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AuthorindexAnderson,T.
W.
,29,102,192Arrow,K.
J.
,4Bachelier,L.
,29Bailey,N.
T.
J.
,2Baraard,G.
A.
,102Bartlett,M.
S.
,1,2,6,299Battin,R.
H.
,4Bharucha-Reid,A.
T.
,2,202Billingsley,P.
,192Birkhoff,G.
D.
,73Birnbaum,A.
,137Birnbaum,Z.
W.
,102Blackwell,D.
,183Brockmeyer,E.
,283Chandrasekhar,S.
,33,152Chapman,D.
,142Chung,K.
L.
,136,221,270,277Cox,D.
R.
,192Cramér,H.
,89,134Darling,D.
A.
,29,102Davenport,W.
B.
,Jr.
,31,150Davis,D.
J.
,31Donsker,M.
D.
,102Doob,J.
L.
,52,70fn,73,96,97Einstein,A.
,1,27,29Epstein,B.
,137Evans,R.
D.
,31,164Feller,W.
,57,126,164,172,196,202,203,221,235,270,291Foster,F.
G.
,260Friedman,B.
,89Fuller,A.
T.
,37Furry,W.
,297Galton,F.
,201Gilbert,E.
N.
,150Ginshick,M.
A.
,136Good,I.
J.
,150,152Goodman,L.
A.
,192Greenwood,M.
,57Grenander,U.
,6Groll,P.
A.
,135Gumbel,E.
J.
,169Gupta,S.
S.
,135Haar,D.
ter,seeterHaarHagstroem,K.
G.
,7fnHalstroem,H.
L.
,283Hannan,E.
J.
,6Hardy,G.
H.
,218Harris,T.
E.
,2Heistrom,C.
W.
,4Jenkins,G.
M.
,111Jensen,A.
,283Kac,M.
,29,102Karlin,S.
,4,173,280,299Kemeny,J.
G.
,241Kendall,D.
G.
,199,221,272,273,304Khinchin,A.
I.
(Khintchine,A.
Ya.
),73,108,110,125Kolmogorov,A.
N.
,291Korff,S.
A.
,164314315AUTHORINDEXLanning,J.
H.
,4Root,W.
L.
,31,150Ledermann,W.
,299Roeenblatt,M.
,6Loève,M.
,52,70fn,73,136,267Rubin,H.
,136Lomax,K.
S.
,169Longuet-Higgins,M.
S.
,40Satterthwaite,F.
E.
,57Lotka,A.
J.
,201Scarf,H.
,4,179Schottky,W.
,1MacFadden,J.
A.
,40Scott,E.
L.
,1,71,130McGregor,J.
L.
,280,299Sevastyanov,B.
A.
,283Machlup,S.
,39Sitgreaves,R.
,136Maguire,B.
A.
,102,137,139Skellam,J.
G.
,32Mandelbrot,B.
,150Slepian,D.
,40Middleton,D.
,4Smith,W.
L.
,179,181,182,183Miller,G.
A.
,191Smoluchowaki,M.
V.
,1Molina,E.
C.
,281Snell,J.
L.
,241Montroll,E.
W.
,29Steffenaen,J.
F.
,201Syski,R.
,4,299,302Neumann,J.
von,setvonNeomaaNeyman,J.
,1,71,130Tak,L.
,182,183Olkin,I.
,18terHaat,D.
,73Ornatein,L.
S.
,97Osborne,M.
F.
M.
,29Uhlenbeck,G.
E.
,97Owen,A.
R.
G.
,168vonNeoman,J.
,73Parten,E.
,8fn,72,74,111Pearson,E.
S.
,102,137,139Perrin,J.
,27,29Pollak,H.
0.
,150Whittle,le,P.
,.
,85Wiener,ener,N.
,28,29,108,110Praia,S.
J.
,257Pratt,J.
W,18Wold,H.
,6Wonham,W.
M.
,37Rónyi,A.
,40Wynn,A.
H.
A.
,102,137,139Reuter,G.
E.
H.
