%!PS (but not EPSF; comments have been disabled) %DVIPSCommandLine: dvips -f spencerf.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1996.12.31:1001 /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X 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b(deg)q(\()p Fm(v)r Fu(\))d Fk(\030)h Fm(D)i Fu(for)f(ev)o(ery)h Fm(v)h Fu(and)f(an)o(y)f(t)o(w)o(o)f(v)o(ertices)i Fm(v)r(;)8 b(w)16 b Fu(ha)o(ving)h Fm(o)p Fu(\()p Fm(D)q Fu(\))f(common)g(edges)h (has)f(a)h(pac)o(king)14 2262 y Fm(P)22 b Fu(with)15 b Fk(j)p Fm(S)s Fk(j)c Fu(=)i Fm(o)p Fu(\()p Fm(n)p Fu(\).)20 b(\(Here)14 b Fm(k)i Fu(is)g(\014xed,)f Fm(N)r(;)8 b(D)13 b Fk(!)g(1)p Fu(.\))20 b(Recen)o(t)15 b(w)o(ork)f(has)h(cen)o(tered)h (on)f(lo)o(w)o(ering)g(the)g(size)h(of)14 2318 y Fk(j)p Fm(S)s Fk(j)e Fu(in)i(terms)f(of)f Fm(D)q Fu(.)20 b(Our)c(main)f (result)h(has)f(also)g(b)q(een)h(sho)o(wn)f(\(indeed,)i(without)e(the)g (logarithmic)h(term)f(for)14 2375 y Fm(k)f Fk(\025)f Fu(3\))h(in)i(our)f(join)o(t)g(pap)q(er)h([1)o(])f(b)o(y)g(quite)h (di\013eren)o(t)g(tec)o(hniques.)p 14 2414 780 2 v 66 2441 a Fh(1)83 2457 y Fg(AMS\(1991\))e(Sub)r(ject)g(Classi\014cation:) 20 b(Primary)14 b(05)p Ff(B)r Fg(40,)f(Secondary)i(60)p Ff(D)q Fg(05)p eop 2 1 bop 14 162 a Fe(the)17 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Fo(\000)p Fs(1)318 526 y Fk(\000)13 b Fm(e)387 533 y Fn(i)401 526 y Fu(,)19 b Fl(else)f Fu(k)o(eep)h Fm(P)660 533 y Fn(i)693 526 y Fu(=)h Fm(P)777 533 y Fn(i)p Fo(\000)p Fs(1)852 526 y Fu(and)f Fm(S)972 533 y Fn(i)1005 526 y Fu(=)g Fm(S)1087 533 y Fn(i)p Fo(\000)p Fs(1)1143 526 y Fu(.)31 b(That)19 b(is,)h(consider)f(the)h(edges)f(in)h(random)14 582 y(sequen)o(tial)g(order)f(and)g(add)h(eac)o(h)f(to)f(the)h(pac)o (king)h(if)f(y)o(ou)g(can.)32 b(W)l(e)19 b(conjecture)g(that)g Fm(E)s Fu([)p Fk(j)p Fm(S)1702 589 y Fn(!)1724 582 y Fk(j)p Fu(])f(meets)h(the)14 639 y(b)q(ounds)14 b(of)f(our)g(Theorem.) 19 b(This)13 b(author)g([6)o(])g(and,)g(indep)q(enden)o(tly)m(,)j(V.)d (R\177)-23 b(odl)14 b(and)f(L.)g(Thoma)g([4)o(])g(ha)o(v)o(e)f(sho)o (wn)14 695 y(that)i Fm(E)s Fu([)p Fk(j)p Fm(P)204 702 y Fn(!)226 695 y Fk(j)p Fu(])e Fk(\030)333 677 y Fn(N)p 317 684 61 2 v 317 711 a(k)q Fs(+1)396 695 y Fu(or)i(equiv)m(alen)o (tly)i(that)e Fm(E)s Fu([)p Fk(j)p Fm(S)891 702 y Fn(!)913 695 y Fk(j)p Fu(])e(=)h Fm(o)p Fu(\()p Fm(N)5 b Fu(\).)18 b(View)o(ed)d(in)g(this)g(ligh)o(t)g(w)o(e)e(are)h(no)o(w)g(lo)q(oking) h(at)14 752 y(a)g(second)g(order)g(term,)f(just)g(ho)o(w)h(close)g(to)f (a)h(\\p)q(erfect)g(pac)o(king")g(can)g(w)o(e)g(get.)k(Unfortunately)l (,)c(this)g(natural)14 808 y(algorithm)e(has)g(eluded)i(more)d (re\014ned)i(analysis.)20 b(W)l(e)13 b(feel)h(it)f(w)o(ould)g(b)q(e)h (most)e(in)o(teresting)i(ev)o(en)f(to)f(pro)o(v)o(e)h(that)14 864 y(the)i(exp)q(onen)o(t)h(of)f Fm(D)h Fu(is)g(the)f(correct)g(one,)g (that)574 967 y Fm(E)s Fu([)p Fk(j)p Fm(S)665 974 y Fn(!)688 967 y Fk(j)p Fu(])d(=)h Fm(O)q Fu(\()p Fm(N)5 b(D)909 948 y Fo(\000)p Fs(1)p Fn(=k)q Fs(+)p Fn(o)p Fs(\(1\))1072 967 y Fu(\))236 b(\(??\))502 b(\(1\))85 1069 y(No)o(w)17 b(w)o(e)h(de\014ne)h(the)f Fl(r)n(e)n(altime)h(r)n(andom)g(gr)n(e)n(e)n (dy)f(algorithm)p Fu(.)29 b(W)l(e)18 b(let)g(time)h Fm(t)f Fu(go)g(con)o(tin)o(uously)h(starting)14 1125 y(from)d(zero.)24 b(The)17 b(pac)o(king)h Fm(P)j Fu(=)16 b Fm(P)635 1132 y Fn(t)666 1125 y Fu(will)j(v)m(ary)d(with)h(time)g(as)g(will)h Fm(S)1239 1132 y Fn(t)1269 1125 y Fu(=)d Fm(V)21 b Fk(\000)1414 1093 y Fj(S)1456 1125 y Fm(P)1485 1132 y Fn(t)1500 1125 y Fu(.)j(W)l(e)17 b(let)g Fm(H)1722 1132 y Fn(t)1753 1125 y Fu(denote)g(the)14 1182 y(restriction)g(of)f Fm(H)k Fu(to)c Fm(S)429 1189 y Fn(t)443 1182 y Fu(.)24 b(If)16 b(b)o(y)h(time)f Fm(t)h Fu(edge)g Fm(e)e Fk(\022)g Fm(S)947 1189 y Fn(t)978 1182 y Fu(has)h(not)g(y)o(et)g(b)q(een)i(b)q(orn)e (then)h(it)g(is)g(b)q(orn)f(in)i(the)e(next)14 1238 y Fm(dt)g Fu(with)f(probabilit)o(y)i Fm(e)427 1222 y Fn(t)455 1220 y(dt)p 447 1227 47 2 v 447 1254 a(k)q(D)498 1238 y Fu(.)j(When)c Fm(e)g Fu(is)g(b)q(orn)f(it)h(is)g(added)g(to)f Fm(P)6 b Fu(.)20 b(In)c(particular,)g(all)h Fm(e)1569 1222 y Fo(0)1596 1238 y Fu(with)e Fm(e)1720 1222 y Fo(0)1742 1238 y Fk(\\)c Fm(e)i Fk(6)p Fu(=)g Fk(;)i Fu(are)14 1294 y(no)g(longer)h(considered.)85 1351 y(Observ)o(e)d(that)f(the)h (edges)g(are)g(b)q(eing)h(b)q(orn)f(in)h(a)e(random)h(order.)18 b(Th)o(us)13 b(if)h(w)o(e)e(con)o(tin)o(ue)i(this)f(pro)q(cess)g(un)o (til)14 1407 y Fm(H)j Fu(has)d(no)g(edges)g(the)f(distribution)j(of)d Fm(S)j Fu(will)g(b)q(e)e(precisely)i(that)d(of)g(the)h(discrete)g (random)g(greedy)f(algorithm.)14 1464 y(It)i(will)i(b)q(e)f(more)f(con) o(v)o(enien)o(t,)g(ho)o(w)o(ev)o(er,)f(to)h(stop)g(the)g(pro)q(cess)g (at)g(time)g Fm(!)h Fu(=)e(ln)8 b Fm(D)q Fu(.)20 b(W)l(e)14 b(no)o(w)f(giv)o(e)i(a)f(heuristic)14 1520 y(guide)h(whic)o(h)g(should) g(motiv)m(ate)f(the)g(full)h(pro)q(cess)f(w)o(e)g(de\014ne)h(later.)k (Let)c(deg)1373 1531 y Fn(t)1388 1520 y Fu(\()p Fm(v)r Fu(\))e(b)q(e)h(\(for)f Fm(v)i Fk(2)e Fm(S)1715 1527 y Fn(t)1729 1520 y Fu(\))h(the)g(degree)14 1577 y(of)h Fm(v)i Fu(in)f Fm(H)196 1584 y Fn(t)226 1577 y Fu(and)f Fl(supp)n(ose)g Fu(all)h(deg)611 1588 y Fn(t)626 1577 y Fu(\()p Fm(v)r Fu(\))c Fk(\030)h Fm(f)5 b Fu(\()p Fm(t)p Fu(\))p Fm(D)q Fu(.)20 b(There)15 b(w)o(ould)h(b)q(e)g Fk(\030)d Fm(k)q(f)1323 1560 y Fs(2)1342 1577 y Fu(\()p Fm(t)p Fu(\))p Fm(D)1433 1560 y Fs(2)1466 1577 y Fu(pairs)j(\()p Fm(e;)8 b(e)1659 1560 y Fo(0)1670 1577 y Fu(\))14 b(where)i Fm(e)f Fu(is)h(an)14 1633 y(edge)f(con)o(taining)g Fm(v)g Fu(and)g Fm(e)483 1617 y Fo(0)508 1633 y Fu(is)g(an)f(edge)h(in)o (tersecting)g Fm(e)p Fu(,)f(but)g(not)g(at)f Fm(v)r Fu(.)19 b(Eac)o(h)14 b Fm(e)1414 1617 y Fo(0)1440 1633 y Fu(is)h(b)q(orn)f(in)h (the)f(next)h Fm(dt)f Fu(with)14 1690 y(probabilit)o(y)j Fm(e)268 1673 y Fn(t)296 1672 y(dt)p 288 1679 V 288 1705 a(k)q(D)355 1690 y Fu(and)f(if)g(b)q(orn)g(diminishes)i(deg)q(\()p Fm(v)r Fu(\))d(b)o(y)g(one)h(for)f(eac)o(h)h(\()p Fm(e;)8 b(e)1362 1673 y Fo(0)1373 1690 y Fu(\).)21 b(\(If)16 b Fm(e)f Fu(itself)i(is)f(b)q(orn)g(then)g Fm(v)i Fu(is)14 1746 y(remo)o(v)o(ed)c(from)f Fm(H)t Fu(.