiener process can be coupled, namely manifested on the same probability space in a dependent way that forces them to be quite close. The simplest such coupling is the Skorokhod embedding, but there exist more precise couplings, such as Komlós–Major–Tusnády approximation theorem.

The convergence of a random walk toward the Wiener process is controlled by the central limit theorem, and by Donsker's theorem. For a particle in a known fixed position at t = 0, the central limit theorem tells us that after a large number of independent steps in the random walk, the walker's position is distributed according to a normal distribution of total variance:

\sigma ^{2}={\frac {t}{\delta t}}\,\varepsilon ^{2},

where t is the time elapsed since the start of the random walk, $\varepsilon$ is the size of a step of the random walk, and $\delta t$ is the time elapsed between two successive steps.

This corresponds to the Green function of the diffusion equation that controls the Wiener process, which suggests that, after a large number of steps, the random walk converges toward a Wiener process.

In 3D, the variance corresponding to the Green's function of the diffusion equation is:

\sigma ^{2}=6\,D\,t

By equalizing this quantity with the variance associated to the position of the random walker, one obtains the equivalent diffusion coefficient to be considered for the asymptotic Wiener process toward which the random walk converges after a large number of steps:

D={\frac {\varepsilon ^{2}}{6\delta t}}

(valid only in 3D)

Remark: the two expressions of the variance above correspond to the distribution associated to the vector ${\vec {R}}$ that links the two ends of the random walk, in 3D. The variance associated to each component $R_{x}$ , $R_{y}$ or $R_{z}$ is only one third of this value (still in 3D).

For 2D:^[12]

D={\frac {\varepsilon ^{2}}{4\delta t}}

For 1D:^[13]

D={\frac {\varepsilon ^{2}}{2\delta t}}

Gaussian random walk

A random walk having a step size that varies according to a normal distribution is used as a model for real-world time series data such as financial markets. The Black–Scholes formula for modeling option prices, for example, uses a Gaussian random walk as an underlying assumption.

Here, the step size is the inverse cumulative normal distribution $\Phi ^{-1}(z,\mu ,\sigma )$ where 0 ≤ z ≤ 1 is a uniformly distributed random number, and μ and σ are the mean and standard deviations of the normal distribution, respectively.

If μ is nonzero, the random walk will vary about a linear trend. If v_s is the starting value of the random walk, the expected value after n steps will be v_s + nμ.

For the special case where μ is equal to zero, after n steps, the translation distance's probability distribution is given by N(0, nσ²), where N() is the notation for the normal distribution, n is the number of steps, and σ is from the inverse cumulative normal distribution as given above.

Proof: The Gaussian random walk can be thought of as the sum of a series of independent and identically distributed random variables, X_i from the inverse cumulative normal distribution with mean equal zero and σ of the original inverse cumulative normal distribution:

Z =

\sum _{i=0}^{n}{X_{i}}

but we have the distribution for the sum of two independent normally distributed random variables, Z = X + Y, is given by

N(μ_X + μ_Y, σ²_X + σ²_Y) (see here).

In our case, μ_X = μ_Y = 0 and σ²_X = σ²_Y = σ² yield

N(0, 2σ²)

By induction, for n steps we have

Z ~ N(0, nσ²).

For steps distributed according to any distribution with zero mean and a finite variance (not necessarily just a normal distribution), the root mean square translation distance after n steps is

{\sqrt {E|S_{n}^{2}|}}=\sigma {\sqrt {n}}.

But for the Gaussian random walk, this is just the standard deviation of the translation distance's distribution after n steps. Hence, if μ is equal to zero, and since the root mean square(rms) translation distance is one standard deviation, there is 68.27% probability that the rms translation distance after n steps will fall between ± σ ${\sqrt {n}}$ . Likewise, there is 50% probability that the translation distance after n steps will fall between ± 0.6745σ ${\sqrt {n}}$ .

Anomalous diffusion

In disordered systems such as porous media and fractals $\sigma ^{2}$ may not be proportional to $t$ but to $t^{2/d_{w}}$ . The exponent $d_{w}$ is called the anomalous diffusion exponent and can be larger or smaller than 2.^[14] Anomalous diffusion may also be expressed as σ_r² ~ Dt^α where α is the anomaly parameter. Some diffusions in random environment are even proportional to a power of the logarithm of the time, see for example Sinai's walk or Brox diffusion.

Number of distinct sites

The number of distinct sites visited by a single random walker $S(t)$ has been studied extensively for square and cubic lattices and for fractals.^[15]^[16] This quantity is useful for the analysis of problems of trapping and kinetic reactions. It is also related to the vibrational density of states,^[17]^[18] diffusion reactions processes^[19] and spread of populations in ecology.^[20]^[21] The generalization of this problem to the number of distinct sites visited by $N$ random walkers, $S_{N}(t)$ , has recently been studied for d-dimensional Euclidean lattices.^[22] The number of distinct sites visited by N walkers is not simply related to the number of distinct sites visited by each walker.

