On the time a diffusion process spends along a line

Stochastic Processes and their Applications North-Holland 229 47 (1993) 229-247 On the time a diffusion along a line process spends Nils Lid Hjort University of Oslo, Norway Rafail Z. Khasminskii I.P. P. I., Moscow, Russian Federahon Received 6 December 1991 Revised manuscript received 28 October 1992 For an arbitrary diffusion process X with time-homogeneous drift and variance parameters p(x) and u*(x), let V, be l/c times the total time X(l) spends in the strip [a + bt -+E, a + bt +f~]. The limit V as .C+ 0 is the full halfline version of the local time of X(t) - a - bt at zero, and can be thought of as the time X spends along the straight line x = a + bl. We prove that V is either infinite with probability 1 or distributed as a mixture of an exponential and a unit point mass at zero, and we give formulae for the parameters of this distribution in terms of p( .), a( .), n, b, and the starting point X(0). The special case of a Brownian motion is studied in more detail, leading in particular to a full process V(b) with continuous sample paths and exponentially distributed marginals. This construction leads to new families of bivariate and multivariate exponential distributions. Truncated versions of such ‘total relative time’ variables are also studied. A relation is pointed out to a second order asymptotics problem in statistical estimation theory, recently investigated in Hjort and Fenstad (1992a, 1992b). Brownian motion * diffusion process * exponential process * local time * multivariate exponential distribution * second order asymptotics for estimators zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE 1. Introduction and summary Consider a time-homogeneous diffusion process X with dX(l)= p(X( t)) dt+ a(X(t)) dW(t), using zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM W( .) to denote a standard Brownian motion. In other words, X is a Markov process with continuous paths and with the property that X( t + h) - X(t), r2(x)h + o(h). for given X(t) = x, has mean value p (X)/I + o( h) and variance Consider m v, =A 1(1X(t)-a-brJ~&}dt, &I 0 (1.1) the total amount of time spent by the process in the narrow strip [a + bt -4.q a + bt + $e], divided by E. We show in Section 2 that this variable has a well-defined limit V, and find its distribution, in terms of a, b, b( *), (T( ), and the starting point X(0) = x. Under certain conditions the variable is infinite almost surely, and in the opposite case the variable is distributed as a mixture of an exponential and a unit Corr~.~prldence to: Dr. N.L. Hjort, Department Blindern, N-03 I6 Oslo, Norway. 0304.4149/93/$06.00 0 1993-Elsevier of Mathematics Science Publishers and Statistics. University B.V. All rights reserved of Oslo. P.B. 1053 N. L. Hjort, R.Z. Khasminskii / Time along a line 230 point mass at zero. Explicit found. The variable formulae for the parameters is a pure exponential of this distribution the result is remarkable, in view of the large class of diffusion processes; X can have both Gaussian and non-Gaussian sample paths. The spends V variable can be thought are also in the case X(0) = a. The simplicity of as the total along the line x = a + bt, and is related relative of in particular time the zyxwvutsrqponmlkjihgfe X( . ) process to what is sometimes called the local time at zero of the X(t) - a - bt process. Usually such local times are studied and used for a limited time interval [0, r] only, however. A special case of the construction above is that of a Brownian motion and a = 0, giving V,(b)=imeasure{tsO: W(t)E[bt--$c, bt+$e]}. (1.2) That the limit V(b), the time Brownian motion spends along x = bt, is simply exponential with parameter IbJ, follows from the general result of Section 2, but is proved in a more direct fashion in Section 3, using moment convergence. This second approach lends itself more easily to the simultaneous study of several relative times. In Section 4 we prove full process convergence of {V,(b): b#O} towards a {V(b): b # 0) with continuous sample paths and with exponentially distributed marginals. Its covariance and correlation structure this construction leads to new families of bivariate is also found. and multivariate In particular exponential distributions. In Section 5 a more general variable V( c, b) is studied, defined as the total relative time during [c, a~) that W(t) spends along bt. The distribution of V( c, b) is again a mixture of an exponential and a unit point mass at zero. A simple consequence of this result is a rederivation of a well known formula for the distribution of the maximum of Brownian motion over an interval. Finally some supplementing results and remarks are given in Section 6. In particular some consequences for empirical partial sum processes are briefly discussed. A certain second order asymptotics problem in statistical by serendipity to the present study on total relative motion. Suppose { 0,: n 3 1) is an estimator sequence estimation theory led time variables for Brownian for a parameter 0, where 8, is based on the first n data