\))19 b(On)14 b(a)o(v)o(erage)f(deg)q(\()p Fm(v)r Fu(\))g(is)i(decreased)g(b)o(y)f Fm(k)q(f)1133 1730 y Fs(2)1151 1746 y Fu(\()p Fm(t)p Fu(\))p Fm(D)1242 1730 y Fs(2)1261 1746 y Fm(e)1282 1730 y Fn(t)1310 1728 y(dt)p 1301 1735 V 1301 1762 a(k)q(D)1365 1746 y Fu(=)f Fm(e)1434 1730 y Fn(t)1449 1746 y Fm(f)1476 1730 y Fs(2)1495 1746 y Fu(\()p Fm(t)p Fu(\))p Fm(D)8 b Fk(\001)g Fm(dt)p Fu(.)20 b(If)14 b(this)g(is)h(to)e(b)q(e)14 1803 y Fm(f)5 b Fu(\()p Fm(t)10 b Fu(+)h Fm(dt)p Fu(\))p Fm(D)16 b Fu(then)f(w)o(e)g(w)o(ould)h(need)718 1905 y Fm(f)5 b Fu(\()p Fm(t)10 b Fu(+)g Fm(dt)p Fu(\))j(=)g Fm(f)5 b Fu(\()p Fm(t)p Fu(\))10 b Fk(\000)g Fm(e)1108 1886 y Fn(t)1123 1905 y Fm(f)1150 1886 y Fs(2)1169 1905 y Fu(\()p Fm(t)p Fu(\))p Fm(dt)829 2007 y(f)856 1988 y Fo(0)868 2007 y Fu(\()p Fm(t)p Fu(\))i(=)h Fk(\000)p Fm(e)1036 1988 y Fn(t)1052 2007 y Fm(f)1079 1988 y Fs(2)1097 2007 y Fu(\()p Fm(t)p Fu(\))14 2090 y(so)i(that,)g(as)g Fm(f)5 b Fu(\(0\))13 b(=)h(1,)h(w)o(e)g(w)o(ould)h(ha)o(v)o(e)f Fm(f)5 b Fu(\()p Fm(t)p Fu(\))13 b(=)h Fm(e)904 2074 y Fo(\000)p Fn(t)945 2090 y Fu(.)21 b(Indeed,)c(the)f(c)o(hoice)g(of)f (birth)i(in)o(tensit)o(y)f(w)o(as)f(designed)14 2147 y(so)g(that)f Fm(f)5 b Fu(\()p Fm(t)p Fu(\))16 b(w)o(ould)f(ha)o(v)o(e) g(this)h(particularly)g(con)o(v)o(enien)o(t)g(form.)85 2203 y(Supp)q(ose)h Fm(v)h Fu(has)e(surviv)o(ed)h(to)e(time)i Fm(t)p Fu(.)23 b(It)16 b(lies)h(on)f Fk(\030)f Fm(D)q(e)1084 2187 y Fo(\000)p Fn(t)1141 2203 y Fu(edges,)h(eac)o(h)g(is)h(b)q(orn)f (with)h(probabilit)o(y)g Fm(e)1893 2187 y Fn(t)1921 2185 y(dt)p 1913 2192 V 1913 2219 a(D)q(k)14 2260 y Fu(in)d(the)g(next)f Fm(dt)h Fu(so)f Fm(v)i Fu(is)f(remo)o(v)o(ed)f(from)g Fm(S)j Fu(with)d(probabilit)o(y)1097 2242 y Fn(dt)p 1097 2249 30 2 v 1103 2275 a(k)1132 2260 y Fu(.)19 b(The)14 b(probabilit)o(y)g(that)f Fm(v)i Fu(surviv)o(es)f(to)f(time)h Fm(t)14 2321 y Fu(starting)h(at)f(time)h(zero)g(w)o(ould)g(then)h(b)q (e)f(exp)q([)p Fk(\000)861 2287 y Fj(R)888 2297 y Fn(t)880 2335 y Fs(0)915 2304 y Fn(dt)p 915 2311 V 921 2337 a(k)950 2321 y Fu(])d(=)h Fm(e)1044 2305 y Fo(\000)p Fn(t=k)1120 2321 y Fu(.)20 b(Since)d(w)o(e)d(w)o(an)o(t)g(deg)q(\()p Fm(v)r Fu(\))e Fk(\030)h Fm(D)q(e)1698 2305 y Fo(\000)p Fn(t)1753 2321 y Fu(but)i(deg)q(\()p Fm(v)r Fu(\))14 2378 y(is)20 b(in)o(tegral)f(w)o(e)g(can)g(only)h(hop)q(e)f(to)g(carry) f(this)i(appro)o(ximation)f(through)f(time)i Fm(!)h Fu(=)e(ln)9 b Fm(D)q Fu(.)31 b(A)o(t)18 b(that)h(time)14 2434 y(Pr[)p Fm(v)14 b Fk(2)f Fm(S)183 2441 y Fn(t)197 2434 y Fu(])h(w)o(ould)g(b)q (e)g Fm(e)436 2418 y Fo(\000)p Fn(t=k)525 2434 y Fu(=)f Fm(D)612 2418 y Fo(\000)p Fs(1)p Fn(=k)692 2434 y Fu(.)19 b(By)14 b(Linearit)o(y)h(of)e(Exp)q(ectation)h(w)o(e)g(w)o(ould)g(ha)o (v)o(e)f Fm(E)s Fu([)p Fk(j)p Fm(S)1683 2441 y Fn(!)1705 2434 y Fk(j)p Fu(])f(=)h Fm(N)5 b(D)1872 2418 y Fo(\000)p Fs(1)p Fn(=k)1952 2434 y Fu(.)p eop 3 2 bop 14 162 a Fe(the)17 b(electr)o(onic)h(journal)g(of)f(combina)m (torics)f(4)i(\(no.)j(2\))c(\(1997\),)g(#R19)430 b Fu(3)85 311 y(As)15 b(w)o(e)g(said)h(earlier)g(w)o(e)f(are)g(unable)h(to)f(mak) o(e)g(this)g(argumen)o(t)g(rigorous)g(and)g(it)h(is)f(only)h (conjecture)g(that)14 368 y(the)g(result)f(is)h(correct.)k(W)l(e)15 b(see)h(the)f(basic)h(problem)g(as)f(one)h(of)f(stabilit)o(y)h(of)f(a)g (random)g(system.)k(The)d(v)m(alues)14 424 y(deg)83 435 y Fn(t)98 424 y Fu(\()p Fm(v)r Fu(\))c(are)h(random)g(v)m(ariables)h (that)f(will)i(naturally)f(oscillate)h(around)e(their)h(means.)19 b(The)13 b(di\016cult)o(y)i(is)f(that)14 481 y(once)19 b(some)g(deg)308 492 y Fn(t)322 481 y Fu(\()p Fm(v)364 464 y Fo(0)375 481 y Fu(\))g(are)f(abnormally)i(o\013)e(their)h(mean)g (then)g(it)g(a\013ects)f(the)h(c)o(hange)g(in)h(deg)1701 492 y Fn(t)1715 481 y Fu(\()p Fm(v)r Fu(\).)30 b(\(If)19 b Fm(v)1909 464 y Fo(0)1920 481 y Fm(;)8 b(v)14 537 y Fu(ha)o(v)o(e)15 b(a)f(common)g(edge)i Fm(e)f Fu(then)g(deg\()p Fm(v)690 521 y Fo(0)701 537 y Fu(\))g(a\013ects)f(the)h(n)o(um)o(b)q (er)g(of)f(\()p Fm(e;)8 b(e)1249 521 y Fo(0)1260 537 y Fu(\))15 b(whic)o(h)g(a\013ects)f(the)h(exp)q(ected)h(c)o(hange)14 594 y(of)j(deg\()p Fm(v)r Fu(\).\))30 b(The)19 b Fm(N)k Fu(di\013eren)o(t)c(deg)668 605 y Fn(t)683 594 y Fu(\()p Fm(v)r Fu(\))e(are)i(all)h(oscillating)g(o\013)e(their)h(means)g(and)g (the)g(oscillation)i(of)d(one)14 650 y(can)g(ha)o(v)o(e)f(an)g(adv)o (erse)h(a\013ect)f(on)g(the)h(oscillations)h(of)e(another.)26 b(T)l(o)18 b(handle)g(this)g(problem)h(w)o(e)e(mo)q(dify)h(the)14 707 y(realtime)e(random)f(greedy)g(algorithm)g(b)o(y)h(what)e(w)o(e)h (think)h(of)f(as)g(stabilization)h(mec)o(hanisms.)14 850 y Fv(2)67 b(Stabilization)14 951 y Fu(As)13 b(b)q(efore)h(the)g (basic)g(ev)o(en)o(t)f(is)h(the)f(birth)h(of)f(an)g(edge)h Fm(e)p Fu(.)19 b(If)14 b(b)o(y)f(time)h Fm(t)g(e)f Fu(has)g(not)g(y)o (et)g(b)q(een)i(b)q(orn)e(it)h(is)g(b)q(orn)f(in)14 1008 y(the)h(next)f Fm(dt)g Fu(with)h(probabilit)o(y)h Fm(e)598 991 y Fn(t)626 990 y(dt)p 618 997 47 2 v 618 1023 a(k)q(D)669 1008 y Fu(.)k(That)13 