Applications

Antony Gormley's Quantum Cloud sculpture in London was designed by a computer using a random walk algorithm.

As mentioned the range of natural phenomena which have been subject to attempts at description by some flavour random walks is considerable, in particular in physics^[23]^[24] and chemistry,^[25] materials science,^[26]^[27] biology^[28] and various other fields.^[29]^[30] The following are some specific applications of random walk:

In financial economics, the "random walk hypothesis" is used to model shares prices and other factors. Empirical studies found some deviations from this theoretical model, especially in short term and long term correlations. See share prices.
In population genetics, random walk describes the statistical properties of genetic drift
In physics, random walks are used as simplified models of physical Brownian motion and diffusion such as the random movement of molecules in liquids and gases. See for example diffusion-limited aggregation. Also in physics, random walks and some of the self interacting walks play a role in quantum field theory.
In mathematical ecology, random walks are used to describe individual animal movements, to empirically support processes of biodiffusion, and occasionally to model population dynamics.
In polymer physics, random walk describes an ideal chain. It is the simplest model to study polymers.
In other fields of mathematics, random walk is used to calculate solutions to Laplace's equation, to estimate the harmonic measure, and for various constructions in analysis and combinatorics.
In computer science, random walks are used to estimate the size of the Web. In the World Wide Web conference-2006, Bar-Yossef et al. published their findings and algorithms for the same.
In image segmentation, random walks are used to determine the labels (i.e., "object" or "background") to associate with each pixel.^[31] This algorithm is typically referred to as the random walker segmentation algorithm.

In all these cases, random walk is often substituted for Brownian motion.

In brain research, random walks and reinforced random walks are used to model cascades of neuron firing in the brain.
In vision science, ocular drift tends to behave like a random walk.^[32] According to some authors, fixational eye movements in general are also well described by a random walk.^[33]
In psychology, random walks explain accurately the relation between the time needed to make a decision and the probability that a certain decision will be made.^[34]
Random walks can be used to sample from a state space which is unknown or very large, for example to pick a random page off the internet or, for research of working conditions, a random worker in a given country.

When this last approach is used in computer science it is known as Markov Chain Monte Carlo or MCMC for short. Often, sampling from some complicated state space also allows one to get a probabilistic estimate of the space's size. The estimate of the permanent of a large matrix of zeros and ones was the first major problem tackled using this approach.

Random walks have also been used to sample massive online graphs such as online social networks.
In wireless networking, a random walk is used to model node movement.
Motile bacteria engage in a biased random walk.
Random walks are used to model gambling.
In physics, random walks underlie the method of Fermi estimation.
On the web, the Twitter website uses random walks to make suggestions of who to follow^[35]
Dave Bayer and Persi Diaconis have proven that 7 riffle shuffles are enough to mix a pack of cards (see more details under shuffle). This result translates to a statement about random walk on the symmetric group which is what they prove, with a crucial use of the group structure via Fourier analysis.

Variants of random walks

A number of types of stochastic processes have been considered that are similar to the pure random walks but where the simple structure is allowed to be more generalized. The pure structure can be characterized by the steps being defined by independent and identically distributed random variables.

Random walk on graphs

A random walk of length k on a possibly infinite graph G with a root 0 is a stochastic process with random variables $X_{1},X_{2},\dots ,X_{k}$ such that $X_{1}=0$ and ${X_{i+1}}$ is a vertex chosen uniformly at random from the neighbors of $X_{i}$ . Then the number $p_{v,w,k}(G)$ is the probability that a random walk of length k starting at v ends at w. In particular, if G is a graph with root 0, $p_{0,0,2k}$ is the probability that a $2k$ -step random walk returns to 0.

Building on the analogy from the earlier section on higher dimensions, assume now that our city is no longer a perfect square grid. When our person reaches a certain junction he picks between the various available roads with equal probability. Thus, if the junction has seven exits the person will go to each one with probability one seventh. This is a random walk on a graph. Will our person reach his home? It turns out that under rather mild conditions, the answer is still yes. For example, if the lengths of all the blocks are between a and b (where a and b are any two finite positive numbers), then the person will, almost surely, reach his home. Notice that we do not assume that the graph is planar, i.e. the city may contain tunnels and bridges. One way to prove this result is using the connection to electrical networks. Take a map of the city and place a one ohm resistor on every block. Now measure the "resistance between a point and infinity". In other words, choose some number R and take all the points in the electrical network with distance bigger than R from our point and wire them together. This is now a finite electrical network and we may measure the resistance from our point to the wired points. Take R to infinity. The limit is called the resistance between a point and infinity. It turns out that the following is true (an elementary proof can be found in the book by Doyle and Snell):

Theorem: a graph is transient if and only if the resistance between a point and infinity is finite. It is not important which point is chosen if the graph is connected.