,299Rice,S.
0.
,150Yule,G.
U.
,57,297-298SubjectindexAbsorbingbarrier,231AbsorbingMarkovchain,241,246Absorbingstate,231,235,236,239,243,244,246definitionof,211,229Absorption,timebefore,238timeto,238-246,263Absorptionprobability,229,231-233,236,237,246Accelerationinchaos,150,152Accessiblestates,208,211,223,224AperiodicMarkovchain,256-258,270-273Aperiodicstate,255Approximatelynormaldistribution,40,88,97,99,103,141,142,157,180,181,267-269,295joint,98Approximatelynormalprocess,98,101Arrivaltimes,143,165Asymptoticallynormaldistribution(seeApproximatelynormaldistribution)Asymptoticallynormalprocess(seeAp-proximatelynormalprocess)Averages,ensemble,72-74Backwardequation,291-293,300Bernoullitrials,15,46,47,50(seealsoCointossing)Betafunction,61Binomialdistribution,13,15,17,21,45-49Birthanddeathprocess,276,278-283,295,299,300,304,306(seealsoPurebirthprocessandPuredeathproc-ess)homogeneous,299,300non-homogeneous,299,300,306Birthprocess,pure(seePurebirthprocess)Blaekwell'stheorem,183BoreIfunction,10nBorelsets,9,10nBranchingprocess,58,199,201,207,216,237,261Breakdown,system(seeFailure,system)Brownianmotion,2,7,27-29,80,95,96,98,129(seealsoWienerprocess)Busychannelsorservers,144-149Campbell'stheorem,149,150,155Cascadeprocess(seeBranchingprocess)Cauchydistribution,21,151CensoredPoissondistribution,283Centrallimittheorem,19,101,141,267,269Cesàromean,convergentein,74,75,247,249Chapman-Kolmogorovequation,194,195,203-205,208,222,250,276,278,290,292Characteristicfunction,definitionof,11Infinitelydivisible,124joint,13,22,25,27,37,53,77,98,116normal,88,89,98stable,151ofastochasticprocesswith(stationary)independentincrement,123,124Characteristicfunctions,tableof,13,14316SUBJECTINDEX317Chebyshev'sinequality,18,67forstochasticprocesces,85Chi-squaredistribution,14,15,24,136,137Chi-squaretest,135Cointoesing,45,46,205,206,237(aualaoBernoullitrials)Communicatingclasses,208-211,222-225,235,245,256classificationof,222,223Communicatingstater,208,224-227,244,245,256,262-264,267,276Communicationtheory,statistical,1,2,84,103,104,109,197,198Compoundexponentialdistribution,60,61CompoundPoissondistribution,57CompoundPoissonprocem,99,128-130,146,156Conditionalcharacteristicfunction,defini-tionof,56,57Conditionalexpectation,41definitionof,continuouscase,5153discretecase,42-44joint,58propertiesof,62-65Conditionalmean(aeeConditionalex-pectation)Conditionalmomentti,definitionof,54-56,58Conditionalprobability,definitionof,con-tinuousoase51,52discretecase,41,42,44propertiesof,continuouscase,51discretecase,45Conditionalprobabilitydensityfunction,definitionof,51Conditionalprobabilitydistributionfune-tion,definitionof,41-43Conditionalprobabilitymaasfunction,definitionof,42,43Conditionalvariance,definitionof,54,55Confidenceinterval,136-139Continuousparameterprocees,definitionof,8describinga,23ergodic,72integralof,79withMarkovproperty,188,190Continuousprobabilitylawa,tableof,14,15Continuousrandomvariable,definitionof,10Convergence,modesof,78inCésaromean,74,75,247,249inmeansquare,78,79,83,91Convergentetheorem,dominated(aceDominatedconvergencetheorem)Convolutionofdistributione,16,17Correlationooefcients,53,94,96matrixof,94Counter,defective,48,50Geiger,30,37,48,50,131,164non-paralyzable,164,165,170,181,182,185nuclearparticle,139,162-166,179,181paralyzable,145,158,164-166,181,182,186typep,164,186Countingprocees,30,36,48,117-199,127,133,137,160,162,164(re.
alroInteger-valuedprocees)renwal(ee.
