From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval
Let be a positive integer, a positive constant and be a sequence of independent identically distributed pseudorandom variables. We assume that the ’s take their values in the discrete set and that their common pseudodistribution is characterized by the (positive or negative) real numbers for any . Let us finally introduce the associated pseudorandom walk defined on by and for . In this paper, we exhibit some properties of . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.
Throughout the paper, we denote by the set of integers, by that of nonnegative integers, and by that of positive integers: , , . More generally, for any set of numbers , we set .
Let be a positive integer, a positive constant, and set . Let be a sequence of independent identically distributed pseudorandom variables taking their values in the set of integers . By pseudorandom variable, we mean a measurable function defined on a space endowed with a signed measure with a total mass equaling the unity. We assume that the common pseudodistribution of the ’s is characterized by the (positive or negative) real pseudo-probabilities for any . The parameters sum to the unity: .
Now, let us introduce the associated pseudorandom walk defined on by and for . The infinitesimal generator associated with is defined, for any function defined on , as Here we consider the pseudorandom walk which admits the discrete -iterated Laplacian as a generator infinitesimal. More precisely, by introducing the so-called discrete Laplacian defined, for any function defined on , by the discrete -iterated Laplacian is the operator given by We then choose the ’s such that which yields, by identification, for any ,
When , is the nearest neighbours pseudorandom walk with a possible stay at its current location; it is characterized by the numbers and . Moreover, if , then ; in this case, we are dealing with an ordinary symmetric random walk (with positive probabilities). If , this is the classical symmetric random walk: and .
Actually, with the additional assumption that for any (i.e., the ’s are symmetric, or the pseudorandom walk has no drift), the ’s are the unique numbers such that
where is an analytical extension of and stands for the th derivative of .
Our motivation for studying the pseudorandom walk associated with the parameters defined by (4) is that it is the discrete counterpart of the pseudo-Brownian motion as the classical random walk is for Brownian motion. Let us recall that pseudo-Brownian motion is the pseudo-Markov process with independent and stationary increments, associated with the signed heat-type kernel which is the elementary solution of the high-order heat-type equation . The kernel is characterized by its Fourier transform: The corresponding infinitesimal generator is given, for any -function , by
We observe that (5) and (7) are closely related to the continuous -iterated Laplacian . For , the operator is the two-Laplacian related to the famous biharmonic functions: in the discrete case, and in the continuous case,
The link between the pseudorandom walk and pseudo-Brownian motion is the following one: when normalizing the pseudorandom walk on a grid with small spatial step and temporal step (i.e., we construct the pseudoprocess where denotes the usual floor function), the limiting pseudoprocess as is exactly the pseudo-Brownian motion.
Now, we consider the first overshooting time of a fixed single threshold or ( being integers) for :
as well as the first exit time from a bounded interval :
with the usual convention that . Hence, when , and , the overshoot at time which is can take the values , that is, . Similarly, when , , and when , . We put , and .
In the same way, we introduce the first overshooting times of the thresholds and ( being now real numbers) for :
as well as the first exit time from a bounded interval : with the similar convention that , and we set, when the corresponding time is finite,
In this paper we provide a representation for the generating function of the joint distributions of the couples , , and . In particular, we derive simple expressions for the marginal distributions of , , and . We also obtain explicit expressions for the famous “ruin pseudoprobabilities” and . The main tool employed in this paper is the use of generating functions.
Taking that the limit as goes to zero, we retrieve the joint distributions of the couples and obtained in [10, 11]. Therein, we used Spitzer’s identity for deriving these distributions. Moreover, we obtain the joint distribution of the couple which is a new and an important result for the study of pseudo-Brownian motion. In particular, we deduce the “ruin pseudo-probabilities” and ; the results have been announced without any proof in a survey on pseudo-Brownian motion , after a conference held in Madrid (IWAP 2010).
In [11, 17, 18], the authors observed a curious fact concerning the pseudodistributions of and : they are linear combinations of the Dirac distribution and its successive derivatives (in the sense of Schwarz distributions). For instance,
The quantity is to be understood as the functional acting on test functions according to . The appearance of the ’s in (15), which is quite surprising for probabilists, can be better understood thanks to the discrete approach. Indeed, the ’s come from the location at the overshooting time of for the normalized pseudorandom walk: the location takes place in the “cluster” of points .
In order to facilitate the reading of the paper, we have divided it into three parts: Part I—some properties of the pseudorandom walk Part II—first overshooting time of a single threshold Part III—first exit time from a bounded interval.
The reader will find a list of notations in Table 2 which is postponed to the end of the paper.