In other words, in a transient system, one only needs to overcome a finite resistance to get to infinity from any point. In a recurrent system, the resistance from any point to infinity is infinite.

This characterization of recurrence and transience is very useful, and specifically it allows us to analyze the case of a city drawn in the plane with the distances bounded.

A random walk on a graph is a very special case of a Markov chain. Unlike a general Markov chain, random walk on a graph enjoys a property called time symmetry or reversibility. Roughly speaking, this property, also called the principle of detailed balance, means that the probabilities to traverse a given path in one direction or in the other have a very simple connection between them (if the graph is regular, they are just equal). This property has important consequences.

Starting in the 1980s, much research has gone into connecting properties of the graph to random walks. In addition to the electrical network connection described above, there are important connections to isoperimetric inequalities, see more here, functional inequalities such as Sobolev and Poincaré inequalities and properties of solutions of Laplace's equation. A significant portion of this research was focused on Cayley graphs of finitely generated groups. In many cases these discrete results carry over to, or are derived from manifolds and Lie groups.

In the context of Random graphs, particularly that of the Erdős–Rényi model, analytical results to some properties of random walkers have been obtained. These include the distribution of first^[36] and last hitting times^[37] of the walker, where the first hitting time is given by the first time the walker steps into a previously visited site of the graph, and the last hitting time corresponds the first time the walker cannot perform an additional move without revisiting a previously visited site.

A good reference for random walk on graphs is the online book by Aldous and Fill. For groups see the book of Woess. If the transition kernel $p(x,y)$ is itself random (based on an environment $\omega$ ) then the random walk is called a "random walk in random environment". When the law of the random walk includes the randomness of $\omega$ , the law is called the annealed law; on the other hand, if $\omega$ is seen as fixed, the law is called a quenched law. See the book of Hughes, the book of Revesz, or the lecture notes of Zeitouni.

We can think about choosing every possible edge with the same probability as maximizing uncertainty (entropy) locally. We could also do it globally – in maximal entropy random walk (MERW) we want all paths to be equally probable, or in other words: for each two vertexes, each path of given length is equally probable. This random walk has much stronger localization properties.

Self-interacting random walks

There are a number of interesting models of random paths in which each step depends on the past in a complicated manner. All are more complex for solving analytically than the usual random walk; still, the behavior of any model of a random walker is obtainable using computers. Examples include:

The self-avoiding walk.^[38]

The self-avoiding walk of length n on Z^d is the random n-step path which starts at the origin, makes transitions only between adjacent sites in Z^d, never revisits a site, and is chosen uniformly among all such paths. In two dimensions, due to self-trapping, a typical self-avoiding walk is very short,^[39] while in higher dimension it grows beyond all bounds. This model has often been used in polymer physics (since the 1960s).

The loop-erased random walk.^[40]^[41]
The reinforced random walk.^[42]
The exploration process.
The multiagent random walk.^[43]

Long-range correlated walks

Long-range correlated time series are found in many biological, climatological and economic systems.

Heartbeat records^[44]
Non-coding DNA sequences^[45]
Volatility time series of stocks^[46]
Temperature records around the globe^[47]

Biased random walks on graphs

Main article: Biased random walk on a graph

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Bibliography

Aldous, David; Fill, Jim, Reversible Markov Chains and Random Walks on Graphs, http://stat-www.berkeley.edu/users/aldous/RWG/book.html
Ben-Avraham D.; Havlin S., Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, 2000.
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Feller, William (1968), An Introduction to Probability Theory and its Applications (Volume 1). ISBN 0-471-25708-7
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Mackenzie, Dana, "Taking the Measure of the Wildest Dance on Earth", Science, Vol. 290, 8 December 2000.
Norris, James (1998), Markov Chains, Cambridge University Press. ISBN 0-521-63396-6
Pólya G.(1921), "Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz", Mathematische Annalen, 84(1–2):149–160, March 1921.
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Toshikazu Sunada (2012), Topological Crystallography --With a View Towards Discrete Geometric Analysis--, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer

External links

Pólya's Random Walk Constants
Random walk in Java Applet
Quantum random walk
Gaussian random walk estimator
Electron Conductance Models Using Maximal Entropy Random Walks Wolfram Demonstrations Project

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

List of topics Category

Authority control	NDL: 00571555

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