points in an i.i.d. sequence, and consider Q8, the number (or strong of times, among n 2 c/a2, where (8, - 012 6. Almost sure convergence consistency) of 0, is equivalent to saying that Q8 is almost surely finite for every 6, and it is natural to inquire about its size. A particular result of Hjort and Fenstad (1992a, Section 7) is that under natural conditions, which include the existence of a normal (0, a’) limit for fi( zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 8, - e), w I{] W( t)j 2 t/v} dt (1.3) S2Qs* dzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Q= Q(c, l/q) = IC as 6 + 0. If {e,,,} and {e,,} are first order equivalent estimator sequences, with the same N(0, u’) limit for &r( 0,., - 0), and Q5,, is the number of s-misses for sequence j, then Q~Qw+ 1 and s2(Q6,1- Q8,*) + 0 in probability. One way of distinguishing N.L. Hjort, R.Z. Khasminskii between the two estimation Qfi,Z. It turns which methods is by studying out that 6 times this difference is that of a constant times / Time along a line second in typical 231 order aspects of Qfi,, - cases has a limit distribution - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR V(c, -l/g), or times the simpler V(c, l/a) V( l/u) - V( - I/(+) if c = c( 6) is allowed to decrease to zero in the definition of Qs,j. Note the connection from Q(c, l/m) of (1.3) to V(c, *l/a). Some further details are in Section 6.3 in the present paper, while further discussion can be found in Hjort and Fenstad (1992a, 1992b). background and 2. The time X spends along a straight line In Section time 2.1 we solve the problem axis. This presented rather immediately in Section for the time spent along leads general to the more a line parallel solution, to the which is 2.2. 2.1. The time X spends along a horizontal line Let X(t) be as in the introductory function a(x) and continuous paragraph, drift function with continuous p(x). and positive For a temporarily diffusion fixed a, define s(y)=exp(l[:E}, also for negative y. The function is often called the scale function are S(z) = jt s(y) dy, or any linear translation of the diffusion process. Two important thereof, quantities cr k+(a) = It is known to drift Iu s(y) dy and k_(a)= a S(Y) dy. i - c.Z that if k+(a) is finite, then there is a positive off towards +a, and vice versa; and similarly (2.1) probability for the process the finiteness of k-(a) corresponds exactly to there being a positive probability for drifting off towards --oo. See for example Karlin and Taylor (1981, Chapter 15.6). If in particular both integrals are infinite then the process is recurrent and visits the line x = a an infinite number of times. The current object of interest is a: V, A Z{IX(t)- alS$&}dt. (2.2) &I0 Let V,(T) be defined similarly, but for the interval [0, 71 only. This is the so-called local time at zero process, Paul Levy’s ‘mesure de voisinage’, for X(t) - a; see for example Karlin and Taylor (1981, Chapter 15.12) and It6 and McKean (1979, Chapter 2 and 6). It is a ‘remarkable and recondite fact’, to quote Karlin and Taylor, that the limit V(T) of V,(T) as E -0 exists for almost every sample path (that is, V, (T, w) converges to a well-defined V( 7, W ) for each w in a subset of probability 232 N.L. Hjort, R.Z. Khasminskii / Time along a line 1 of the underlying probability space). It follows from this local time theory V, = V,(W) converges to a well-defined V= V(a) too, with probability of V as the total relative time X spends along the line x = a. that 1. We think In the following we are able to find the exact distribution of V. The arguments we shall use actually show convergence in distribution of V, to V directly, that is, we do not need or use the somewhat sophisticated local time theory or the almost sure pathwise existence of V to prove that V, has the indicated limit distribution as F goes to zero. If (Y is positive, write V- exp(a) for the exponential g(v) = (Ye-“” for z, 3 0. It has mean l/a and Laplace distribution with density transform E exp(-AV) = Q/(cr +A). 1. Assume Theorem that the X process starts at X(0) = a. Zf k+(a) and k-(a) are both injinite, then V = ~0 with probability one. Otherwise the limit Vof V, is exponentially distributed with parameter Q (a) = &‘(a){ l/ k+( a) + l/k_(a)}. Proof. That V, goes almost from the theory recurrency Suppose of Karlin surely to infinity and Taylor when both integrals (1981, Chapter are infinite follows 15.6). This is connected to the phenomenon mentioned after (2.1) above. next that both k+(a) and k_(a) are finite. For a fixed positive A, study the function = E, exp(-AV,) u,,,(x) = E, exp where the subscript x here and below means that the expectation is conditional starting point X(0)=x, and where fF(x) = E~‘Z{\X-a( SUE}. General results diffusion derivatives processes imply the u*,~ function has two piecewise continuous and satisfies tf12(x)ul;,,(x) see for example Integrating that on for the theory from the limit problem a -38 - Afe(x)u,,,(x) +P(x)&(x) developed to a ++s by Karlin and letting = 0, and Taylor (1981, Chapter E + 0 shows that a solution 15.3). u,,(x) to