It is true that a number of interesting questions remain open in the concrete setting of the random walk on the integers. Feller [S9], Breiman [S1], Chung [S4] in the more general setting of random walk on the real line. New types of random walk problems are now in the pioneering phase, which was unheard of when the first edition was published.
The relevance of harmonic analysis for random walks naturally arises from the invariance of the transition probabilities under translation in the additive group forming the state space. In this sense, the study of random walks leads one to potential theory in a very simple setting. viii PREFACE TO THE FIRST EDITION. As we shall see, both of these kernels have a counterpart in the theory of random walks.
Levy, namely the theory of the distribution function in Levy's Theorie de l'addition des variables aleatoires (1937), which was . first modern book on the subject of random walk.).
INTERDEPENDENCE GUIDE
THE CLASSIFICATION OF RANDOM WALK
Therefore, it is reasonable to call the random walk recurrent if F = 1 and transient if F < 1. In the recurrent case, we first prove a few lemmas (P3, P4, P5) of independent interest, which will be combined later with P2 to provide a complete description (TI) of the fundamental differences between recurrent and transient random walk. At the present stage of development of the theory, it is not even entirely obvious that the random walk of type (a) or (b) above is transient.
Since the random walk is repetitive (this assumption is much stronger than necessary), A ' is not empty. In case (i), it only remains to show that the random walk is strongly aperiodic. Now we want to define T, as the first time the random walk lands on a red point.
HARMONIC ANALYSIS
Ykpk+
The last term tends to zero as a function of the first part of (1), and this shows that A(9) 0 and completes the proof. For the random walk of higher dimensions (d >_ 2) we shall be content to give the analogue of P4 only for the first and second moments, but first it will be convenient to adopt a sensitive notation which agrees with that of D4. Although it will play a minor role in random walk theory, we will sketch the essential components of the traditional analytical Fourier proof of the Central Limit Theorem for identically distributed independent random variables.
PS If P(x,y) is the transition function of a one-dimensional random walk with mean µ = 0, and variance a2 = m2 < oc, then. In the terminology of Chapter I, a random walk with d-dimensional state space R is aperiodic if the group if and the group R are equal. We only need to talk about a d-dimensional random walk when R has dimension d and when P(O,x) is defined for all x in R.
Having shown how one can trivially make an aperiodic random walk periodic, by artificially increasing its state space, we will now show that it is almost as easy to replace a periodic random walk by an aperiodic one, which retains all properties that could possibly be of any interest. If the last coordinate of each element of X is 0, then we are obviously done, in light of the induction hypothesis. Suppose a random step P(x,y) on the state space R = Rd is given and happens to be periodic.
By our construction, the random walk defined by Q on Rk is aperiodic, and has all the essential properties of the periodic random walk defined by P on R. For example, consider the triangular lattice in the plane consisting of all points (complex numbers) z of form. This isomorphism therefore transforms the given process into the bona fide random walk with transition probabilities.
Now we return to Fourier analysis to establish a simple criterion for aperiodicity, in terms of the characteristic function of a random walk. TI A random walk on R (of dimension d), is aperiodic if and only if its characteristic function c6(9), defined for 9 in E (of dimension d), has the following property: ¢(9) = 1 then and only if all coordinates are 0.
Using P4, one obtains the following estimate for the real part of the characteristic function of a random walk. However, with the help of Fourier analytical apparatus, we will be able to obtain even sharper results than (2), in the form of an upper bound on the supremum in (2), which will depend on the dimension d of the random walk. The next lemma is the analogue of the criterion in Ti for strongly aperiodic instead of aperiodic random walk.
Now consider the transition function defined by Q(O,x) = P(0,x + zo), where P(O,x) is the transition function of the given strongly aperiodic random walk. This construction is based on the observation that a random walk with P(0,0) > 0 is aperiodic if and only if it is strongly aperiodic. The random walk is then recurrent and aperiodic, and in this case it is known that the series.
It is known from TI that such a random walk is repeated when a > 1 (for then the first moment m will be finite). We will now prove that the random walk in question is in fact recurrent when an I and transitive when . In the next example we will encounter a specific random walk of the type discussed in E2 with a = 1.