b(edge)h(is)g(added)g(to)e Fm(P)6 b Fu(,)14 b(all)g Fm(v)h Fk(2)e Fm(e)g Fu(are)g(remo)o(v)o(ed)g(from)g Fm(S)i Fu(and)f(all)g Fm(e)1952 991 y Fo(0)14 1064 y Fu(con)o(taining)i(an)o(y)e(suc)o(h)h Fm(v)h Fu(are)f(deleted.)21 b(W)l(e)15 b(add)g(t)o(w)o(o)e(stabilization)j(mec)o(hanisms.)21 b(On)15 b(certain)g(o)q(ccasions)g(w)o(e)14 1121 y Fl(waste)h Fu(a)g(v)o(ertex)f Fm(v)r Fu(.)22 b(When)16 b(this)h(o)q(ccurs)f Fm(v)h Fu(is)g(remo)o(v)o(ed)e(from)h Fm(S)i Fu(and)e(all)h(edges)f Fm(e)g Fu(con)o(taining)h Fm(v)g Fu(are)f(deleted.)14 1177 y(On)j(certain)f(o)q(ccasions)h(when)f(an)g(edge)g Fm(e)g Fu(has)g(b)q(een)h(b)q(orn)f(and)g Fm(v)h Fk(2)f Fm(e)g Fu(w)o(e)f Fl(r)n(evive)h Fm(v)r Fu(.)28 b(When)18 b(this)g(o)q(ccurs)g Fm(v)14 1233 y Fu(is)g(\\put)g(bac)o(k")f(in)o(to) h Fm(S)i Fu(and)e(the)g(edges)g Fm(e)762 1217 y Fo(0)791 1233 y Fk(6)p Fu(=)f Fm(e)h Fu(con)o(taining)h Fm(v)g Fu(are)f(put)g(bac)o(k)f(in)o(to)h Fm(H)t Fu(.)27 b(\(A)18 b(v)o(ertex)f Fm(v)h Fk(2)g Fm(e)g Fu(is)14 1290 y(reviv)o(ed)f(at)e (the)i(momen)o(t)e Fm(e)h Fu(is)h(b)q(orn)f(or)g(not)f(at)h(all.)23 b(More)16 b(formally)g(w)o(e)g(can)g(sa)o(y)f(that)h(when)g Fm(e)g Fu(is)h(b)q(orn)f Fm(e)h Fu(is)14 1346 y(deleted)j(and)e(all)h (nonreviv)o(ed)g Fm(v)g Fk(2)e Fm(e)h Fu(are)g(remo)o(v)o(ed)g(from)f Fm(S)j Fu(as)e(are)g(all)h(edges)f Fm(e)1462 1330 y Fo(0)1492 1346 y Fu(con)o(taining)h(suc)o(h)f Fm(v)r Fu(.)28 b(The)14 1403 y(term)15 b(\\reviv)o(e")g(giv)o(es)g(the)h(sense)f(w)o(e)g(aim)h (for)e(that)h(this)g(o)q(ccurs)h(rarely)l(.\))85 1459 y(Here)g(are)g(the)g(probabilities.)25 b(Supp)q(ose)17 b(deg)878 1470 y Fn(t)892 1459 y Fu(\()p Fm(v)r Fu(\))c(=)i Fm(D)q(e)1075 1443 y Fo(\000)p Fn(t)1126 1459 y Fk(\000)c Fu(\001)16 b(with)g(\001)e Fk(\025)h Fu(0.)22 b(Then)17 b Fm(v)g Fu(is)g(w)o(asted)e(in)i(the)14 1516 y(time)h(in)o(terv)m(al)g ([)p Fm(t;)8 b(t)k Fu(+)f Fm(dt)p Fu(])18 b(with)f(probabilit)o(y)861 1498 y Fs(\001)p 826 1505 99 2 v 826 1531 a Fn(k)q(D)q(e)889 1523 y Fd(\000)p Fc(t)929 1516 y Fm(dt)p Fu(.)27 b(Supp)q(ose)18 b(deg)1258 1527 y Fn(t)1273 1516 y Fu(\()p Fm(v)r Fu(\))d(=)i Fm(D)q(e)1460 1499 y Fo(\000)p Fn(t)1512 1516 y Fu(+)12 b(\000)17 b(with)h(\000)f Fk(\025)f Fu(0.)27 b(If)17 b(an)14 1572 y(edge)d Fm(e)g Fu(con)o(taining)g Fm(v)i Fu(is)e(b)q(orn)g(then)g Fm(v)h Fu(is)f(reviv)o(ed)h(with)f(probabilit) o(y)1289 1554 y Fs(\000)p 1236 1561 127 2 v 1236 1588 a Fn(D)q(e)1280 1580 y Fd(\000)p Fc(t)1316 1588 y Fs(+\000)1367 1572 y Fu(.)19 b(The)14 b(a)f(priori)i(probabilit)o(y)g(that)14 1629 y Fm(v)i Fu(is)f(reviv)o(ed)g(is)f(then)645 1695 y(deg)714 1706 y Fn(t)729 1695 y Fu(\()p Fm(v)r Fu(\))835 1664 y Fm(dt)p 794 1685 126 2 v 794 1726 a(k)q(D)q(e)879 1713 y Fo(\000)p Fn(t)1006 1664 y Fu(\000)p 928 1685 185 2 v 928 1726 a Fm(D)q(e)988 1713 y Fo(\000)p Fn(t)1039 1726 y Fu(+)c(\000)1131 1695 y(=)1222 1664 y(\000)p 1183 1685 105 2 v 1183 1726 a Fm(k)q(D)1247 1713 y Fo(\000)p Fn(t)1293 1695 y Fm(dt)14 1793 y Fu(This)16 b(giv)o(es)f(a)g(con)o(v)o (enien)o(t)h(symmetry:)541 1917 y(Pr[)p Fm(v)g Fu(reviv)o(ed)g(or)f(w)o (asted)o(])d(=)1068 1886 y Fk(j)c Fu(deg)1157 1897 y Fn(t)1172 1886 y Fu(\()p Fm(v)r Fu(\))h Fk(\000)i Fm(D)q(e)1347 1869 y Fo(\000)p Fn(t)1387 1886 y Fk(j)p 1068 1906 332 2 v 1171 1948 a Fm(k)q(D)q(e)1256 1935 y Fo(\000)p Fn(t)1405 1917 y Fm(dt)461 b Fu(\(2\))85 2032 y(Consider)20 b(an)o(y)f Fm(v)j Fu(at)d(time)h Fm(t)p Fu(.)33 b(Supp)q(ose)21 b(deg)895 2043 y Fn(v)915 2032 y Fu(\()p Fm(t)p Fu(\))f(=)g Fm(D)q(e)1102 2016 y Fo(\000)p Fn(t)1156 2032 y Fk(\000)13 b Fu(\001)20 b(with)g(\001)g Fk(\025)g Fu(0.)33 b(In)21 b(the)f(next)f Fm(dt)h Fu(there)14 2089 y(is)g(probabilit)o(y)305 2067 y Fs(deg)355 2075 y Fc(t)369 2067 y Fs(\()p Fn(v)q Fs(\))p 305 2078 108 2 v 310 2104 a Fn(k)q(D)q(e)373 2096 y Fd(\000)p Fc(t)417 2089 y Fm(dt)f Fu(that)f(some)h Fm(e)g Fu(con)o(taining)h Fm(v)g Fu(is)g(b)q(orn)f(\(and)g Fm(v)h Fu(can't)f(b)q(e)g(reviv)o(ed)h(as)f(\001)f Fk(\025)i Fu(0\))e(and)14 2145 y(probabilit)o(y)287 2127 y Fs(\001)p 251 2134 99 2 v 251 2161 a Fn(k)q(D)q(e)314 2152 y Fd(\000)p Fc(t)354 2145 y Fm(dt)d Fu(that)f Fm(v)i Fu(is)f(w)o(asted;)f(so)g (probabilit)o(y)1045 2127 y Fn(dt)p 1045 2134 30 2 v 1050 2161 a(k)1094 2145 y Fu(that)g Fm(v)g Fk(62)f Fm(S)1299 2152 y Fn(t)p Fs(+)p Fn(dt)1369 2145 y Fu(.)19 b(Supp)q(ose)d(deg)1648 2156 y Fn(t)1663 2145 y Fu(\()p Fm(v)r Fu(\))11 b(=)i Fm(D)q(e)1842 2129 y Fo(\000)p Fn(t)1892 2145 y Fu(+)c(\000)14 2202 y(with)16 b(\000)e Fk(\025)g Fu(0.)20 b(Then)d Fm(v)g Fu(cannot)e(b)q(e)i(w)o(asted)e(and)g(the)h(probabilit)o(y)h(that)e (some)g Fm(e)h Fu(con)o(taining)g Fm(v)i Fu(is)e(b)q(orn)g(and)f Fm(v)14 2258 y Fu(is)j(not)e(reviv)o(ed)i(is)356 2236 y Fs(deg)406 2244 y Fc(t)420 2236 y Fs(\()p Fn(v)q Fs(\))p 356 2247 108 2 v 361 2274 a Fn(k)q(D)q(e)424 2265 y Fd(\000)p Fc(t)468 2258 y Fu(\(1)11 b Fk(\000)624 2240 y Fs(\000)p 571 2247 127 2 v 571 2274 a Fn(D)q(e)615 2265 y Fd(\000)p Fc(t)652 2274 y Fs(+\000)703 2258 y Fu(\))k(=)791 2240 y Fn(dt)p 791 2247 30 2 v 797 2274 a(k)826 2258 y Fu(.)25 b(That)16 b(is,)i(for)e Fl(any)g Fu(v)m(alue)j Fm(H)1356 2265 y Fn(t)1387 2258 y Fu(of)d(the)h(pro)q(cess)g(at)f(time)i Fm(t)f Fu(with)14 2314 y Fm(v)d Fk(2)f Fm(S)121 2321 y Fn(t)767 2388 y Fu(Pr[)p Fm(v)h Fk(62)f Fm(S)936 2395 y Fn(t)p Fs(+)p Fn(dt)1018 2388 y Fk(j)f Fm(H)1081 2395 y Fn(t)1096 2388 y Fu(])g(=)1174 2357 y Fm(dt)p 1174 2377 41 2 v 1181 2419 a(k)1906 2388 y Fu(\(3\))p eop 4 3 bop 14 162 a Fe(the)17 b(electr)o(onic)h(journal)g(of)f(combina)m (torics)f(4)i(\(no.)j(2\))c(\(1997\),)g(#R19)430 