must satisfy u;(a+)-&(a-)=2Au,(a)/(z*(a). (2.3) Now let w(x, a) be the probability that the process after start in X(0) = x succeeds in reaching the level x = a in a finite amount of time. If this happens then V starting from x is equal in distribution to a V starting from a, because of the strong Markov property and the postulated time-homogeneity. And if it does not happen then V = 0. Hence u,(x) = E, e-*” = w(x, a)E, ee”“+{l- = w(x, a)uA(a)+ I- w(x, a)}E em0 w(x, a). (2.4) / Time along a line zyxwvutsrqponmlkjihgfedcba 233 N.L. Hjort, R.Z. Khasminskii This equation is also reached if one more carefully starts with V,-equations and then lets E + 0. But w(x, a) can be found explicitly, since its satisfies iv’(x) w”(x, a) + p(x)w’(x, a) = 0 with boundary conditions w(--CO, a) = 0, w(a, a) = 1, W(CO,a) = 0. Differentiation here is w.r.t. x and a is still fixed. The solution is zyxwvutsrqponmlkjihgfed k+(x)/k+(a) if _xS a, k- (x)/k_(u) if .xS u, 1 w(x, a) = in terms of the (2.1) functions. l/k-(u) in terms In particular, of the transience (2.5) ~‘(a+, determining a) = -l/k+(u) quantities and ~‘(a-, a) = (2.1). This can now be used in (2.3) to make (2.4) more explicit: ~;(a+)= lead to {-l/k+(u) in the end u*(a)= ~‘(a+, - u){u,(u)-1) l/k_(u)}{ and uA (a) - ~;(a-)= 1) = 2hu, (a)/~‘( a). And solving this produces Q(U) l/k+(a)+ l/k-(a) much as the previous one. Now +CO is attracting for w(x, a) become w(x, a) = This case can be handled w(--CO, a) = 1, w(u, a) = 1, w(co, a) =O, giving k+(x)/k+(u) if x 2 a, 1 if x S u. The proof a(u)=fa2(u)/k+(u). similarly. actually very but ---COis not, and the boundary as (2.6) (2.3) and (2.4) are still valid, and we find after similar arguments with parameter finite is handled u){u,+(u)-1) l/k+(u)+l/k~(u)+2A/a2(u)=a(u)+h’ with the a(u) parameter as given in the theorem. Assume next that k+(u) is finite but k_(u) infinite. conditions solution ~‘(a-, The final that V is exponential case of k+(u) infinite and k-(u) Cl gives the distribution of V for an arbitrary starting point x, namely (2.7) v~{x(O)=x}-w(x,~)Exp(~(~))+{1-w(x,~~}~,, in which 6” is a unit point probabilistical interpretation, mass at zero. The weight w(x, a) here has a direct and is given in (2.5) for the case of two attracting boundaries and in (2.6) for the case of only l tooattracting, with a similar modification for the case of k+(u) infinite but k_(u) finite. In the case of (2.6) we see that V- Exp(cu(u)) for any starting point to the left of a. 2.2. The time X spends along a general line from a general starting point The generalisation to a result about the (1.1) variable is now immediate. Just consider the new process X*(t) = X(t) - bt, which is a diffusion with p*(x) = p(x) - b and N.L. Hjort, R.Z. Khasminskii / Time along a line 234 the same a(x)‘. The previous result is valid for the time X*(t) spends along the horizontal line x* = a. We need Q(Y) = exp[-jz 2{p(x) - b}/a2(x) dx] as well as co k+(a, b) = I* a U(Y) dy and k-(a, b) = --ui ~,,L,(Y) dy. (2.8) We find the following. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA start at X(0)=x, and suppose one or both of the two Then V, of (1.1) converges in distribution to the mixture Theorem 2. Let the process integrals (2.8) are finite. Vl{X(O)=x}-w(x,a,b)Exp(a(a,b))+{l-w(x,a,b)}& of an exponential a(a, (2.9) and a unit point mass at zero. Here b)=ia2(a){k+(a, b)- ‘+k_(a, b)- ‘} and s,./,(y) dylk+(a, b) ifxza, w(x, a, b) = \I I U(Y) dylk-(a, b) -co are jinite. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI If one of them is injinite, replace the corresponding if both denominators ratio with ifx s a, 1. 2.3. Example: Let us apply Total time for Brownian the general theorem motion to the case of X = W , the standard Brownian motion process, which has p(x) = 0 and a(x) = 1. We allow an arbitrary starting point W( 0) =x. Take b positive and consider the total relative time V, of (1.1). Then VJW(O)=x) + v- Q(b) if ~?=a, e-2h(a-*) Exp( b) + { I- if x s a. eP2h(a- x)}&, (2.10) There is a symmetric result for negative b, involving an exponential with parameter Exp((bj) when the starting point is a. An when lb1 . Notice in particular that V b = 0 then V is infinite with probability one; see Section 6.1 for a more informative result. 3. Moment convergence proof In the following we stick to the Brownian motion, and for simplicity take it to start at W (0) = 0. For b # 0, let us consider m I{ W(t) E bt +$E} dt zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE V,(b) =t I0 N.L. Hjort, R.Z of (1.2) in more detail. Khasminskii / Time along 235 a line That V(b) Ev(lbl) V,(b) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA +d is already a consequence of the general theorem, and indeed above. We now offer a different proof, by demonstrating of all moments. This is sufficient since the exponential by its moment sequence. lends itself more easily Section 4. a special appropriate distribution (3.1) case of (2.10) convergence is determined In addition to having some independent merit this proof to the study of simultaneous convergence aspects; see For the first moment, that zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM observe u El’,(b) Pr{bt-i&d =$ W(t)Gbt+is}dt I” = I0 “L(br) where A(x) = +(x/~“?)