A random walk problem

1960 ◽  
Vol 56 (4) ◽  
pp. 390-392 ◽  
Author(s):  
J. Gillis
Keyword(s):  
Random Walk ◽  
Legendre Polynomial ◽  
Rectangular Lattice ◽  
Nearest Neighbour ◽  
Two Dimensional ◽  
Image Position ◽  
The Right ◽  
Step Turn ◽  
Previous Step

We consider a random walk on a two-dimensional rectangular lattice in which steps are strictly between nearest neighbour points. The conditions of the walk are that the walker must, at each step, turn either to the right or to the left of his previous step with respective probabilities ½(1+α), ½(1−α), (≤ α ≤ 1). To fix the ideas it is assumed that he starts from the origin and the probability of each of the four possible starting directions is ¼. If Ar denotes the probability of return to the origin after r steps we shall show thatwhere β = ½(α + α−1) and Pn is the nth Legendre polynomial. It is clear that Ar is zero for r ≢ 0(mod 4).

The transient behaviour of the simple random walk in the plane

1988 ◽  
Vol 25 (01) ◽  
pp. 58-69 ◽  
Author(s):  
D. Y. Downham ◽  
S. B. Fotopoulos
Keyword(s):  
Random Walk ◽  
Transient Process ◽  
Asymptotic Form ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Two Dimensional ◽  
Transient Behaviour ◽  
Practical Implications ◽  
Asymptotic Forms

For the simple two-dimensional random walk on the vertices of a rectangular lattice, the asymptotic forms of several properties are well known, but their forms can be insufficiently accurate to describe the transient process. Inequalities with the correct asymptotic form are derived for six such properties. The rates of approach to the asymptotic form are derived. The accuracy of the bounds and some practical implications of the results are discussed.

On the right lower circular density of two-dimensional Jordan curves in the plane

1968 ◽  
Vol 64 (1) ◽  
pp. 67-70
Author(s):  
D. G. Larman
Keyword(s):  
Jordan Curve ◽  
Two Dimensional ◽  
End Points ◽  
Jordan Curves ◽  
End Point ◽  
Dimensional Measure ◽  
Image Position ◽  
The Right

We use J(a, b) to denote a Jordan curve of positive two-dimensional measure in the plane, with end-points a and b. If υ is a point of J(a, b), we define the right lower arc density at υ bywhere J( υ, υ′) is the largest arc, whose left-end point is υ, which is contained in the disc c(υ, r).

Wald's identity and absorption probabilities for two-dimensional random walks

1965 ◽  
Vol 61 (3) ◽  
pp. 747-762 ◽  
Author(s):  
V. D. Barnett
Keyword(s):  
Random Walk ◽  
Double Series ◽  
Moment Generating Function ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Region ◽  
Discrete Random Variables ◽  
Image Position ◽  
Wald's Identity ◽  
Absorption Probabilities

SummarySuppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), whereand we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.

A correlated random walk model for two-dimensional diffusion

1984 ◽  
Vol 21 (2) ◽  
pp. 233-246 ◽  
Author(s):  
Robin Henderson ◽  
Eric Renshaw ◽  
David Ford
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Exact Expression ◽  
Dispersion Matrix ◽  
Two Dimensional ◽  
Correlated Random Walk ◽  
Dimensional Diffusion ◽  
Step Transition ◽  
Lattice Random Walk ◽  
Previous Step

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

The transient behaviour of the simple random walk in the plane

1988 ◽  
Vol 25 (1) ◽  
pp. 58-69 ◽  
Author(s):  
D. Y. Downham ◽  
S. B. Fotopoulos
Keyword(s):  
Random Walk ◽  
Transient Process ◽  
Asymptotic Form ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Two Dimensional ◽  
Transient Behaviour ◽  
Practical Implications ◽  
Asymptotic Forms

For the simple two-dimensional random walk on the vertices of a rectangular lattice, the asymptotic forms of several properties are well known, but their forms can be insufficiently accurate to describe the transient process. Inequalities with the correct asymptotic form are derived for six such properties. The rates of approach to the asymptotic form are derived. The accuracy of the bounds and some practical implications of the results are discussed.