Thus T is the first time the simple random walk in the plane visits the diagonal Re (x) = Im (x). Therefore, we obtained sufficient conditions for repetition and transscience of the "bulb random walk":. Note that it is consequently possible to define an aperiodic repeating random walk on the additive group of rational numbers.
Show that the two-dimensional random walk is transient if m < oo and if the mean vector ,u # 0. A particularly elegant continuous analogue of the simple two-dimensional random walk is the random flight investigated by Lord Rayleigh (see [103], p . 419).
TWO-DIMENSIONAL RECURRENT RANDOM WALK
This is done, as in the first half of the proof, by decomposing HA(x, y) according to the possible values of T. The proof of (b) depends on the interpretation of gA(x, y) as the expected number of visits by the random walk xn with xo = x to the peak before the time T of the first visit to the set A. In the case of recurrent aperiodic random walk, P4 will be significantly strengthened at the beginning of the next section.
But now the substitution of formula (1) for HB(x, y) and of equation (2) for g(y,y) in equation (3) completes the proof. The proof will be Fourier analytical, and we adopt the notation of Chapter II, and let 0(6) denote the characteristic function of the random walk. Here 12(x) is the integral over the circular disk 101 5 IT and 12(x) is the contribution of the rest of the square.
A similar result can be expected in the case of the random walk, and so we make two conjectures, each reflecting some essential feature of the classical result. The next step consists in solving (3) and we will show that every solution of (3) is of the form Letting T = TB denote the time of the first visit of the random walk to B, one can write, for x and y in R - B and n z I,.
Now we take for the starting point of the random walk xo = z = r + is, a point in the first quadrant R - C and ask for the probability. It will be the first time in this chapter that we can get a result about the time-dependent behavior of the random walk. Therefore, the random walk analog of equation (1) must relate to the convergence rate of the sequence.
Therefore, equation (7) holds, and in the light of the discussion after the proof of TI, various extensions of TI are valid for simple random walk. E2 For simple random walk, we shall use the above result to obtain a new version of the famous double point problem. Now it is quite clear, from the symmetry of the simple random walk, that Px[T > it, Tx > n] has the same value for each of the four points x where P(O,x) > 0.
For simple random walk in the plane, let T denote the time of the first double point (the time of the first visit of x to a point that has been occupied for about n 0).
RANDOM WALK ON A HALF-LINE'
Similarly, the repetition of the random walk will not be as important here as in Chapter III. When p = q the random walk is recurrent so that (1) holds since every point is subsequently visited with probability one, and hence every subset B of R. When p > q the random walk is transitive, but (1 ) is still true since every point to the right of the initial point is visited with probability one.
For the case where (1) fails, we must of course resort to a transient random walk, and the Bernoulli random walk with p < q is indeed an example. We will see, and not surprisingly, that (1) is equivalent to the statement that a random walk visits the half-line B infinitely often with probability one, regardless of the starting point, and an extremely simple necessary and sufficient condition for (1) to hold will be given in the theorem You of this section. In this process, we will need simple but quite powerful methods of Fourier analysis or, alternatively, the theory of complex variables.
Briefly, we shall switch back to the notation and terminology in D3.1, where we describe the random walk x with xo = 0, which. This completes the proof of the first part, and the argument for T' is omitted since it is exactly the same. In fact, it is interesting to observe that no use was made of the definition of T at all, except for its property of being a stopping point.
Since we have not used any of the properties of our special stopping time T so far, it seems that we are still far from our goal of computing E[tTe'8ST] and similar characteristic functions as explicitly as possible. 2 For an even shorter proof of P1 and P4 below, in the spirit of Fourier analysis, see Dym and McKean [S7; pp. The results of the classical analysis that we now introduce will be useful in working with the very special type of Fourier series that we have already encountered, namely the Fourier series of the form.
Uniqueness arises from the fact that an analytic function in IzI < 1 is determined by its limits.' The proof for '1e and fe is the same, where IzI >_ I replaces IzI 5 1. El For one-dimensional symmetric random walk, certain simplifications of the formulas in D3, P4 and P5 are worth examining in detail.