/Ji goes to j: is the density dt = l//b/. $(bJi)/fi (3.2) dt+O(E), function Next consider for W (t). Accordingly the pth moment. Elf,(b) One finds FV,(b)” 1” m =-Ic” = o -..I p! . . i where f;, Gaussian when ,..., ,,,(T, Pr{W(t,)Ebt,*ie,..., . ft I . . . , x,,) and Markovian W(f,)EbS,f~E}dt,...dtr, 0 I,..., r,,W, . . ., 4,) dt,. . .dt,+O(&), r,<...<r,, is the density function of ( zyxwvutsrqponmlkjihgfedcbaZYXW W(t,), . . . , W (t,,)). By the properties of W ( .) this density can in fact be written t, <. . . < tp. To carry out the p-dimensional for (x,, . . . , x,,), and transform to new variables integration, insert (bt, , . . . , bt,,) u, = t, , Ui = ti - t,_, for i = 2, . . . , p. The result is then that EV,(b)“+p! 4(bJu,). ~ 1 for eachp. But this is manifestly the moment . .4(bJu,) vfiq sequence dul . + .du,, =p!(l/jbOP of Exp(lbJ), proving (3.1). 0 The case of b = 0 is different, since W spends an infinite amount of time along the time axis. An interesting ]N(O, 1)j limit result for the relative time in fe during [0, T] is in Section 6.1 below. N.L. Hjorr, R.Z. Khasminzkii / Time along a line 236 zyxwvutsrqponmlkjihgfedcbaZYX 4. The exponential process We have seen that V,( 6) goes to an exponentially manner we should find bivariate and distributed multivariate and in the same V(b), exponential distributions by considering two or more b’s at the same time. This requires verification of simultaneous convergence in distribution of ( V,(b), V,(c)) and similar quantities. This section indeed demonstrates process some of the properties of the limiting convergence process. of V,( . ) to V( . ), and studies 4.1. Process convergence The first main Theorem. result is as follows. There tially distributed in distribution is a well- defined marginals stochastic process and with the property paths, and V, (. ) converges on the C- space C[ bO, b,] of continuous functions not containing b # 0) with exponen- . , . , V( b,)) is the limit . . . , V,( b,)) for each finite set of non-null indexes b,. There of (V,(b,), exists a version of V with continuous topology V = { V(b): that ( V( b,), 1) in the uniform to V( on [ bO, b,], for each interval zero. Proof. Consider two rays bt and ct and their associated total relative time variables Using the Cramer-Weld theorem in conjunction with the moment convergence method we see that convergence of EV,( b)“V,( c)” to the appropriate limit, for eachp and 4, is sufficient. But this can be proved by slight elaborations on the techniques of Section 3. By Fubini’s theorem, (V,(b), V,(c)). 1 =-_ I{ W (s,) E bs, *is,. FP+q W (t,)Ect,*$e and its expected value is seen to converge ,..., . . , W (s,,) E bs, +fe, W (t,)~ct&}ds,~~~dt~, to EV(b)PV(c)q cs =I 0 4 mf; ,,...,.s,,,,,,...,r,,(bs,, 0 . . , bsp, ct,, . . . , ctq) ds,. * .dt, (4.1) by Lebesgue’s theorem on dominated convergence. Note that the integral is over all of [0, oO)p+y and that a simple expression like (3.3) for the density of ( W (s,), . . , W ( tq)) is only valid when the time-points are ordered, so the factual integration in (4.1) is difficult to carry through (but possible; see Section 4.3 below). What is important at the moment is however the mere existence of this and other similar limits of product moments for the V, (. )-process. We may conlude that all finite-dimensional distributions converge to well-detined limits. That these tinite-dimensional distributions also constitute a Kolmogorov-consistent system is a by-product of the tightness condition verified below. N.L. Hjort, R.Z. Khasminskii / Time along a line 231 The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA V,( - )-process has continuous paths in b # 0 for each E, since W( . ) is continuous. In order to prove process convergence on C[b,, of the { V, (. )} f amily as E if Vt( b) = V,( -b), then the processes V$( . ) and V,( . ) have characteristics, so it suffices to consider the positive part of we need to demonstrate results in Shorack lim:_ip tightness and Wellner E{ V,(b+ (1986, p. 52) it is enough h) - V,(b)}4s for all h 2 0 and for all b with b and Kh* b,] for a given interval goes to zero. Note that identical distributional the process. Following to verify that for some K, (4.2) 0 < b,< b, . By the b + h in [b,, b,], where arguments for finite-dimensional convergence used above the left hand side of (4.2) is equal to m,(h) = E{ V(b + h) - V(b)}4. This is seen to be a smooth function of h with finite derivatives arguments can in fact be furnished where at zero. Ingenious and rather elaborate Taylor expansion to prove that 6 = h/b, so that m,(h) = K,(b)h*+ K3(b)h3+* . ., for local constants K,(b) that are continuous functions of b (as long as b # 0). This is dominated by a common Kh2 for all b and b+ h in the interval under consideration. This verifies (4.2), and incidentally at the same time verifies the so-called sure continuity of the sample paths, Kolmogorov see Shorack condition and Wellner for almost (1968, Chapter 2, Section 3). Using the moment formula in Section 4.3 below one may in fact calculate the left hand side of (4.2) explicitly, and a fair amount of analysis leads to m,(h) = 24.352h2/b6+O(h3). this level of detail, 4.2. Dependence The proof above however. 