Renwalcountingprocees)Covariance,definitionof,16Covariancefunction,forbusyservers,149definitionof,71derivativeof,84-87,92ofanergodicprocees,74-76inharmonieanalysis,109-115ofanormalprocees,91,92,94,96sample,109ofawhitenoire,113-115Covariancekernel,ofanapproximatelynormalproceee,101definitionof,67ofthederivativeofaprocees,84ofanergodicprocees,74exereisesofthe,76,77,96ofafilteredPoissonpraxes,148inharmonieanalysen,111,112oftheincrementproceesofthePo~prooess,68,69ofanintegratedWienerprocees,81ofanormalproces.
,89,92,95ofaPoissonprocees,67,68roleof,instochaaticintegral,7981,87ofa8tationaryproces.
,70-72ofaWienerproces.
,67,68Covariancematrix,88,89,92Covariancestationaryproces*,inbusyserversproblem,149Chebyshevinequalityfor,85,86definitionof,70derivativeof,84-87,91ergodic,75generaldiscussionof,66,71,74,77,96,108inharmonieanalysis,108-116integralof,87,108normallydistributed,91,94Cramér-vonMisestest,100,102,141,143(acealaoGoodneesoffittest)Customerleesratio,286,288318SUBJECTINDEXDeadtime,ofnon-paralyzablecounters,164,165,170,185ofnuclearparticlecounters,163,164,181,182ofparalyzablecounters,164-166,186Deathproces,pure,299,300Decisiontheory,43DecompositionofMarkovchain(seeMarkovchain,decompositionof)Defectivecounter,48,50Densityfunction,probability,10,14Derivativeofastochasticprocess,83,84,87Differenteequations,49,50,239stochastic,193Differentiabilityinmeensquare,83-85,91-92Differentialequations,stochastic,84,113Diracdeltafunction,113Discretebranchingprocess(eeeBranchingprocess)DiscreteparameterMarkovchain(secMarkovchain,discreteparameter)Discreteparameterprocess,definitionof,7describing,22ergodic,72,73withMarkovproperty,188Discreteprobabilitylaws,tableof,13,15Discreterandomvariable,definitionof,10DishonestMarkovprocees,92Distributionfunction,probability,10,72Dominatedconvergentetheorem,128,218,251proofof,173,174Dot-dashproces,36,38,40D-statistic(secKolmogorov-Smirnovtest)Durationofrandomwalk,239Ecology,2,32,48Eigenvalues,theoryof,196Ensembleaverages,72-74ErgodicMarkovchain(setMarkovchain,irreducible,positiverecurrent)Ergodicprocees,74,75Ergodictheorem,72,73Ergodicity,geometric,272,273Erlangianprobabilitydistribution,199Erlang'slossformula,283Essentialstates,224Estimation,ofensembleaverages,72,83parameter,3,6,35,143,268,298withconfidenceintervals,136-139withmethodofmoments,134,139methodeof,135ofthespectraldensityfunction,111,116oftransitionprobabilitymatrix,207Event,random,definitionof,8Evolutionaryprocess,69Excesslife,asymptoticdistributionof,184distributionof,171-173meenof,179,186Expectation(eeeMean)Exponentialdistribution,14,15,20,21,26,31,33,62,82,98,132,153,170,282,285characterizationof,123,282compound,60,61ofexcesslife,173ofinter-arrivaltimes,98,118,133,135,137,142,163,173-177,199,281,283,287oflifetime,31,60,61,163ofthesamplespectraldensityfunction,116ofservicetime,82,148,149,190,199,206,281,286,287Extinctionprobabilities,201,202,207,216,305Extrapolation,3,43,113,114Extremevaluedistribution,169Failurefunction,conditionalrateof,168,169Failureprocess,289Failures,system,5,31,137,162,168,184,192,205,247,286,289Familyofevents,8,9Fatou'slemma,250,273F-distribution,14,136Filter,104-107,110-115FilteredPoissonprocess(eetPoissonproc-ess,filtered)FiniteMarkovchain(secMarkovchain,finite)Firstentrance,methodof,214,215Firstpassageprobabilities,213,226,227,229,231,242,247