b Fu(4)14 311 y(Indeed,)17 b(\(3\))d(is)h(the)h(purp)q(ose)g(of)e(our)h (stabilization.)22 b(W)l(e)15 b(deduce)716 422 y(Pr[)p Fm(v)f Fk(2)f Fm(S)885 429 y Fn(t)900 422 y Fu(])f(=)h Fm(e)994 399 y Fo(\000)1026 369 y Fj(R)1053 379 y Fc(t)1045 417 y Fb(0)1078 388 y Fc(dt)p 1078 393 28 2 v 1083 409 a(k)1124 422 y Fu(=)g Fm(e)1193 403 y Fo(\000)p Fn(t=k)1906 422 y Fu(\(4\))14 517 y(Let)i Fm(X)j Fu(b)q(e)d(an)o(y)f(random)g(v)m (ariable)i(that)e(dep)q(ends)i(only)f(on)f(the)h(history)g(of)f(the)g (pro)q(cess)h(up)g(to)f(time)h Fm(s)p Fu(.)20 b(Then)710 613 y Fm(E)s Fu([)p Fm(X)t Fk(j)p Fm(w)11 b Fk(2)i Fm(S)930 620 y Fn(t)945 613 y Fu(])f(=)h Fm(E)s Fu([)p Fm(X)t Fk(j)p Fm(w)e Fk(2)i Fm(S)1238 620 y Fn(s)1255 613 y Fu(])638 b(\(5\))14 708 y(The)13 b(reason)g(is)g(that)f(an)o(y)h (history)g(up)g(to)f(time)h Fm(s)h Fu(with)f Fm(w)g Fk(2)g Fm(S)1089 715 y Fn(s)1119 708 y Fu(has)g(precisely)i(the)e(same)f (probabilit)o(y)j Fm(e)1821 692 y Fo(\000)p Fs(\()p Fn(t)p Fo(\000)p Fn(s)p Fs(\))p Fn(=k)14 765 y Fu(of)g(b)q(eing)h(extended)h (to)d(a)h(history)g(up)h(to)e(time)i Fm(t)f Fu(with)h Fm(w)d Fk(2)g Fm(S)1105 772 y Fn(t)1120 765 y Fu(.)14 907 y Fv(3)67 b(The)22 b(Big)h(Picture)14 1009 y Fu(W)l(e)15 b(set)784 1065 y Fm(!)g Fu(=)e(ln)8 b Fm(D)j Fk(\000)g Fm(K)f Fu(ln)f(ln)f Fm(D)713 b Fu(\(6\))14 1145 y(\()p Fm(K)20 b Fu(a)e(suitably)g(large)g(constan)o(t\))e(and)i(con)o(tin)o (ue)g(the)g(pro)q(cess)g(\(starting)f(at)g(time)h(zero\))f(to)g(time)h Fm(!)r Fu(.)27 b(Call)18 b Fm(e)14 1202 y Fu(a)f Fl(false)g(birth)g Fu(if)h Fm(e)f Fu(is)g(b)q(orn)h(at)e(time)h Fm(t)h Fu(but)f(at)f(some) h(time)g Fm(t)1083 1185 y Fo(0)1111 1202 y Fm(<)f(t)h Fu(some)g Fm(v)1335 1185 y Fo(0)1362 1202 y Fk(2)f Fm(e)h Fu(w)o(as)f(reviv)o(ed)i(when)g(some)e Fm(e)1952 1185 y Fo(0)14 1258 y Fu(w)o(as)h(b)q(orn.)27 b(The)18 b(n)o(um)o(b)q(er)f (of)g(false)h(births)g(is)g(at)f(most)g(the)g(n)o(um)o(b)q(er)h(of)f (reviv)m(als)i(since)g(w)o(e)e(can)h(asso)q(ciate)f Fm(e)14 1315 y Fu(with)d(that)e(reviv)m(al)j Fm(t)373 1298 y Fo(0)385 1315 y Fm(;)8 b(v)430 1298 y Fo(0)454 1315 y Fu(with)14 b Fm(t)572 1298 y Fo(0)597 1315 y Fm(<)e(t)i Fu(maximal)g(and)g(this)f(asso)q(ciation)h(is)g(injectiv)o(e.)21 b(F)l(alse)13 b(births)h(actually)g(do)14 1371 y(o)o(v)o(erlap)i (previous)h(births.)24 b(\(An)o(throp)q(omorphically)18 b(sp)q(eaking,)f(though,)f(the)h(pro)q(cess)f(do)q(es)h(not)f(kno)o(w)g (that)14 1427 y(a)h(birth)g(is)g(false.\))25 b(The)17 b(set)g(of)f(b)q(orn)h(edges)h Fm(e)e Fu(whic)o(h)i(are)f(not)f(false)h (births)h(giv)o(es)f(the)g(pac)o(king)g Fm(P)1776 1411 y Fo(\003)1812 1427 y Fu(that)f(w)o(e)14 1484 y(desire.)21 b(Set)15 b Fm(S)270 1467 y Fo(\003)301 1484 y Fu(=)e Fm(V)20 b Fk(\000)442 1452 y Fj(S)484 1484 y Fm(P)519 1467 y Fo(\003)538 1484 y Fu(.)85 1540 y(F)l(or)12 b(eac)o(h)h(v)o (ertex)g Fm(w)g Fu(let)h Fm(S)s(U)5 b(RV)639 1547 y Fn(w)677 1540 y Fu(b)q(e)14 b(the)f(indicator)h(for)e Fm(w)h Fk(2)g Fm(S)1187 1547 y Fn(!)1211 1540 y Fu(;)g Fm(W)6 b(AS)s(T)g(E)1418 1547 y Fn(w)1458 1540 y Fu(the)13 b(n)o(um)o(b)q(er)g(of)f(times)i (\(zero)14 1597 y(or)j(one\))g(that)g Fm(w)h Fu(is)f(w)o(asted;)h Fm(RE)s(V)8 b(I)t(V)h(E)742 1604 y Fn(w)786 1597 y Fu(the)17 b(n)o(um)o(b)q(er)h(of)f(times)g Fm(w)h Fu(is)g(reviv)o(ed.)27 b Fm(S)1523 1580 y Fo(\003)1559 1597 y Fu(consists)17 b(of)g(surviving)14 1653 y(v)o(ertices,)e(w)o(asted)g(v)o(ertices,)g (and)g(v)o(ertices)h(in)g(false)f(births)h(so)468 1749 y Fk(j)p Fm(S)512 1730 y Fo(\003)530 1749 y Fk(j)c(\024)603 1708 y Fj(X)621 1796 y Fn(w)671 1749 y Fm(S)s(U)5 b(RV)800 1756 y Fn(w)836 1749 y Fu(+)10 b Fm(W)c(AS)s(T)g(E)1062 1756 y Fn(w)1098 1749 y Fu(+)11 b(\()p Fm(k)g Fu(+)f(1\))p Fm(RE)s(V)e(I)t(V)h(E)1484 1756 y Fn(w)14 1875 y Fu(As)15 b(constan)o(ts)g(do)g(not)g(concern)g(us)h(w)o(e)f(de\014ne)637 1970 y Fm(LO)q(S)s(S)763 1977 y Fn(w)802 1970 y Fu(=)e Fm(W)6 b(AS)s(T)g(E)1031 1977 y Fn(w)1067 1970 y Fu(+)11 b Fm(RE)s(V)d(I)t(V)i(E)1315 1977 y Fn(w)14 2066 y Fu(so)15 b(w)o(e)g(can)g(b)q(ound)h(more)f(con)o(v)o(enien)o(tly)637 2161 y Fk(j)p Fm(S)681 2143 y Fo(\003)699 2161 y Fk(j)e(\024)773 2121 y Fj(X)790 2208 y Fn(w)840 2161 y Fm(S)s(U)5 b(RV)969 2168 y Fn(w)1005 2161 y Fu(+)10 b(\()p Fm(k)h Fu(+)g(1\))p Fm(LO)q(S)s(S)1316 2168 y Fn(w)85 2287 y Fu(No)o(w)16 b(Linearit)o(y)h(of)f(Exp)q(ectation)h(comes)g(in)o(to)f(pla)o(y)l(.)25 b(The)17 b(exp)q(ectation)g(of)f(this)h(sum)g(is)g(the)f(sum)h(of)f (the)14 2344 y(exp)q(ectations)f(so)f(that)f(it)h(su\016ces)h(to)e (appropriately)i(b)q(ound)g Fm(E)s Fu([)p Fm(S)s(U)5 b(RV)1279 2351 y Fn(w)1304 2344 y Fu(],)13 b Fm(E)s Fu([)p Fm(LO)q(S)s(S)1519 2351 y Fn(w)1544 2344 y Fu(])h(for)f(a)h(giv)o(en)h Fm(w)q Fu(.)k(F)l(rom)14 2400 y(\(4\))491 2457 y Fm(E)s Fu([)p Fm(S)s(U)5 b(RV)670 2464 y Fn(w)695 2457 y Fu(])12 b(=)h(Pr[)p Fm(w)g Fk(2)g Fm(S)947 2464 y Fn(!)971 2457 y Fu(])f(=)h Fm(e)1065 2438 y Fo(\000)p Fn(!)q(=k)1163 2457 y Fu(=)g Fm(D)1250 2438 y Fo(\000)p Fs(1)p Fn(=k)1337 2457 y Fu(ln)1375 2435 y Fn(K=k)1448 2457 y Fm(D)p eop 5 4 bop 14 162 a Fe(the)17 b(electr)o(onic)h(journal)g(of)f(combina)m (torics)f(4)i(\(no.)j(2\))c(\(1997\),)g(#R19)430 b Fu(5)14 311 y(No)o(w)15 b(it)g(su\016ces)h(to)e(sho)o(w)703 368 y Fm(E)s Fu([)p Fm(LO)q(S)s(S)879 375 y Fn(w)904 368 y Fu(])e(=)h Fm(O)q Fu(\()p Fm(D)1070 349 y Fo(\000)p Fs(1)p Fn(=k)1157 368 y Fu(ln)1195 346 y Fn(c)1219 368 y Fm(D)q Fu(\))630 b(\(7\))14 451 y(Fix)22 b Fm(w)h Fu(and)f(consider)h Fm(E)s Fu([)p Fm(LO)q(S)s(S)613 458 y Fn(w)638 451 y Fu(].)40 b(F)l(or)22 b(ev)o(ery)g Fm(t)g Fu(\(2\))f(giv)o(es)h(the)g (probabilit)o(y)i Fm(w)e Fu(is)h(w)o(asted)e(or)h(reviv)o(ed.)14 508 y(Ho)o(w)o(ev)o(er,)14 b(this)i(is)f(conditional)i(on)e Fm(w)f Fk(2)f Fm(S)760 515 y Fn(t)789 508 y Fu(whic)o(h)j(o)q(ccurs)g (with)f(probabilit)o(y)i Fm(e)1417 491 y Fo(\000)p Fn(t=k)1493 508 y Fu(.)j(Th)o(us)434 635 y Fm(E)s Fu([)p Fm(LO)q(S)s(S)610 642 y Fn(w)635 635 y Fu(])12 b(=)708 578 y Fj(Z)750 588 y Fn(!)731 672 y(t)p Fs(=0)815 605 y Fm(e)836 588 y Fo(\000)p Fn(t=k)p 801 625 126 2 v 801 667 a Fm(k)q(D)q(e)886 653 y Fo(\000)p Fn(t)931 635 y Fm(E)s Fu([)p Fk(j)c Fu(deg)1069 646 y Fn(t)1084 635 y Fu(\()p Fm(w)q Fu(\))h Fk(\000)h Fm(D)q(e)1268 617 y Fo(\000)p Fn(t)1309 635 y Fk(j)i(j)h Fm(w)g Fk(2)g Fm(S)1477 642 y Fn(t)1491 635 y Fu(])p Fm(dt)14 754 y Fu(W)l(e)i(shall)i(sho)o(w)478 810 y Fm(E)s Fu([)p Fk(j)8 b Fu(deg)616 821 y Fn(t)630 810 y Fu(\()p Fm(w)q Fu(\))h Fk(\000)i Fm(D)q(e)815 791 y Fo(\000)p Fn(t)856 810 y Fk(j)h(j)g Fm(w)h Fk(2)g Fm(S)1023 817 y Fn(t)1038 810 y Fu(])f(=)h Fm(O)q Fu(\(\()p Fm(D)q Fu(\()p Fm(t)c Fu(+)i(1\))p Fm(e)1373 791 y Fo(\000)p Fn(t)1413 810 y Fu(\))1431 791 y Fs(1)p Fn(=)p Fs(2)1483 810 y Fu(\))405 b(\(8\))14 893 y(W)l(e)16 b(note)g(that)g(\()p Fm(t)11 b Fu(+)g(1\))427 877 y Fs(1)p Fn(=)p Fs(2)478 893 y Fu(\()p Fm(D)q(e)556 877 y Fo(\000)p Fn(t)597 893 y Fu(\))615 877 y Fo(\000)p Fs(1)p Fn(=)p Fs(2)692 893 y Fm(e)713 877 y Fo(\000)p Fn(t=k)806 893 y Fu(is)16 b(maximized)i(at)d Fm(t)g Fu(=)f Fm(!)19 b Fu(where)d(it)g(is)h(at)e (most)h(ln)1697 872 y Fs(1)p Fn(=)p Fs(2)1756 893 y Fm(D)h Fu(so)f(that,)14 950 y(giv)o(en)g(\(8\),)e(\(7\))g(holds)i(with)f Fm(c)e Fu(=)600 932 y Fs(3)p 600 939 17 2 v 600 966 a(2)621 950 y Fu(.)14 1093 y Fv(4)67 b(Phan)n(tom)23 b(Edges)14 1195 y Fu(Giv)o(en)14 b(deg)213 1206 y Fn(t)227 1195 y Fu(\()p Fm(w)q Fu(\))f(what)g(do)g(w)o(e)h(exp)q(ect)g(of)f(deg)810 1206 y Fn(t)p Fs(+)p Fn(dt)880 1195 y Fu(\()p Fm(w)q Fu(\)?)19 b(Let)13 b Fm(e)h Fu(b)q(e)g(an)g(edge)f(con)o(taining)i Fm(w)f Fu(at)f(time)g Fm(t)p Fu(.)20 b(Roughly)14 1251 y(sp)q(eaking)13 b(eac)o(h)f Fm(v)i Fk(2)f Fm(e)p Fu(,)g Fm(v)h Fk(6)p Fu(=)f Fm(w)g Fu(is)f(remo)o(v)o(ed)g(with)g(probabilit)o (y)1107 1233 y Fn(dt)p 1107 1240 30 2 v 1113 1267 a(k)1154 1251 y Fu(so)f Fm(e)h Fu(is)h(remo)o(v)o(ed)e(with)i(probabilit)o(y)g Fm(dt)p Fu(.)19 b(Then)14 1308 y(deg)83 1318 y Fn(t)98 1308 y Fu(\()p Fm(w)q Fu(\))d(w)o(ould)i(drop)f(b)o(y)h(deg)562 1318 y Fn(t)576 1308 y Fu(\()p Fm(w)q Fu(\))p Fm(dt)f Fu(in)h(time)g Fm(dt)f Fu(giving)i(exp)q(onen)o(tial)g(deca)o(y)l(.)27 b(Renormalizing,)19 b Fm(e)1789 1291 y Fn(t)1812 1308 y Fu(deg)1880 1318 y Fn(t)1895 1308 y Fu(\()p Fm(w)q Fu(\))14 1364 y(w)o(ould)d(b)q(e)g(a)e(martingale.)85 1420 y(W)l(ell,)19 b(not)e(exactly)l(.)28 b(The)18 b(condition)h(that)e Fm(w)h Fu(itself)h(surviv)o(es)f(has)f(a)h(\(small\))g(e\013ect.)27 b(F)l(or)17 b(one)g(thing,)i(it)14 1477 y(ma)o(y)c(happ)q(en)i(that)f (an)f Fm(e)h Fu(con)o(taining)h Fm(w)g Fu(is)f(b)q(orn)g(and)g Fm(w)h Fu(is)f(reviv)o(ed.)23 b(It)16 b(is)g(helpful)i(then)e(to)g (think)g(of)g(that)f Fm(e)14 1533 y Fu(as)g(a)g Fl(phantom)i(e)n(dge)e Fu(whic)o(h)h(then)f(exp)q(eriences)j(exp)q(onen)o(tial)e(deca)o(y)l(.) k(F)l(ormally)c(w)o(e)f(de\014ne)775 1635 y Fm(P)6 b(H)t(AN)923 1642 y Fn(t)950 1635 y Fu(=)998 1595 y Fj(X)1016 1684 y Fn(t)1029 1676 y Fd(0)1066 1635 y Fm(e)1087 1617 y Fo(\000)p Fs(\()p Fn(t)p Fo(\000)p Fn(t)1178 1604 y Fd(0)1188 1617 y Fs(\))1906 1635 y Fu(\(9\))14 1772 y(where)18 b(the)f(sum)g(is)h(o)o(v)o(er)e(all)i(those)f(times)h Fm(t)799 1756 y Fo(0)827 1772 y Fk(\024)e Fm(t)i Fu(when)f Fm(w)h Fu(has)f(b)q(een)i(reviv)o(ed.)26 b(\(If)17 b Fm(w)h Fu(hasn't)f(b)q(een)h(reviv)o(ed)14 1829 y Fm(P)6 b(H)t(AN)162 1836 y Fn(t)189 1829 y Fu(=)13 b(0.\))19 b(Note)c Fm(P)6 b(H)t(AN)20 b Fu(is)15 b(nev)o(er)h(negativ)o(e.)k(W)l (e)15 b(de\014ne)h(the)g(adjusted)f(degree)h Fm(X)1603 1836 y Fn(t)1632 1829 y Fu(b)o(y)748 1931 y Fm(X)786 1938 y Fn(t)812 1931 y Fu(=)d(deg)929 1942 y Fn(t)944 1931 y Fu(\()p Fm(w)q Fu(\))c(+)i Fm(P)6 b(H)t(AN)1217 1938 y Fn(t)14 2033 y Fu(and)16 b(normalize)g(b)o(y)f(setting)892 2089 y Fm(Z)923 2096 y Fn(t)950 2089 y Fu(=)e Fm(e)1019 2070 y Fn(t)1034 2089 y Fm(X)1072 2096 y Fn(t)1883 2089 y Fu(\(10\))14 2173 y(so)i(that)566 2229 y Fk(j)8 b Fu(deg)655 2240 y Fn(t)669 2229 y Fu(\()p Fm(w)q Fu(\))i Fk(\000)g Fm(D)q(e)854 2210 y Fo(\000)p Fn(t)895 2229 y Fk(j)i(\024)h Fm(e)989 2210 y Fo(\000)p Fn(t)1030 2229 y Fk(j)p Fm(Z)1074 2236 y Fn(t)1098 2229 y Fk(\000)d Fm(D)q Fk(j)g Fu(+)g Fm(P)c(H)t(AN)1398 2236 y Fn(t)14 2313 y Fu(so)15 b(that)f(\(8\))h (will)i(follo)o(w)e(from)579 2415 y Fm(E)s Fu([)p Fk(j)p Fm(Z)673 2422 y Fn(t)696 2415 y Fk(\000)10 b Fm(D)q Fk(j)j(j)f Fm(w)h Fk(2)g Fm(S)948 2422 y Fn(t)963 2415 y Fu(])f(=)h Fm(O)q Fu(\(\()p Fm(D)q Fu(\()p Fm(t)c Fu(+)i(1\))p Fm(e)1298 2396 y Fn(t)1312 2415 y Fu(\))1330 2396 y Fs(1)p Fn(=)p Fs(2)1382 2415 y Fu(\))483 b(\(11\))p eop 6 5 bop 14 162 a Fe(the)17 b(electr)o(onic)h(journal)g(of)f(combina)m (torics)f(4)i(\(no.)j(2\))c(\(1997\),)g(#R19)430 b Fu(6)14 311 y(and)16 b(the)f(relativ)o(ely)h(easier)567 413 y Fm(E)s Fu([)p Fm(P)6 b(H)t(AN)765 420 y Fn(t)791 413 y Fk(j)12 b Fm(w)i Fk(2)f Fm(S)934 420 y Fn(t)948 413 y Fu(])f(=)h Fm(O)q Fu(\(\()p Fm(D)q Fu(\()p Fm(t)d Fu(+)g(1\))p Fm(e)1283 395 y Fo(\000)p Fn(t)1323 413 y Fu(\))1341 395 y Fs(1)p Fn(=)p Fs(2)1393 413 y Fu(\))472 b(\(12\))14 516 y(W)l(e)15 b(sho)o(w)g(\(11\))f(b)o(y)h(emplo)o(ying)h(the)g (general)f(inequalit)o(y)i Fm(E)s Fu([)p Fk(j)p Fm(W)6 b Fk(j)p Fu(])11 b Fk(\024)i Fm(E)s Fu([)p Fm(W)1323 499 