0 circumvented the need for information on structure In order to investigate this to some extent we calculate Let O<b<c and - c<O<d. Then Cov{V(b), WI=; & covariances and correlations. (4.3a) and Cov{ V(- c), V(d)} =$ A+; A- ;. (4.3b) N.L. Hjort, R.Z. Khasminskii 238 To prove this, consider / Time along a line the case of two positive parameters. Then by previous arguments m EV,(b) V,(c) = ‘x Pr{W(s)Ebsf$a, JJ0 + JJ S<, M,,(k ct) +fF,r(c~, bt)l ds dt .+-Ads where (3.3) (x, y) = (A, The rest 1*/y *)} is used J;;), of the dy = $fi( tary integrations, again. W(t)~ctf$~}dsdt/e* 0 Now dt, first to (s, U) = (s, t - s) and transform then to to get calculation is carried out using 1/ k) exp( - kl). This formula and is valid for positive k the formula j: exp{-i( zyxwvutsrqponmlkj k2y 2 + can be proved by clever but elemenand 1. One finds 1 -_ -exp(-f(b*+4c(c-b))x2}+iexp(-fc2x2) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR =I’+11 c2c- b The first similarly. formula bc’ in (4.3) follows from this, and the other case is handled form, Then using (b, c) = (b, b + h) 0 It is convenient to give formulae (4.3) in another in the first case and (-c, d) = (-c, kc) in the second. and Cov{V(- c), and the correlation corr{V(b), V(kc)}=l- coefficients V(b+h(}=& -3 c2 (k+2)(2k+ 1) ’ become (4.4a) N.L. Hjorr, R.Z. Khasminskii / Time along a line 239 and corr{V(-c), For small 3k V(kc)I=-(k+2)(2k+1). h it is worth noting E{V(b+h)- (4.4b) that V(b)}=&-$+I, 2 4 E{V(b+h)-V(b)}2=L+-b* (b+h)2 b(b+2h)% + 4.3. Bivariate and multivariate We have constructed exponential a full exponential 4 zyxwvutsrqponmlkjihgfedcbaZ distributions process, and in particular ( V(b,), . . . , V( 6,)) is a random vector with dependent and exponential marginals. These bivariate and multivariate exponential classes of distributions appear to be new. See Block (1985), for example, for a review of the field of multivariate exponential distributions, and see Section 6.5 below for a couple of other processes with exponential marginals. Formula (4.4) shows that if values p, > 0, p2 > 0, p E (0, 1) are given, then a pair of dependent exponentials ( V( b,), V( b2)) can be found with EV(b,) p2, and correlation p. The class of bivariate exponential distributions = p,, EV(bJ rich in the sense of achieving all positive correlations. The negative correlation (4.4) starts out at zero for k small, decreases to -f for k = 1, and then climbs towards zero again when k grows, so negative correlations between -f and cannot be attained. Note that the maximal and V(- b). In order to study the bivariate moment sequence negative distribution (4.1) explicitly, correlation for ( V(b), V(c)) occurs between we = is accordingly calculate for the case of 0 < b < c. The technique in up -1 V(b) its double is to split the integral into n ! = (p + q) ! parts, corresponding to all different orderings of the n = p + q time indexes, the point being that a formula like (3.3) for the density of ( zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA W (t,), . . . , W (t,)) can be exploited for each given ordering. These orderings can be grouped into (i) types of paths, say (e, t,, . . . , e,,t,,) where t, <. . . < t, and ej is equal to b in exactly p cases and equal to c in exactly q cases. There are p!q! different paths for given locations for the p b’s and q c’s, so the full integral can be written C p!q!g(path), where the sum is over all (i) classes of paths and g(path) is the contribution for a specific path of the appropriate type. It remains to calculate the g-terms of various types, i.e. to evaluate ... I I o<r,<...<,,, .L,,...,l,,(eltl,. . . , en&,) dt,. . .dt, for a path with e,‘s equal to b or c. Stameniforous integrations, similar strenuous than those used to prove (4.3), show in the end that g(path)=(~)i(“l(~)i”‘. . .(?-)““‘, to but more (4.5) N.L. Hjort, R.Z. Khasminskii / Time along a line 240 where the path when read backwards, i.e. looking through (e,, . . . , e,) in the notation above, has i(0) b’s first, then i(1) c’s, then i(2) b’s, etc. Furthermore b,,= b, b, = c, and b,= b+j(c-b). i(l)+. . * + i(n) = n. And g(path) terms. To illustrate types Note that of is equal EV(b)pV(c)y this somewhat * .=p, i(O)+i(2)+. cryptic .