A correlated random walk model for two-dimensional diffusion

1984 ◽  
Vol 21 (02) ◽  
pp. 233-246 ◽  
Author(s):  
Robin Henderson ◽  
Eric Renshaw ◽  
David Ford
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Exact Expression ◽  
Dispersion Matrix ◽  
Two Dimensional ◽  
Correlated Random Walk ◽  
Dimensional Diffusion ◽  
Step Transition ◽  
Lattice Random Walk ◽  
Previous Step

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

scholarly journals Risk theory and Wiener processes

1972 ◽  
Vol 7 (1) ◽  
pp. 96-99 ◽  
Author(s):  
H. Bohman
Keyword(s):  
Random Walk ◽  
Risk Process ◽  
Risk Theory ◽  
Practical Applications ◽  
The Difference ◽  
The Individual ◽  
Image Position ◽  
The Right ◽  
Risk Processes ◽  
First Time

We will in this paper consider the risk process from the point of view of random walk in one dimension. The particle starts out at the origin. Each claim is equivalent to a step in the random walk. The length of the step is equal to the amount of the claim minus the amount of the premium which has been obtained since the preceding claim. If the difference is positive the particle advances to the right and if the difference is negative to the left. At distance U to the right from the origin there is a barrier. The problem is to find the distribution function of X, the time it takes the particle to cross the barrier for the first time.In most practical applications of risk theory U is large in comparison to the individual steps of the particle. We will in this paper assume that U is large in comparison to the individual steps and draw certain conclusions about the risk processes from this assumption.The individual steps of the particle have a certain distribution. The corresponding characteristic function is ϕ. For reasons which will be seen later we will consider ϕ to be a function of it = θ instead of t. This means thatThe mean value and the standard deviation of each step is equal to m and σ respectively. We now writeWe now define two random variables X and Y.X = time to cross the barrier for the first timeY = X σ2/U2.

On the Expected Number of Visits of a Particle before Absorption in a Correlated Random Walk

1973 ◽  
Vol 16 (3) ◽  
pp. 389-395 ◽  
Author(s):  
G. C. Jain
Keyword(s):  
Random Walk ◽  
Correlated Random Walk ◽  
Expected Number ◽  
Unit Distance ◽  
Straight Line ◽  
Particle Move ◽  
Number Of Visits ◽  
The Right ◽  
Previous Step

Let a particle move along a straight line a unit distance during every interval of time τ. During the first interval τ it moves to the right with probability ρ1 and to the left with probability ρ2 = 1 - ρ1. Thereafter at the end of each interval τ, the particle with probability p continues its motion in the same direction as in the previous step and with probability q = l - p reverses it.

Some arithmetic properties of the Legendre polynomials

1957 ◽  
Vol 53 (2) ◽  
pp. 265-268 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Complex Multiplication ◽  
Elliptic Functions ◽  
Arithmetic Properties ◽  
Special Values ◽  
Right Hand ◽  
Image Position ◽  
The Right ◽  
Divisibility Properties

1. Good (4) has proved the formulawhere Pn(x) is the Legendre polynomial of degree n and t is any integer greater than n. The form of the right-hand side suggests that (1) may be of use in deriving arithmetic properties of Pn(x).Elsewhere (1) the writer indicated a connexion between divisibility properties of Pm(a) for special values of a and the complex multiplication of elliptic functions. If p = 2m + 1 is an odd prime, put

Residual Gas Reactions in the Electron Microscope: I. Qualitative Observations on the Water Gas Reaction

1968 ◽  
Vol 26 ◽  
pp. 292-293
Author(s):  
Richard E. Hartman ◽  
Roberta S. Hartman ◽  
Peter L. Ramos
Keyword(s):  
Electron Beam ◽  
Electron Microscope ◽  
Gas Phase ◽  
Absolute Temperature ◽  
Residual Gas ◽  
Gas Reactions ◽  
Image Position ◽  
The Right ◽  
Water Gas ◽  
Thinning Rate

The action of water and the electron beam on organic specimens in the electron microscope results in the removal of oxidizable material (primarily hydrogen and carbon) by reactions similar to the water gas reaction .which has the form:The energy required to force the reaction to the right is supplied by the interaction of the electron beam with the specimen.The mass of water striking the specimen is given by:where u = gH2O/cm2 sec, PH2O = partial pressure of water in Torr, & T = absolute temperature of the gas phase. If it is assumed that mass is removed from the specimen by a reaction approximated by (1) and that the specimen is uniformly thinned by the reaction, then the thinning rate in A/ min iswhere x = thickness of the specimen in A, t = time in minutes, & E = efficiency (the fraction of the water striking the specimen which reacts with it).

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