y Fs(2)1340 516 y Fu(])1353 499 y Fs(1)p Fn(=)p Fs(2)1420 516 y Fu(and)j(sho)o(wing)608 618 y Fm(E)s Fu([\()p Fm(Z)707 625 y Fn(t)730 618 y Fk(\000)11 b Fm(D)q Fu(\))833 599 y Fs(2)864 618 y Fk(j)h Fm(w)h Fk(2)g Fm(S)1006 625 y Fn(t)1021 618 y Fu(])f(=)h Fm(O)q Fu(\()p Fm(D)q Fu(\()p Fm(t)d Fu(+)g(1\))p Fm(e)1338 599 y Fn(t)1352 618 y Fu(\))513 b(\(13\))85 720 y(W)l(e)18 b(think)g(of)f(\(13\))g(as)g(the)h(core)g (of)f(our)h(argumen)o(t.)26 b(The)18 b(idea)h(will)g(b)q(e)g(that)e Fm(Z)1525 727 y Fn(t)1557 720 y Fu(is)h(a)g(con)o(tin)o(uous)g(time)14 776 y(martingale.)i(But)13 b(not)f(exactly)l(.)20 b(Essen)o(tially)l(,) 15 b(conditioning)g(on)e Fm(w)g Fu(surviving)i(means)e(the)g(edges)g Fm(e)g Fu(con)o(taining)14 833 y Fm(w)k Fu(are)f(not)g(b)q(orn)h(so)f (the)g(v)o(ertices)h Fm(v)h Fu(on)e(suc)o(h)g(edges)h(ha)o(v)o(e)f (sligh)o(tly)h(less)g(c)o(hance)g(of)f(b)q(eing)i(remo)o(v)o(ed.)k(But) 17 b(it)14 889 y(will)f(b)q(e)f(close)g(enough.)20 b(Indeed,)c(this)f (motiv)m(ates)f(our)h(c)o(hoice)g(\(6\))e(of)h Fm(!)j Fu(since)e(w)o(e)f(w)o(an)o(t)g(the)g(di\013erence)i(of)e(one)14 946 y(in)i(the)f(degree)h(to)f(ha)o(v)o(e)f(negligible)k(e\013ect.)14 1089 y Fv(5)67 b(Almost)22 b(a)g(Martingale)14 1190 y Fu(W)l(e)15 b(w)o(an)o(t)g(to)f(sho)o(w)h(\(13\))f(for)g(a)h(giv)o(en)h Fm(t)d Fk(\024)g Fm(!)r Fu(.)20 b(W)l(e)15 b(shall)h(examine)g Fm(X)1234 1197 y Fn(s)1267 1190 y Fu(for)e(0)f Fk(\024)g Fm(s)g Fk(\024)g Fm(t)p Fu(.)14 1247 y Fi(Claim:)21 b Fu(Let)15 b(0)d Fk(\024)h Fm(s)g(<)g(t)j Fu(and)f(let)h Fm(H)655 1254 y Fn(s)688 1247 y Fu(b)q(e)f(an)o(y)g(v)m(alue)i(with)e Fm(w)f Fk(2)f Fm(H)1184 1254 y Fn(s)1201 1247 y Fu(.)20 b(Then)533 1349 y Fm(E)s Fu([)p Fm(X)621 1356 y Fn(s)p Fs(+)p Fn(ds)705 1349 y Fk(\000)11 b Fm(X)789 1356 y Fn(s)819 1349 y Fk(j)h Fm(H)882 1356 y Fn(s)900 1349 y Fm(;)c(w)k Fk(2)h Fm(S)1037 1356 y Fn(t)1052 1349 y Fu(])f(=)h Fk(\000)p Fm(X)1198 1356 y Fn(s)1216 1349 y Fm(ds)d Fu(+)g Fm(\013X)1383 1356 y Fn(s)1401 1349 y Fm(ds)14 1451 y Fu(with)16 b(0)c Fk(\024)h Fm(\013)g Fk(\024)329 1433 y Fs(1)p 296 1440 83 2 v 296 1467 a Fn(D)q(e)340 1458 y Fd(\000)p Fc(s)383 1451 y Fu(.)85 1507 y(The)h Fm(\013)g Fu(represen)o(ts)g(an)f(\\error)g(term")g (caused)i(b)o(y)e(the)h(e\013ectiv)o(e)g(degree)g(loss.)20 b(Applying)c(\(5\))d(it)h(su\016ces)g(to)14 1564 y(sho)o(w)502 1620 y Fm(E)s Fu([)p Fm(X)590 1627 y Fn(s)p Fs(+)p Fn(ds)675 1620 y Fk(\000)c Fm(X)758 1627 y Fn(s)788 1620 y Fk(j)i Fm(H)851 1627 y Fn(s)869 1620 y Fm(;)c(w)k Fk(2)h Fm(S)1006 1627 y Fn(s)p Fs(+)p Fn(ds)1082 1620 y Fu(])g(=)g Fk(\000)p Fm(X)1229 1627 y Fn(s)1246 1620 y Fm(ds)d Fu(+)h Fm(\013X)1414 1627 y Fn(s)1432 1620 y Fm(ds)406 b Fu(\(14\))14 1704 y(with)16 b(0)c Fk(\024)h Fm(\013)g Fk(\024)329 1686 y Fs(1)p 296 1693 V 296 1719 a Fn(D)q(e)340 1711 y Fd(\000)p Fc(s)383 1704 y Fu(.)85 1760 y(If)h(an)h(edge)g Fm(e)f Fu(with)h Fm(w)f Fk(2)e Fm(e)j Fu(is)g(b)q(orn)g(and)g Fm(w)g Fu(is)g(reviv)o(ed)g(then)g(the)f(new)h(term)f(in)h Fm(P)6 b(H)t(AN)1629 1767 y Fn(s)p Fs(+)p Fn(ds)1720 1760 y Fu(balances)15 b(the)14 1817 y(loss)h(in)g(deg)222 1828 y Fn(s)p Fs(+)p Fn(ds)306 1817 y Fm(w)q Fu(.)k(\(The)15 b(edge)h(is)f(coun)o(ted)h(as)f(a)g(phan)o(tom)f(edge.\))20 b(No)o(w)15 b(consider)h(the)f(con)o(tribution)h(to)f(the)14 1873 y(exp)q(ectation)h(when)g(no)f(suc)o(h)g Fm(e)h Fu(is)f(b)q(orn.)21 b(Automatically)477 1975 y Fm(P)6 b(H)t(AN)625 1982 y Fn(s)p Fs(+)p Fn(ds)714 1975 y Fu(=)13 b Fm(P)6 b(H)t(AN)910 1982 y Fn(s)927 1975 y Fm(e)948 1957 y Fo(\000)p Fn(ds)1022 1975 y Fu(=)13 b Fm(P)6 b(H)t(AN)1218 1982 y Fn(s)1245 1975 y Fk(\000)11 b Fm(P)6 b(H)t(AN)1439 1982 y Fn(s)1456 1975 y Fm(ds)14 2077 y Fu(so)15 b Fm(P)6 b(H)t(AN)20 b Fu(has)15 b(no)g(error)f(term.)20 b(Dealing)c(with)f(deg) 958 2088 y Fn(s)p Fs(+)p Fn(ds)1034 2077 y Fu(\()p Fm(w)q Fu(\))g(is)g(somewhat)g(more)f(tec)o(hnical.)85 2134 y(Let)h Fm(v)i Fu(b)q(e)f(a)e(v)o(ertex)h(sharing)g(a)g(common)g(edge)g Fm(e)h Fu(with)f Fm(w)q Fu(.)20 b(Supp)q(ose)c(deg)1378 2145 y Fn(s)1395 2134 y Fu(\()p Fm(v)r Fu(\))c(=)h Fm(D)q(e)1575 2117 y Fo(\000)p Fn(s)1629 2134 y Fk(\000)d Fu(\001)15 b(with)g(\001)e Fk(\025)g Fu(0.)14 2190 y(There)18 b(are)f(deg)295 2201 y Fn(s)312 2190 y Fu(\()p Fm(v)r Fu(\))11 b Fk(\000)h Fu(1)17 b(edges)h Fm(e)615 2174 y Fo(0)643 2190 y Fk(6)p Fu(=)f Fm(e)g Fu(con)o(taining)h Fm(e)g Fu(that)f(migh)o(t)g(b)q(e)h(b) q(orn)g(and)f Fm(v)i Fu(migh)o(t)f(b)q(e)g(w)o(asted)e(so)h Fm(v)14 2247 y Fu(has)g(probabilit)o(y)337 2229 y Fn(dt)p 337 2236 30 2 v 343 2262 a(k)372 2247 y Fu(\(1)11 b Fk(\000)509 2229 y Fs(1)p 476 2236 83 2 v 476 2262 a Fn(D)q(e)520 2254 y Fd(\000)p Fc(s)563 2247 y Fu(\))17 b(of)g(b)q(eing)i(remo)o(v)o (ed.)26 b(Supp)q(ose)18 b(deg)1231 2258 y Fn(s)1249 2247 y Fu(\()p Fm(v)r Fu(\))d(=)i Fm(D)q(e)1436 2230 y Fo(\000)p Fn(s)1491 2247 y Fu(+)12 b(\000)18 b(with)f(\000)g Fk(\025)g Fu(0.)26 b(There)14 2303 y(are)15 b(deg)159 2314 y Fn(s)177 2303 y Fu(\()p Fm(v)r Fu(\))9 b Fk(\000)h Fu(1)15 b(edges)h Fm(e)472 2287 y Fo(0)497 2303 y Fk(6)p Fu(=)d Fm(e)i Fu(con)o(taining)h Fm(v)h Fu(that)e(migh)o(t)g(b)q(e)h(b)q(orn)g(and)f Fm(v)i Fu(then)f(m)o(ust)f(not)f(b)q(e)i(reviv)o(ed)h(so)d Fm(v)14 2360 y Fu(has)h(probabilit)o(y)333 2342 y Fn(dt)p 333 2349 30 2 v 339 2375 a(k)368 2360 y Fu(\(1)9 b Fk(\000)525 2342 y Fs(1)p 469 2349 129 2 v 469 2375 a Fn(D)q(e)513 2367 y Fd(\000)p Fc(s)551 2375 y Fs(+\000)602 2360 y Fu(\))15 b(of)g(b)q(eing)i(remo)o(v)o(ed.)i(In)d(an)o(y)f(case)g(it)h (has)f(probabilit)o(y)1612 2342 y Fn(dt)p 1612 2349 30 2 v 1618 2375 a(k)1647 2360 y Fu(\(1)9 b Fk(\000)i Fm(\013)p Fu(\))k(of)g(b)q(eing)14 2416 y(remo)o(v)o(ed)g(with)g(0)e Fk(\024)g Fm(\013)g Fk(\024)509 2398 y Fs(1)p 477 2405 83 2 v 477 2432 a Fn(D)q(e)521 2423 y Fd(\000)p Fc(s)564 2416 y Fu(.)p eop 7 6 bop 14 162 a Fe(the)17 b(electr)o(onic)h(journal)g(of)f(combina)m (torics)f(4)i(\(no.)j(2\))c(\(1997\),)g(#R19)430 b Fu(7)85 311 y(Let)17 b Fm(e)h Fu(b)q(e)g(an)g(edge)f(con)o(taining)i Fm(w)q Fu(.)26 b(No)17 b(ev)o(en)o(t)h(\(w)o(e'v)o(e)e(excluded)k(the)d (birth)h(of)f Fm(e)h Fu(already\))f(can)h(remo)o(v)o(e)14 368 y(t)o(w)o(o)c Fm(v)r(;)8 b(v)169 351 y Fo(0)192 368 y Fk(2)13 b Fm(e)i Fu(since)h(they)f(share)g(only)g(the)g(one)g(common) g(edge)g Fm(e)p Fu(.)20 b(\()p Fm(H)e Fu(is)e(simple.