=q, i(l)+i(3)+*. and i(O)+ to p! q! times the sum of all such formula, try EV( b)2 V( c)‘. There are ($ = 6 paths, corresponding to (6, b, c, c), (b, c, b, c), (b, c, c, b), (c, b, b, c), (c, b, c, b), (c, c, b, b), and each of these has weight 2!2! = 4. Their are respectively (0,2,2,0,0), (0, 1, zyxwvutsrqponm 1, zyxwvutsrqponm 1, l), (i(O), i(l), . . . , i(4)) representations (1,2, l,O, 0), (0, 1,2, l,O), (1, l,l, l,O), (2,2,0,0,0>. Accordingly 1 EV(b)2V(c)2=4 bfb;+ 1 b,b2b3b4+ b,bfb,+ where b,= b, b, = c,. . . , bq= b+4(c-b). We have not been able to produce an explicit 1 - 1 b,b,b2b3+ bib; ~ b,b;b,+ formula density of ( V(b), V(c)), but at least an expression generating function. It becomes E exp{sV(b)+ 1 - for the joint can be found ’ probability for its joint moment tV(c)} hoop+& (;) = n- s”t4E{ V(b)pV(c)q} . =n;,+l,+~~n&lsptq . . = c sptq p=o,qzo C i(Oj,i(l~,,i(p+qj($) where again the inner sum is over all 0, + i( 0), i( 1 ), the multiplicities above. One p!q!dpath) paths ., i(p + q) q) i(‘)’ ’ * (&) ““‘(i) ! /p!q! types of paths with have even-sum ‘ip+q)’ p (4’6) b’s and q c’s, and p and odd-sum q, as explained can similarly establish formulae for product moments of more than two investigate other aspects of the multivariate exponential distributions V( b)‘s, and associated with the V( .) process. We remark that these distributions can be simu- lated, with some effort, through using V,( b jzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON with a small E, and this is one way of computing bivariate and multivariate probabilities when needed. Another way would be via numerical inversion of the joint moment generating function. zyxwvutsrqponmlkjihgfedcba 5. Total relative As a generalisation w = bt during time along a line after time c of (1.2), consider the total relative time spent along the ray t 2 c, i.e. I{bt- ;E< W (t)sbt+&}dt. (5.1) 241 zyxwvutsrqponmlkjihgfedcbaZYXWVUTS N.L. Hjort, R.Z. Khasminskii / Time along a line The story told in the final paragraph of Section these variables. them is that The main result about 1 is one motivation for studying UC, b)-k(l+h) E~P(I~J)+(~-~(I~I~))S,, (5.2) VF(~, b, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB +d where again &, is degenerate at zero and k(u) = 2(1- Q(u)). Note that k((b(&) when c = 0, so that (5.2) indeed contains our earlier result (3.1). = 1 It is possible to prove this by establishing a differential equation for the Laplace transform of V(c, b) with appropriate boundary conditions, and then solve, as in Section 2, but it is as convenient to prove moment simplicity. It takes one moment to show that convergence. Take b>O for EV,(c, b) -+f,(b)Jcc =-I Cc zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC dt = 4(bfi)/J?dt c zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI Ic 1 b cr 244.x) dx = k(b&)/b. hJ; And when p 2 2 we find +p! ... J J =p!Jc;i J$: (‘__I~_ .c., s J A,,.._, r,,W,>. . . , 4,) dt, . . .dtp ;m... WJu,) Ju, 0 1 pp’ k(b&) ZZp!bb The Laplace 0 transform function ,...du, . (5.3) of this limit distribution m (-h)‘pp,k(b&) E exp( -A V) = 1 + C ,,=, p! =l+k(b~)-----= which is recognised du G as the moment ’ 1 b -‘lb l+hlb generating candidate becomes p-’ 0b k(b&) function A+ 1 - k(bv”-& of the mixture with probability k(b&) is an exponential with parameter 1 - k(b&) is equal to zero. This proves (5.2). b variable that and with probability Remark. Let us briefly discuss a specific consequence, namely that Pr{ V,(c, b) = 0} in this situation converges to Pr{ V(c, b) = 0}, which is 1 - k(b&) = 2@(b~‘?) - 1. But having VF(c, 6) = 0 in the limit means that W (t) stays away from bt during [c, a), and it cannot stay above the curve all the time since W (f)/t goes to zero. Hence 2@(b&) - 1 is simply the probability that W (t) < bt during all of [c, ~0) or 242 N.L. Hjort, R.Z. Khasminskii / Time along a line Pr{max,,,. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA W (t)/ t < b}. Using finally the transformation W *(t) = t W ( 1/ t) to another Brownian motion one sees that (5.4) We have in other words rederived The distribution lem discussed a classic distributional result for Brownian of V(c, - 1) - V(c, 1) comes up in the statistical in Section Section 6). When c = 0 this is a difference between two unit intercorrelation -5. The case c > 0 is more complicated. Then (0 (V(c, -l), estimation 1; see also Section 6.3 below and Hjort and Fenstad (U-190) V(c, 1)) = (o, u,) (Up,, U,) exponentials with probability no,, , with probability v,~, with probability 7~~~) with probability r,, , motion. prob(1992b, with (5.5) in which U_, and U, are unit exponentials with a certain dependence structure. that W (t) stays between --t and t during [c, co), Furthermore roO is the probability n,,, is the probability that W (t) comes below --t but is never above t, rO, is the probability that W (t) comes above f but is never below -t, and n,, is the probability that W (t) experiences both W (t) < - t and W (t) > t during [c, 03). When c = 0 then r,, is 1 and the others are zero. In the positive case these probabilities can be found in terms of H(u), the probability that maxO~.Y~, 1W (s)/ c u, by the transformation arguments used to reach (5.4). One finds 7r“O=H(&), ~o,=7r,o=2@(~)-1-7roo, = Pr{maxo~ps, %-,,=l-Tr00-7r01-Tr10, ) W (s)\ s u}. A classic alternating series expression in which H(u) for H(u) example, can be found in Shorack and Wellner (1986, Chapter 2, Section 2), for and a new way of deriving this formula is by calculating all product moments zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA EV(-C)~ V(C)~ and then study the analogue of (4.6). This would be analogous to the way in which (5.4) was proved above, but the present case is much more laborious. Here we merely EV(c, -l)V(c, from which 6.1. l)=?r,,EU_,U,=~k(3&), the correlation 6. Supplementing note that between CL, and (I, also can be read off results Total relative time along the time axis The variable V,(b) of (1.2) is infinite when b = 0. But consider 7 {-+ES W (t)<&} dt, (6.1) N.L. Hjort, R.Z. Khasminskii / Time along a line the relative time along the time axis during as E + 0 and T + 03, as follows, using 243 [0, T]. The moment sequence converges (3.3) once more: I E(V,,TY = g$2 ... i i O<ll<...