2. Part I—Some Properties of the Pseudorandom Walk
2.1. Pseudodistribution of and
We consider the pseudorandom walk related to a family of real parameters satisfying for any and . Let us recall that the infinitesimal generator associated with is defined by
In this section, we look for the values of , , for which the infinitesimal generator is of the form (5). Next, we provide several properties for the corresponding pseudorandom walk.
Suppose that can be extended into an analytical function . In this case, we can expand Therefore, Since , we see that the expression (5) of holds if and only if the ’s satisfy the equations
Proposition 1. The numbers , , satisfying (19), are given by In particular, .
Proof. First, we recall that the solution of a Vandermonde system of the form , , is given by
and, for any ,
In the notation of and that of forthcoming determinants, we adopt the convention that when the index of certain entries in the determinant lies out of the range of , the corresponding column is discarded. That is, for and , the respective determinants write
It is well-known that, for any ,
In the particular case where for , we have, for any , that
Therefore, the solution simply writes
Now, we see that system (19) is a Vandermonde system with the choices , , and for , . With these settings at hands, we explicitly have
and the result of Proposition 1 ensues.
Finally, the value of is obtained as follows: by using the fact that , We find it interesting to compute the cumulative sums of the ’s: for , The last displayed sum is classical and easy to compute by appealing to Pascal’s formula which leads to a telescopic sum: Thus, for , Observe that this sum is nothing but . Next, we compute the total sum of the ’s: by using the fact that , As previously mentioned, there is an interpretation to this sum: this is the total variation of the pseudodistribution of . We can also explicitly determine the generating function of : for any , We sum up below the results we have obtained concerning the pseudodistribution of .
Proposition 2. The pseudodistribution of is determined, for , by or, equivalently, by The total variation of the pseudodistribution of is given by The generating function of is given, for any , by In particular, the Fourier transform of admits the following expression: for any , by
Remark 3. For , we have ; that is, the pseudorandom walk does not stay at its current location. If , it can be easily seen, by using the identity , that . On the other hand, for any , it is clear that . In Table 1 and Figures 1 and 2, we provide some numerical values and (rescaled) profiles of the pseudodistribution of for and and several values of .
In the sequel, we will use the total variation of as an upper bound which we call : Set for any . We notice that and, more precisely, Let us denote this bound by : In view of (40) and (42), since , we see that .
Proposition 4. The pseudodistribution of is given, for any , by Actually, the foregoing sum is taken over the such that . We also have that
Proof. By the independence of the ’s which have the same pseudoprobability distribution, we plainly have that Hence, by inverse Fourier transform, we extract that By writing , we get for the integral lying in (47) that By plugging (48) into (47), we derive (43). Next, we write, for , that If , then the term in sum (49) corresponding to vanishes and The second sum in the foregoing equality is easy to compute: If , then the term in sum (49) corresponding to is and By using the convention that if , we see that the second sum above also coincides with (51). Formula (44) ensues in both cases.
Proposition 5. The upper bound below holds true: for any positive integer and any integer , Assume that . The asymptotics below holds true: for any ,
Proof. Let us introduce the usual norms of any suitable function :
and recall the elementary inequalities .
It is clear from (46) that, for any integer , This proves (53). Next, by (46), since , we have, for any , that The assumption entails that for any . We see that on , and on for any . Hence, Now, choose for a positive . We have that
which clearly entails, for large enough , .Thus, if , which proves (54).
If , . In this case, the same holds true upon splitting the integral into .
Remark 6. A better estimate for can be obtained in the same way: Nevertheless, we will not use it. We also have the following inequality for the total variation of :
Proposition 7. For any bounded function defined on ,
Proof. Recall that we set for any . We extend these settings by putting for . We have that The foregoing sum can be easily evaluated as follows: which proves (62).
2.2. Generating Function of
Let us introduce the generating functions, defined for complex numbers , , by We first study the problem of convergence of the foregoing series. We start from If and , then If we choose , such that , and (which is equivalent to , or ), then the double sum defining the function is absolutely summable. If , then
If we choose such that , then the same conclusion holds.
Now, we have that and, thanks to (38), we can state the following result.
Proposition 8. The double generating function of the , , , is given, for any complex numbers , such that , by In particular, for any and such that ,
On the other hand, By substituting in the foregoing equality, we get the Fourier series of the function : from which we extract the sequence of the coefficients . Indeed, since , we have that and where is the circle of radius 1 centered at the origin and counter clockwise orientated. Then, referring to (70), we obtain, for any satisfying , that where is the polynomial given by
We are looking for the roots of which lie inside the circle . For this, we introduce the th roots of : for ; .
From now on, in order to simplify the expression of the roots of , we make the assumption that is a real number lying in (and then ). The roots of are those of the equations , , where They can be written as with
We notice that . Because of the last coefficient in the polynomial , it is clear that the roots and are inverse: .
Let us check that for any . Straightforward computations yield that where