,259,266definitionof,213eninequalityon,224,225Firstpassagetime,189,213,246,265meen,242-244,253recurrentcase,259,264Fourierintegral,108-112Fourierseries,108Fouriertransform,12,106-109Forwardequation,291-293,300F-probabilitylaw,14,136Frequencyresponsefunction,105-108,110-113F-test,139Fubini'stheorem,273,275Functionalequations,solutionof,121-123SUBJECTINDEX319Furryprocess,296,297,299,302,306Galaxies,distributionof,129,130Gambler'sruin,229,231,234,235Gammadistribution,14,15,17,20,57,61,132,135,199,285ofinter-arrivaltimes,162,175,177,181,199ofwaitingtimes,133,134,174,285Gammafunction,62Gaussiandistribution(seeNormaldistri-bution)Geigercounter(seeCounter,Geiger)GeneralizedPoissonprocess(seePoissonprocess,generalized)Geometriedistribution,13,15,21,132,139,284,299,303characterizationof,123Geometrieergodicity,272,273Goodnessoffittest,99,102,135Harmonies,105,106Hazardfunction,168,169HomogeneousMarkovchain(seeMarkovchain,homogeneous)HornogeneousMarkovprocess,189HomogeneousPoissonprocess,124-126Hypothesistesting,3,135,137,139,142-1441denticallydistributedrandomvariables,definitionof,10Identicallydistributedstochasticproc-esses,definitionof,25ImbeddedMarkovchain(seeMarkovchain,imbedded)Impulseresponsefunction,105,113,156InerementprocessofaPoissonprocess,68,87Independentincrements,integer-valuedproceswith,37,124,125,191ofthepurebirthprocess,296stochasticprocesseswith,28,47,65,81,117,130,296covariancekernelof,68definitionof,26,27Independentrandomvariables,criteriafor,13,15Independentstochasticprocesses,defini-tionof,25Indicatorfunction,definitionof,45Inessentialstates,224,243Infinitelydivisiblelaws,124Inputfunction,104-108,110-114Integer-valuedprocess,117,123,160,299(seealsoCountingprocess,One-minus-0neprocessandTwo-valuedprocess)definitionof,30withindependentincrements,37,124,125,191withstationaryindependentincrements,118,126,130(seealsoPoissonproc-ess,generalized)Integralofstochasticprocessas(seeSto-chasticintegral)Intensityfunction,168,169IntensityfunctionofaPoissonproces,125IntensityfunctionsofaMarkovchain,289-292,305,306Intensityofpassage,278,289,293,294Intensityoftransition(seeTransitionin-tensity)Inter-arrivaltimes,138,139,168,170,266definitionof,117,132,133,266exponentiallydistributed,118,133-135,142,173-177,199,281,283,287gammadistributed,174-177,181,199mean,148,185,186,258,265,282ofanon-homogeneousPoissonprocesa,138inqueues,5,199,206,207,258,265,281-283,287inarenewalcountingproceas,163,165,166,171-183,185,186Inversefunction,definitionof,21Inversionformula,12IrreducibleMarkovchain(seeMarkovchain,irreducible)Jointcharacteristicfunction(seeChar-acteristicfunction,joint)Jointlydistributedrandomvariables,12Jointlynormaldistribution,approximate-ly,98n-dimensional,88-94two-dimensional,53,54,59,95,96,116Jointlynormalprobabilitydensityfunc-tion,54,88,89,94Keyrenewaltheorem,183,184,186Kolmogorov-Smirnovtest,100-102,141,143(secalsoGoodnessoffittest)Kolmogorov'sbackwardequation,291-293,300Kolmogorov'sforwardequation,291-293,300Lagrange'smetbod,302Laplace-Stieltjestranaform,177,178,180,183Latticerandomvariable,182,185,186320SUBJECTINDEXLawoflargenumbers,18,267Lifetesting,137Lineargrowthprocess,non-homogeneous,302Lockingtime(seeDeadtime)Long-rundistribution,250ofbirthanddeathprocess,280,282,283,287determinationof,261,262definitionof,248existenceof,255-258,265,277,284forqueues,284,286Machinefailures(seeFailures,system)Managementscience,1,4,144,146Markovchain,187-199,203-217,220-273,276-306absorbing,241,246aperiodic,256-258,270-273continuousparameter,definitionof,190,193decompositionof,209-211,222,223,263definitionof,187-189discreteparameter,definitionof,189,193finite,absorptiontimeof,239,241,246decompositionof,222,223definitionof,196firstpassageprobabilitiesof,229,253irreducible,249,251,253,255,257,262,270,271,273,277stationarydistributionof,251transitionprobabilitymatrixof,232,246,255,257horoogeneous,208,212definitionof,193,194imbedded,198,206intensityfunctionsof,289,292,293transitionprobabilitymatrixof,196-198,204,206,207,256two-state,197,198,207,256,293imbedded,190,273oftheGI/M/1queue,206,265oftheM/G/1queue,190,198,199,258,260irreducible,236,249,277classificationof,258,260definitionof,235,276withdoublystochastictransitionprobabilitymatrix,255,256,262geometricallyergodic,272withalong-rundistribution,255,256,257,258,265,277non-eurrent,235-237,249,253,258,260,261,265null-recurrent,258,260,265periodic,256,258,263positiverecurrent,245,252,253,256,258,260-262,264-266,277inqueueingtheory,258recurrent,225,235,236,242-244,246,252,258,260,270stationarydistributionof,251,252,258,265,277multiple,191,192non-homogeneous,288withstationarytransitionprobabilities(secMarkovchain,homogeneous)two-state,197,207,213,293,256Markovprocess,117,187-189,191,192classificationof,188definitionof,187,188homogeneous,189pathological,292withstationarytransitionprobabilities,189Markovproperty,194,214Martingale,65Mean,definitionof,13Meanrecurrencetime(seeRecurrencetime,mean)Meansquare,convergentein,78,79,80,91Meansquaredifferentiability,83-85,91-92Meansquareerror,114Meanvaluefunction,68,72,76,79-83,85,92,95definitionof,67derivativeof,83,156ofacovariancestationaryprocess,71,94ofahomogeneousPoissonprocess,68,125,143oftheincrementprocessofthePoissonprocess,68,69ofanon-homogeneousPoissonprocess,125,138,156,158,297ofanormalprocess,67,95,101ofarenewalcountingprocess,170,177-181,183,185ofastochasticprocesswithstationaryindependentincrements,77Means,tableof,13,14Methodoffustentrance,214,215Momentfunction,second,179Moment-generatingfunction,definitionof,11Moments,17,18MultipleMarkovchain,191,192Multiplicativeprocess(seeBranchingprocess)8UBJECTINDEX321Negativebinomialdistribution,13,15,17relatedtopurebirthprocesses,297relatedtothecompoundPoissonandgeometricdistributions,57,58relatedtothegeneralizedPoissondis-tribution,127relatedtorenewalcountingprocesses,179Non-homogeneousbirthanddeathproc-ess,299,300,306Non-homogeneouslineargrowthprocess,302Non-homogeneousMarkovchain,288Non-homogeneousPoissonprocess(neePoissonproces,non-hoxnogeneous)Non-Markovchain,205Non-Markovprocess,203Non-paralyzablecounter,164,165,170,181,182,185Non-recurrentclans,221-225,235Non-recurrentMarkovchain,235-237,249,253,258,260,261,265Non-recurrentstate,221-229,235,238,239,241-247,252,263definitionof,221Non-returnstate,208-211Normaldistribution,14,15,17,19-21,54,60,91,93,95,96,141,169asymptotically(saApproximatelynor-maldistribution)jointly(zeeJointlynormaldistribution)ofanormalproces,88relatedtoafilterodPoissonprocess,151instochasticprocesnes,26,66,76,77,192Normalprobabilitydensityfunction,jointly,54,88,89,94Normaiproces,66,88-99,101,102,113,114,116approximately,98,101definitionof,89,90.