<l”<T [ cw~* p+l)] dr, . . . dt,, * .&-$fO(R J...JO<X,<...<X,,<l -I”. . Xl +,p! 4(OjP zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED ~(xP-xP_,)-“2(1-xP)odx,~ p! =---z (2n) The limit distribution I-(;)pr(l) 1 T(ip+l) candidate V, p! 2P’2 T(fp+l)’ has consequently EVF = ($jp(2p)!/p!, which means that Vz gets moment generating function (1 - 2tjP”‘. So Vi is a ,Y: (since the distribution of a chi-squared is determined by its moments), i.e. V, is a \N(O, l)]. It was not necessary here to send T to infinity, = W ( ct)/v? gives a new Brownian motion) ( W *(t) of T. of V,,i- as E + 0 is independent One generalisation of this is in the following h(W(t)/~) dr= $ since the scaling property for W implies that the limit distribution direction. Instead j-7”’ I?( W*(r)) of (6.1), look at dr, where h(x) is any function with bounded support, and where W * in the second expression is another Brownian motion obtained from the first one by transformation. The case considered earlier is h(x) = 1(1x1 c$}. It can be shown that V,,, + alN(0, lj1 in distribution as T/E’+ CO, where a is a constant depending on h. This is not easy to prove via the moment convergence technique, but can be established using methods from Khasminskii (1980). 6.2. Implications for partial sum processes Let us first point out that an alternative construction of our total relative time variables is to use I{bt G W (t) G bt + E} instead of I{ W ( t) E bt *$E} in (1.2) and (5.1). Results of previous sections hold equally for this alternative definition of V,(b) and V,(c, b), and this is a bit more convenient in Section 6.3 below. Now suppose zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA X, , X2, . . . are i.i.d. with mean 5 and variance 02, and consider the normalised partial sum process W ,,,(t) = m- “2 CL”:’ (X, -5)/a. In particular Y, = (X, -&)/CT and S, for their partial sums, and = WJ- m, writing W ,( *) converges to Brownian motion by Donsker’s theorem. Motivated by (1.2) w,(nlm) 244 N.L. Hjorr, R.Z. Khasminskii / Time along a line and (5.1) we define (6.2) where (cm) denotes the smallest integer exceeding or equal to cm. It is clear that this variable is close to V,(c, b) for large m, and should accordingly converge in distribution to V( c, b) of (4.2) when m + ~0 and .Z+ 0. zyxwvutsrqponmlkjihgfedcbaZYXWVUT Assume that the Xi’s have a$nite third absolute moment. If c > 0 isjixed, V(c, b) ifonly e(m)+0 as m+O. And then V,+,Jc, b) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB ‘d Proposition. V,,,C,l(c(m),b) provided e(m)+O, +d c(m)+O, V(b)-Ew(lbl) mc(m)+a, (6.3) and e(m)/c(m)“2-+0. Proof. This can be proved in various ways and under various conditions. One feasible possibility is to demonstrate moment convergence of E{ V,,,,,,(c, b)}” towards the right hand side of (5.3), for each p. One basically needs the smallest n in the sum to grow towards infinity, so that the central limit theorem and Edgeworth- Cram& expansions can begin to work, and the largest of all e(m)terms to go to zero, so that Taylor expansions can begin to work; see the middle term in (6.2). When c is fixed then the sum is over all n 2 mc, and it suffices to have F(m) + 0 as m + ~0. To reach V(b) = V(0, b) in the limit we need the stated behaviour for e(m) and c(m). We have used the third moment assumption to bound the error r(t) in the Edgeworth expression G,(t) = B(t) + r( t) for the distribution of T,, = and this is helpful when it comes &?(J?,-[)/u; one has Ir,(t)l~~n-“~/(l+lt1)~, to verifying convergence. conditions when employing Lebesgue’s theorem on dominated 0 We may conclude that the total relative time along bn/m for the normalised partial sum process has a limit distribution, which is either exponential or of the mixture type (4.2). The middle expression also invites V,,,, to be thought of as the total relative time for the normalised T, process along the square root boundary bm. The result is also valid for T,, = A( 0, - 0)/a in a more general estimation theory setup; see Hjort and Fensatd (1992a, 1992b). The result of Section 6.1 has also implications for partial sum processes. One can prove that N.L. Hjort, R.Z. Khnsminskii / Time along a line when E + 0 and m-1’2CL, m +a, 1{lS,l~$} the random under suitable conditions. has th e a b so 1u t e normal 245 This implies for example that limit, as does rn-“* x7=, Z{Si = 0} for walk process. 