\))21 b(Th)o(us)15 b Fm(e)g Fu(is)g(remo)o(v)o(ed)g(with)14 424 y(probabilit)o(y)i(\(1)9 b Fk(\000)i Fm(\013)p Fu(\))p Fm(dt)k Fu(with)g(0)e Fk(\024)g Fm(\013)f Fk(\024)759 406 y Fs(1)p 726 413 83 2 v 726 440 a Fn(D)q(e)770 432 y Fd(\000)p Fc(s)814 424 y Fu(.)19 b(By)d(Linearit)o(y)g(of)f(Exp)q (ectation)522 515 y Fm(E)s Fu([deg)640 525 y Fn(s)p Fs(+)p Fn(ds)716 515 y Fu(\()p Fm(w)q Fu(\))9 b Fk(\000)i Fu(deg)910 525 y Fn(s)927 515 y Fu(\()p Fm(w)q Fu(\)])h(=)h Fk(\000)8 b Fu(deg)1181 525 y Fn(s)1199 515 y Fu(\()p Fm(w)q Fu(\)\(1)h Fk(\000)h Fm(\013)p Fu(\))p Fm(ds)14 605 y Fu(with)16 b(0)c Fk(\024)h Fm(\013)g Fk(\024)329 587 y Fs(1)p 296 594 V 296 620 a Fn(D)q(e)340 612 y Fd(\000)p Fc(s)383 605 y Fu(.)20 b(As)15 b Fm(P)6 b(H)t(AN)20 b Fu(is)15 b(p)q(ositiv)o(e)i(or)d(zero,)h(deg)1098 616 y Fn(s)1116 605 y Fu(\()p Fm(w)q Fu(\))c Fk(\024)i Fm(X)1283 612 y Fn(s)1316 605 y Fu(and)411 695 y Fm(E)s Fu([)p Fm(X)499 702 y Fn(s)p Fs(+)p Fn(ds)583 695 y Fk(\000)e Fm(X)667 702 y Fn(s)684 695 y Fu(])h(=)h Fk(\000)p Fm(X)830 702 y Fn(s)848 695 y Fm(ds)d Fu(+)h Fm(\013)d Fu(deg)1054 706 y Fn(s)1072 695 y Fu(\()p Fm(w)q Fu(\))p Fm(ds)k Fu(=)h Fk(\000)p Fm(X)1320 702 y Fn(s)1338 695 y Fm(ds)d Fu(+)g Fm(\013X)1505 702 y Fn(s)1523 695 y Fm(ds)14 785 y Fu(where)16 b(the)f(new)g Fm(\013)h Fu(still)g(satis\014es)g(0)c Fk(\024)h Fm(\013)g Fk(\024)829 767 y Fs(1)p 796 774 V 796 801 a Fn(D)q(e)840 793 y Fd(\000)p Fc(s)883 785 y Fu(.)20 b(This)c(completes)g(\(14\))e(and)h(hence)h(the)g(Claim.)14 842 y Fl(R)n(emark.)k Fu(The)15 b(ab)q(o)o(v)o(e)f(claim)h(can)g(also)f (b)q(e)h(stated)f(and)g(pro)o(v)o(en)g(without)g(the)h(use)g(of)e (in\014nitesimals,)k(giving)e(a)14 898 y(b)q(ound)g(on)f Fm(E)s Fu([)p Fm(X)303 905 y Fn(s)p Fs(+\001)p Fn(s)395 898 y Fk(\000)7 b Fm(X)475 905 y Fn(s)493 898 y Fu(].)19 b(In)14 b(that)f(case)h(there)g(w)o(ould)g(b)q(e)h(an)f(additional)h (additiv)o(e)g(term)e Fm(O)1682 905 y Fn(H)1713 898 y Fu(\(\(\001)p Fm(s)p Fu(\))1826 882 y Fs(2)1844 898 y Fu(\))h(with)14 955 y(the)j(implicit)i(constan)o(t)d(dep)q(enden)o(t)j (on)e(the)g(h)o(yp)q(ergraphs)g Fm(H)t Fu(.)24 b(Letting)17 b(\001)p Fm(s)f Fk(!)g Fu(0)g(the)h(results)g(b)q(elo)o(w)h(w)o(ould)14 1011 y(b)q(e)e(the)f(same.)85 1068 y(W)l(e)g(normalize)h(with)g Fm(Z)505 1075 y Fn(s)538 1068 y Fu(giv)o(en)f(b)o(y)g(\(10\).)k(Then) 333 1158 y Fm(E)s Fu([)p Fm(Z)414 1165 y Fn(s)p Fs(+)p Fn(ds)499 1158 y Fk(\000)10 b Fm(Z)575 1165 y Fn(s)593 1158 y Fu(])i(=)h(\()p Fm(e)705 1139 y Fn(s)733 1158 y Fu(+)e Fm(e)800 1139 y Fn(s)817 1158 y Fm(ds)p Fu(\)\()p Fm(X)936 1165 y Fn(s)963 1158 y Fk(\000)g Fm(X)1047 1165 y Fn(s)1064 1158 y Fm(ds)f Fu(+)h Fm(\013X)1232 1165 y Fn(s)1249 1158 y Fm(ds)p Fu(\))f Fk(\000)h Fm(e)1389 1139 y Fn(s)1406 1158 y Fm(X)1444 1165 y Fn(s)1474 1158 y Fu(=)i Fm(\013Z)1582 1165 y Fn(s)1600 1158 y Fm(ds)238 b Fu(\(15\))14 1248 y(whic)o(h,)20 b(as)f Fm(\013)g Fu(is)h(small,)g (justi\014es)f(our)g(statemen)o(t)f(that)g Fm(Z)1057 1255 y Fn(s)1094 1248 y Fu(is)h(almost)g(a)f(martingale.)32 b(W)l(e)19 b(close)g(with)g(t)o(w)o(o)14 1305 y(rough)e(upp)q(er)h(b)q (ounds)g(that)f(shall)h(b)q(e)g(con)o(v)o(enien)o(t)f(later.)26 b(As)17 b Fm(\013)g Fu(is)h(alw)o(a)o(ys)e(nonnegativ)o(e)i Fm(E)s Fu([)p Fm(Z)1712 1312 y Fn(t)p Fs(+)p Fn(dt)1781 1305 y Fk(j)p Fm(Z)1825 1312 y Fn(t)1839 1305 y Fu(])d Fk(\025)i Fm(Z)1950 1312 y Fn(t)14 1361 y Fu(for)e(all)h Fm(t)f Fu(so)g(for)g(an)o(y)g Fm(s)411 1345 y Fo(0)435 1361 y Fk(\024)e Fm(s)837 1418 y(E)s Fu([)p Fm(Z)918 1425 y Fn(s)935 1418 y Fk(j)p Fm(Z)979 1425 y Fn(s)995 1416 y Fd(0)1008 1418 y Fu(])f Fk(\025)h Fm(Z)1112 1425 y Fn(s)1128 1416 y Fd(0)14 1495 y Fu(As)i Fm(\013)e Fk(\024)208 1477 y Fs(1)p 176 1484 81 2 v 176 1511 a Fn(D)q(e)220 1502 y Fd(\000)p Fc(t)276 1495 y Fu(w)o(e)i(ha)o(v)o(e)g(in)h(the)f (other)g(direction)516 1611 y Fm(E)s Fu([)p Fm(Z)597 1618 y Fn(t)p Fs(+)p Fn(dt)665 1611 y Fk(j)p Fm(Z)709 1618 y Fn(t)724 1611 y Fu(])d Fk(\024)h Fm(Z)828 1618 y Fn(t)850 1552 y Fj(\022)881 1611 y Fu(1)d(+)994 1580 y Fm(dt)p 964 1601 101 2 v 964 1642 a(D)q(e)1024 1629 y Fo(\000)p Fn(t)1070 1552 y Fj(\023)1113 1611 y Fk(\024)j Fm(Z)1192 1618 y Fn(t)1214 1611 y Fu(exp)1291 1552 y Fj(\022)1357 1580 y Fm(dt)p 1326 1601 V 1326 1642 a(D)q(e)1386 1629 y Fo(\000)p Fn(t)1432 1552 y Fj(\023)14 1720 y Fu(so)i(for)g(an)o (y)f Fm(s)246 1704 y Fo(0)271 1720 y Fk(\024)f Fm(s)480 1795 y(Z)511 1802 y Fn(s)527 1794 y Fd(0)552 1795 y Fk(\024)g Fm(E)s Fu([)p Fm(Z)681 1802 y Fn(s)698 1795 y Fk(j)p Fm(Z)742 1802 y Fn(s)758 1794 y Fd(0)770 1795 y Fu(])g Fk(\024)g Fm(Z)875 1802 y Fn(s)891 1794 y Fd(0)911 1795 y Fu(exp)q([)994 1738 y Fj(Z)1035 1748 y Fn(s)1016 1832 y(s)1032 1824 y Fd(0)1095 1765 y Fm(dt)p 1065 1785 V 1065 1827 a(D)q(e)1125 1814 y Fo(\000)p Fn(t)1171 1795 y Fu(])f(=)h Fm(Z)1275 1802 y Fn(s)1291 1794 y Fd(0)1304 1795 y Fm(e)1325 1777 y Fs(\()p Fn(e)1354 1764 y Fc(s)1370 1777 y Fo(\000)p Fn(e)1412 1764 y Fc(s)1426 1756 y Fd(0)1439 1777 y Fs(\))p Fn(=D)1883 1795 y Fu(\(16\))85 1898 y(Our)h(c)o(hoice)h (of)f Fm(!)i Fu(assures)e(that)g(\()p Fm(e)697 1882 y Fn(s)722 1898 y Fk(\000)9 b Fm(e)787 1882 y Fn(s)803 1869 y Fd(0)816 1898 y Fu(\))p Fm(=D)14 b Fu(is)h(small)g(so)f(emplo)o (ying)h(the)f(inequalit)o(y)i Fm(e)1651 1882 y Fn(x)1684 1898 y Fk(\024)d Fu(1)8 b(+)h(2)p Fm(x)k Fu(v)m(alid)14 1955 y(for)i(0)d Fk(\024)h Fm(x)g(<)g Fu(1)i(w)o(e)f(rewrite)i(\(16\))e (as)637 2086 y Fm(Z)668 2093 y Fn(s)684 2085 y Fd(0)709 2086 y Fk(\024)f Fm(E)s Fu([)p Fm(Z)838 2093 y Fn(s)855 2086 y Fk(j)p Fm(Z)899 2093 y Fn(s)915 2085 y Fd(0)928 2086 y Fu(])f Fk(\024)h Fm(Z)1032 2093 y Fn(s)1048 2085 y Fd(0)1061 2086 y Fu([1)c(+)i(2)1180 2056 y Fm(e)1201 2039 y Fn(s)1228 2056 y Fk(\000)g Fm(e)1295 2039 y Fn(s)1311 2027 y Fd(0)p 1179 2076 145 2 v 1232 2118 a Fm(D)1329 2086 y Fu(])541 b(\(17\))14 2174 y(and)16 b(our)e(c)o(hoice)j(of)d Fm(!)j Fu(further)e(assures)619 2265 y Fm(Z)650 2272 y Fn(s)666 2263 y Fd(0)691 2265 y Fk(\024)e Fm(E)s Fu([)p Fm(Z)820 2272 y Fn(s)837 2265 y Fk(j)p Fm(Z)881 2272 y Fn(s)897 2263 y Fd(0)909 2265 y Fu(])g Fk(\024)g Fm(Z)1014 2272 y Fn(s)1030 2263 y Fd(0)1043 2265 y Fu([1)c(+)i Fm(O)q Fu(\(ln)1225 2243 y Fo(\000)p Fn(K)1291 2265 y Fm(D)q Fu(\)])14 2355 y(for)k(all)h Fm(s)168 2338 y Fo(0)180 2355 y Fm(;)8 b(s)p Fu(.)19 b(Recall)e Fm(Z)422 2362 y Fs(0)453 2355 y Fu(=)c Fm(D)q Fu(.)20 b(This)c(assures)f(the)g(v)o (ery)g(rough,)g(but)g(useful)873 2445 y Fm(E)s Fu([)p Fm(Z)954 2452 y Fn(s)971 2445 y Fu(])d Fk(\024)h Fu(2)p Fm(D)778 b Fu(\(18\))p eop 8 7 bop 14 162 a Fe(the)17 b(electr)o(onic)h(journal)g(of)f(combina)m (torics)f(4)i(\(no.)j(2\))c(\(1997\),)g(#R19)430 b Fu(8)14 311 y Fv(6)67 b(The)22 b(V)-6 b(ariance)14 413 y Fu(Our)16 b(ob)s(ject)e(here)i(will)h(b)q(e)f(to)e(sho)o(w)h(\(13\))f(in)i(the)f (form)719 515 y Fm(E)s Fu([\()p Fm(Z)818 522 y Fn(t)841 515 y Fk(\000)c Fm(D)q Fu(\))944 496 y Fs(2)962 515 y Fu(])h Fk(\024)h Fm(cD)q Fu(\()p Fm(t)d Fu(+)g(1\))p Fm(e)1245 496 y Fn(t)1883 515 y Fu(\(19\))14 617 y(where,)15 b(for)g(de\014niteness,)h(w)o(e)f(set)926 673 y Fm(c)e Fu(=)g(80)14 757 y(W)l(e)i(actually)h(sho)o(w)f(the)g(follo)o(wing.)14 813 y Fi(Lemma:)20 b Fl(If)14 b Fm(E)s Fu([\()p Fm(Z)362 820 y Fn(s)389 813 y Fk(\000)c Fm(D)q Fu(\))491 797 y Fs(2)509 813 y Fu(])j Fk(\024)g Fm(cD)q Fu(\()p Fm(s)c Fu(+)i(1\))p Fm(e)798 797 y Fn(s)830 813 y Fu(for)k(all)h Fm(s)d Fk(\024)g Fm(t)i Fl(then)g Fm(E)s Fu([\()p Fm(Z)1275 820 y Fn(t)1299 813 y Fk(\000)10 b Fm(D)q Fu(\))1401 797 y Fs(2)1419 813 y Fu(])i Fm(<)h(cD)q Fu(\()p Fm(t)d Fu(+)h(1\))p Fm(e)1703 797 y Fn(t)1717 813 y Fu(.)85 870 y(Assume)21 b(this)h(Lemma)g(and)f(consider)i(the)e(function)h Fm(f)5 b Fu(\()p Fm(t)p Fu(\))23 b(=)g Fm(E)s Fu([\()p Fm(Z)1328 877 y Fn(t)1356 870 y Fk(\000)14 b Fm(D)q Fu(\))1462 853 y Fs(2)1481 870 y Fu(])f Fk(\000)i Fm(cD)q Fu(\()p Fm(t)f Fu(+)g(1\))p Fm(e)1775 853 y Fn(t)1790 870 y Fu(.)38 b Fm(f)27 b Fu(is)21 b(a)14 926 y(con)o(tin)o(uous)13 b(function)h(for)e(0)g Fk(\024)h Fm(t)g Fk(\024)g Fm(!)i Fu(and)e Fm(f)5 b Fu(\(0\))12 b(=)h Fk(\000)p Fm(cD)g(<)g Fu(0.)19 b(If)13 b(some)f Fm(f)5 b Fu(\()p Fm(t)1341 933 y Fs(1)1360 926 y Fu(\))13 b Fm(>)f Fu(0)h(then)g(b)o(y)g(the)g(In) o(termediate)14 983 y(V)l(alue)k(Theorem)g(some)f Fm(f)5 b Fu(\()p Fm(t)511 990 y Fs(2)529 983 y Fu(\))14 b(=)h(0)h(and)h(b)o(y) f(con)o(tin)o(uit)o(y)g(there)h(w)o(ould)f(b)q(e)h(a)f(minimal)i Fm(t)f Fu(with)f Fm(f)5 b Fu(\()p Fm(t)p Fu(\))15 b(=)f(0.)23 b(But)14 1039 y(then)16 b Fm(f)5 b Fu(\()p Fm(s)p Fu(\))12 b Fk(\024)h Fu(0)i(for)g Fm(s)d Fk(\024)h Fm(t)j Fu(so)f Fm(f)5 b Fu(\()p Fm(t)p Fu(\))12 b Fm(<)h Fu(0,)i(a)g(con)o (tradiction.)20 b(Hence)c(all)g Fm(f)5 b Fu(\()p Fm(t)1320 1046 y Fs(1)1339 1039 y Fu(\))12 b Fk(\024)h Fu(0,)i(whic)o(h)h(is)g (precisely)h(\(19\).)85 1096 y(Note)c Fm(Z)224 1103 y Fs(0)256 1096 y Fu(=)g Fm(D)q Fu(,)g(constan)o(t.)19 b(Our)c(idea)f(is)h(that)e Fm(Z)930 1103 y Fn(s)948 1096 y Fu(,)h(0)e Fk(\024)h Fm(s)g Fk(\024)g Fm(t)p Fu(,)i(is)f(almost)g(a)f (con)o(tin)o(uous)i(time)f(martingale.)14 1217 y Fa(6.1)56 b(The)18 b(SplitUp)14 1303 y Fu(W)l(e)d(split)i([0)p Fm(;)8 b(t)p Fu(])13 b(in)o(to)j(in)o(terv)m(als)g([)p Fm(s;)8 b(s)h Fu(+)i Fm(ds)p Fu(])k(and)g(write)735 1405 y Fm(Z)766 1412 y Fn(t)791 1405 y Fk(\000)c Fm(D)i Fu(=)936 1365 y Fj(X)958 1452 y Fn(s)996 1405 y Fu(\()p Fm(Z)1045 1412 y Fn(s)p Fs(+)p Fn(ds)1131 1405 y Fk(\000)d Fm(Z)1207 1412 y Fn(s)1225 1405 y Fu(\))14 1540 y(with)k Fm(s)h Fu(from)e(0)g(to)h Fm(t)7 b Fk(\000)g Fm(ds)15 b Fu(in)g(steps)f(of)f Fm(ds)p Fu(.)19 b(\(Again)14 b(w)o(e)g(can)g(a)o(v)o(oid)g (in\014nitesimals)i(b)o(y)e(making)g(these)g(steps)g(\001)p Fm(s)14 1597 y Fu(and)i(letting)f(\001)p Fm(s)e Fk(!)g Fu(0)i(at)g(the)g(end.\))20 b(Squaring)c(and)f(taking)h(exp)q(ectation) 696 1699 y Fm(E)s Fu([\()p Fm(Z)795 1706 y Fn(t)819 1699 y Fk(\000)10 b Fm(D)q Fu(\))921 1680 y Fs(2)939 1699 y Fu(])j(=)g Fm(V)c(AR)h Fu(+)h Fm(C)s(O)q(V)14 1801 y Fu(where)16 b(the)f(squared)g(terms)g(giv)o(e)g(the)h(\\v)m(ariance") 709 1903 y Fm(V)9 b(AR)k Fu(=)875 1862 y Fj(X)897 1950 y Fn(s)943 1903 y Fm(E)s Fu([\()p Fm(Z)1042 1910 y Fn(s)p Fs(+)p Fn(ds)1126 1903 y Fk(\000)e Fm(Z)1203 1910 y Fn(s)1221 1903 y Fu(\))1239 1884 y Fs(2)1257 1903 y Fu(])613 b(\(20\))14 2035 y(and)16 b(the)f(crossterms)f(giv)o(e)h(the)h(\\co)o(v)m(ariance") 523 2137 y Fm(C)s(O)q(V)22 b Fu(=)13 b(2)723 2097 y Fj(X)745 2184 y Fn(s)794 2097 y Fj(X)790 2186 y Fn(s