Nuclearpartielecounter,139,162-166,179,181Null-recurrentMarkovchain,258,260,265Null-recurrentstate,245Occupationtime,189,221,265,268definitionof,211,212One-minus-oneprocess,36,37,77(zeealloTwo-valuedproces,)Operationsresearch,1,4,144,146Ornstein-Uhlenbeckprocess,96,97,115Orderstatistics,140,141,143Outputfunction,104-108,110-114Paralyzablecountcr,14-5,158,164-166,181,182,186Parameterestimation(zeeEstimation,parameter)Passageintensity,278,289,293,294PathologicalMarkovprocess,292Period,ofacommunicatingclans,256,262,263ofastate,255,256,262,263PeriodicMarkovchain,256,258,263Periodicstate,255,256Persistentstate(zeeRecurrentstate)Planchereltheory,108,109Poissoncountingprocees(neeCountingprocess)Poissondistribution,13,15,17,20,21,26,31,97,102,127ofarrivals,82,134,135,138,144,147,149,152,153,158censored,283compound,57ofacompoundPoissonproces,57,129,132inestimationandtesting,136,137,139,141,142,144andafilteredPoissonprocess,147,148,151-153ofparticlearrivalsatcounters,162,170,179,181,182,185,186ofaPoissonprocess,30-32,34,67,119,124ofapurebirthproces,296inqueueing,149,190,282ofarenewalcountingprocees,174,176Poissonprocess,8,15,26,31,33-35,69,77,117,131-136,143,155,159,192,288asymptoticnormalityof,103axiomaticderivationof,118-120,123andthebinomialdistribution,49inaBrownianmotion,80,98compound,99,128-130,146,156covariancekernelof,68,69definitionof,30inestimationandtesting,135,136,141,143filtered,144-147,150-152,157,159,186,297,306extended,152,153generalized,156filterednon-homogeneous,156generalized,32,118,126,127,130,131axiomaticderivationof,127homogeneous,124-126incrementproces,of,68,87intensityfunctionof,125322SUBJECTINDEXmeanvaluefunctionof,68,69,126,143non-homogeneous,118,124-126,138,143,156,157,297axiomaticderivationof,125,126withnon-stationaryincrements(seePoissonprocess,non-homogeneous)inparticlecounterproblems,132,138,139,158,163,166inapurebirthprocess,296,305underrandomselection,47,48,50asarandomtelegraphsignal,36,37asarenewalprocess,163,174samplefunctionof,29samplemeanof,86,inashotnoiseprocess,149inspace,31,32,34withstationaryincrements,124-126truncated,298andtheuniformdistribution,140,141Poissonsquarewave,77PositiverecurrentMarkovchain,245,252,253,256,258,260-262,264-266,277Positiverecurrentstate,245,253Prediction,3,43,113,114Probabilitydensityfunctions,definitionof,10tableof,14Probabilitydistributionfunction,defini-tionof,10ofastocbasticprocess,72Probabilityfunction,8-10transition(suTransitionprobabilityfunction)Probabilitylaws,definitionof,10tableof,13-15Probabilitymaasfunctions,definitionof,10tableof,13Purebirthprocess,293,295,300withconstantbirthrate,118,296,298wittiimmigration,305withlinearbirthrate,296,297,299,302,306Puredeathprocess,299,300Quadraticmean,convergentein,78,79,83,91Queues,190,191,262,273,281-283,288classificationof,5descriptionof,199finiteserver,281-288infiniteserver,144,145,147-149samplemeanof,82singleserver,198,199,284,288G1/M/1,206,265M/G/1,258two-server,288Radon-Nikodyntheorem,52Randomevent,definitionof,8Randomtelegraphsignal,37,77,80,86,115definitionof,36Randomvariables,definitionof,8-10identicallydistributed,10independent,13,15jointlydistributed,12kindsof,10uncorrelated,16Randomwalk,204,230,232,233,237,242,254,280withabsorbingbarrier,231approximatingabirthanddeathproc-ess,280definitionof,23,229durationof,239,241firstpassagetimein,243,244non-current,225,236,253null-recurrent,259positiverecurrent,245,253,254,259,280propertiesof,258,259recurrent,225,236,244,259withreflectingbarrier,231Rayleighdistribution,25Realizationofastochasticprocess(seeSamplefunction)Recurrencecriterion,225,235Recurrencetime,265-268mean,242-247,252-255,267,268Recurrentclans,221-227,235RecurrentMarkovchain(neeMarkovchain,irreducible,recurrent)Recurrentstate,221-229,235,245-247,252,253,263,264,267,268definitionof,221null,245positive,245,253Recursiverelations,195,200,201Reflectingbarrier,231Regressionanalysis,114Reliability,system,38,94,293Renewalcountingprocess,162,164,168,170,176,178,186,270asymptoticnormalityof,180,181definitionof,160delayed,162,163,178,179withexponentialinter-arrivaltienes,118,163,174,176withgammadistributedinter-arrivaltirnes,176-177,181meanvaluefunctionof,170,171,174,178-181,183,186momentsof,179SUBJECTINDEX323inparticlecounterproblems,162-168,179samplefunctionof,160,161Renewalequation,170-172,178,183solutionof,173,179,184,185Renewaltheorem,key,183,184,186Responsefunction,ofafilteredPoissonprocess,145-147frequency,105-108,110-113impulse,105,113,156Returnstate,208,209,255Sampleaverages,72-74Samplecovariancefunction,109Sampledescriptionspace,8,9,45Samplefunction,72,74,85ofcovariancestationaryprocesces,108definitionof,24ofaPoissonprocess,29ofarandomtelegraphsignal,36ofarenewalcountingproces,160,161Samplemaan,ofacontinuousparameterstochasticproces,75,76,78ergodic,73-75meanandvarianceof,82,86Samplespectraldensityfunction,109,116Scalingcircuits,162,185Schwarz'sinequality,75Scintillationcounter(teeCounter,non-paralyzable)Secondmomentfunction,179Secondorderstationaryproces,(seeCo-variancestationaryprocees)Semi-conductornoise,38,293Servicetimes,82,148,149,190ofafinite-eerverqueue,281-283,286,287ofaninfnite-eervenqueue,147ofqueues,5,199ofasingle-serverqueue,206,258,265Shotnoire,2,7,31asafilteredPolssopprocesti,144non-stationary,157aswhitenoire,115Shotnoiseprocens,115,149,150Sinewave,76,115Spatialdistributionofplant,andtinmalcommunities,2,32,48Spectralanalysis,111Spectraldensityfunction,110-116sample,109,116Spectraldistributionfunction,110Spectrum,103,104,111(teealsoSpectraldensityfunction)Stablecharacteristicfunction,151Standarddeviation,definitionof,11Standardizedrandomvariable,definitionof,97,98Statespace,decompositionof,209-211definitionof,188Stationarydistribution,ofaMarkovchain,definitionof,248,265determinationof,265exstenceof,249-252,258,277forqueues,283,284,287ofarandomwalk,254Stationaryindependentincrements,ofacompoundPoissonprocees,99,130ofageneralizedPoissonprocees,126,127ofaPoissonproces,30,33,118,120,136stochaaticprocessieswith,33,35,77,121asymptoticnormalityof,103characteristicfunctionof,123,124definitionof27(seealsoPoissonprocess,generalized)ofaWienerprocees,28StationaryMarkovchain(seeMarkovchain,homogeneous)Stationaryprocess,69,72,113,114covariance(auCovariancestationaryproces,)strictly,70-73,77,249Stationarytransitionprobabilities,Mar-kovchainwith(seeMarkovchain,homogeneous)Markovprocestiwith,189Statisticalcommunicationtheory,1,2,84,103,104,109,197,198Statisticaldecisiontheory,43Statisticalmechanica,1,2;27,73Statisticalspectralanalysis,111Steadystate,189Stieltjesintegral,42Stirling'sformula,225Stochasticdifferentialequations,84,113Stochasticintegral,79,80,295notionof,39,78,79,80Stochasticper,approximatelynor-mal,98,101classificationof,7,8continuousparameter(reeContinuousparameterpee)covanancestationary(seeCovariancestationaryprocees)definitionof,6,7,22-24derivativeof,83,84,87discreteparameter(seeDiacreteparam-eterproces)ergodic,74,75evolutionary,69identicallydistributed,25independenceof,25324SUBJECTINDEXinteger-valued,117,123,160,299Stationary,69,72,113,114strictlystationary,70-73,77,249two-valued,36,38,40Successruns,237Systemfailures(seeFailures,system)Systemreliability,38,94,293Telegraphsignal,random,37,77,80,86,115Testinghypotheses,3,135,137,139,142-144Thermalnoise,2,7,27ThiefofBaghdadproblem,50Timebeforeabsorption,238Timetoabsorption,238-246,263Time-of-deliverylag,4Timeseries,71,111,114definitionof,5economie,6Timeseriesanalysis,1,5,6,43,84,103,109Transientstate(seeNon-recurrentstate)Transitionintensity,289,293ofabirthanddeathprocesti,281-283definitionof,277,278Transitionprobabilityfunction,289,196,198,213,221,238,265,277,278asymptoticbehaviorof,247,272ofabirthanddeathprocess,278,283,299,300,306ofabranchingproces,200,201intheChapman-Kolmogorovequation,203,276,300definitionof,188,193,194determinationof,195differentialequationsfor,289,291-293,295ofafiniteMarkovchain,196,197limittheoremsfor,219one-step,193,200,207,277ofapurebirthprocess,295,296ofapuredeathproces,299Transitionprobabilitygeneratingfunc-tion,300-306Trausitionprobabilitymatrix,196,225,235,237,243,263,270intheChapman-Kolmogorovequation,195definitionof,194doublystochastic,255,256,262estimationof,207ofafiniteabsorbingMarkovchain,246ofafiniteMarkovchain,196,'210,211,241,257,258ofafiniterandomwalk,232,233,242ofahomogeneousMarkovchain,196-198,204,206,207ofarandomwalk,230,236,243-245,253,259,280ofatwo-stateMarkovchain,197,198,207TruncatedPoissonprocess,298Two-stateMarkovchain,197,207,213,293,256Two-valuedprocess,36,38,40TypeIcounter,164,165,170,181,182,185TypeIIcounter,158,164-166,181,182186Uncorrelatedrandomvariables,definitionof,16Uniformdistribution,14,15,19,21,76,77,100,132andthePoissondistribution,140,141,143,154,158Vacuumtube,2,30,31,150,162Variance,definitionof,11Variances,tableof,13,14Waitingtimes,5,15,50,132,133,137,138inMarkovchains,26-269andorderstatistics,140,143inqueues,283,285-287inrenewalprocesses,166,174,175,178Weaklystationaryprocess(weCovariancestationaryprocess)Weibuildistribution,169Whitenoire,113-115Wide-eensestationaryproceas(reeCo-variancestationaryprocesti)Wiener-Khintchinerelations,110Wienerprocess,8,26-29,40covariancekernelof,67,68,76definitionof,28ingoodnessoffittests,99-103integrated,81,87asamartingale,65meenvaluefunctionof,67,76asamodelforBrownianmotion,27,29,98,99,129asanormalprocess,88,91andtheOrnstein-Uhlenbeckprocess,97processesrelatedto,95,96,102,103W.
-statietic(seeCramér-vonMisestest)Yuleproces,297Zerocrossingproblem,40Zero-onelaw,215Zero-oneprocess,38,39(reealloTwo-valuedprocess)