6.3. Second order asymptotics for the number To show how the total relative time variables of S-errors for Brownian motion are related to the estimation theory problems described in Section 1, consider the structurally simple case of i.i.d. variables X, with mean [ and standard deviation u, and where (n/ (n + k))X,, is used to estimate 8. Consider Q8 (k), the number of times I( n/ (n + k))X,, - 512 6, counted among n 2 c/S2. Then 6*Q,(k) tends to Q = Q(c, l/a) of (1.2), for each choice of k, and 6* times Q,(k) - Ql(O) goes to zero. This follows from results in Hjort and Fenstad (1992a). But 6{Q,(k) - Q8(0)} can be written A, - Bg, after some analysis, where dt A,=& and where m m = l/6*. With F = m-“2k[/c These variables resemble (T Jmu those considered mu ’ in (6.2) and (6.3). we have A8 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA +d W~V ,,,t c , -l/a ) a nd 4 & =d WuV m ,,(c , l/a ), where ‘&;d’ signifies that the difference the result of Section 6.2 that S{Q,(k) goes to zero in probability. It follows from - %(O)) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG +d Q/4 Vc, -l/a) W C, l/a)} as 6 + 0. (6.4) This is also true with c = 0 in the limit, i.e. with k[/a{ V(-l/a)V(l/a)} on the right hand side, provided c = c(6) = 6 is used in the definition of Q,(k) and Q,(O). Note the relevance of (5.5) for the present problem. Hjort and Fenstad (1992b) also work with the direct Q(O) and similar variables. These converge to explicit expected functions value of Q8( k) of k (and other parameters, in more general situations), which can then be minimised to single out estimator sequences with the second order optimality property of having the smallest expected number of s-errors. This is done in Hjort and Fenstad (1992b), in several situations. We remark that the skewness y = E(X, - [)‘/(r3 is not involved in (6.4), but is prominently present in the limit of E{ Q8( k) - Q8(0)}, and its minimisation. 6.4. Relative To generalise time along other curves our framework, consider (6.5) 246 N.L. Hjort, R.Z. Khasminskii / Time along a line where x = b(s) is some curve of interest cases there is a distributional and g(t) a possible limit as E + 0, and perhaps scaling factor. the first couple In many of moments can be obtained. The limit distribution is simple only for cases that can be transformed back to (1.1) and (5.1), however. For an example, we note that the total relative time an Ornstein-Uhlenbeck to be exponential, 6.5. for example, Other exponential process X(t) spends with suitable g(t) in (6.5). and gamma along be’ can be shown processes (i) If U(b) = lb\ V(b), then U(b) is unit exponential for each b. In particular its marginal mean and variance are constant, and Cov{ U(b), U( b + h)} = b/( b + 2h). (ii) By adding independent copies of V( . ) (or U( . )) zyxwvutsrqponmlkjihgfedcbaZYXWV we get processes with marginals that are gamma distributed. This leads in particular to bivariate and multivariate gamma distributions or chi-squared distributions (with even-numbered degrees of freedom only). (iii) There are other processes that share with V and U the property exponentially distributed marginals. One example is V*(b) = f{ W ,(b)‘+ of having W ,(b)*}, where W , and W2 are independent Brownian motions. This is a Markov process, while our V(b) process is not. The possible correlations of (V*(b,), , . . , V*(b,)) span a smaller space than it those of (V(b,), . . . , V(b,)), indicating that the V” process may be less adequate when it comes to building multivariate exponential models. (iv) And yet another process with exponential marginals is provided by V**(b) = max,,,{ W (t) - bt}, defined for positive b. This process is studied by Cinlar (1992). If X(t) = W (t) - bt, then Cinlar’s paper is concerned with the maximum this process and where the maximum occurs, whereas the present paper concerned with the amount of time such a process spends along value of has been a line. Acknowledgements This paper was written at the Mathematical Sciences Research where we spent some pleasant weeks as invited participants Programs. This was made possible through generous support Foundation grant 8505550. Institute at Berkeley, in the 1991 Statistics from National Science References H.W. Block, Multivariate exponential distributions, in: S. Kotz and N.L. Johnson, Encyclopedia of Statistical Science (Wiley, New York, 1985). E. Cinlar, Sunset over Brownistan, Stochastic Process. Appl. 40 (1992) 45-53. N.L. Hjort and G. Fenstad, On the last time and the number of times an estimator is more than E from its target value, Ann. Statist. 20 (1992a) 469-489. N.L. Hjort, R.Z. Khasminskii / Time along a line 247 N.L. Hjort and G. Fenstad, Some second order asymptotics for the number of times an estimator is more than E from its target value, Statist. Res. Rept., Univ. of Oslo (Oslo, 1992b). K. It6 and H.P. McKean Jr., Diffusion Processes and their Sample Paths (Springer, Berlin, 1979, 2n ed.). S. Karlin and H.M. Taylor, A Second Course in Stochastic Processes (Academic Press, Toronto, 1981). R.Z. Khasminskii, Stability of Stochastic Differential Equations (Sijthoff and Noordhoff, Amsterdam, 1980). G.R. Shorack and J.A. Wellner, Empirical Processes with Applications to Statistics (Wiley, New York, 1986).