Meta tags:

Headings (most frequently used words):

white, noise, time, applications, mathematical, vector, contents, statistical, properties, practical, definitions, generation, informal, use, see, also, references, external, links, music, electronics, engineering, computing, tinnitus, treatment, work, environment, discrete, continuous, series, analysis, and, regression, random, transformations,

Text of the page (most frequently used words):
the (203), noise (170), white (113), and (89), displaystyle (88), process (56), #random (54), that (40), signal (34), with (33), time (29), for (28), are (27), zero (24), vector (24), edit (23), gaussian (22), distribution (22), theorem (21), independent (19), used (19), this (18), model (16), 978 (16), not (16), sound (15), variables (15), values (15), theory (14), variance (14), other (14), isbn (14), such (14), mathrm (14), from (13), can (13), some (13), also (13), have (13), finite (13), non (12), probability (12), mathematical (12), value (12), continuous (12), variable (12), sigma (12), power (12), function (12), will (12), series (11), analysis (11), has (11), which (11), mean (11), its (11), statistical (10), stochastic (10), one (10), discrete (10), each (10), uncorrelated (10), over (10), definition (10), density (9), applications (9), must (9), normal (9), use (8), data (8), processing (8), space (8), spectrum (8), ratio (8), engineering (8), tinnitus (8), their (8), covariance (8), matrix (8), often (8), but (8), varphi (8), mathbb (8), any (8), flat (8), statistically (8), wikipedia (7), may (7), different (7), retrieved (7), markov (7), distributed (7), pdf (7), cognitive (7), doi (7), music (7), equal (7), frequencies (7), delta (7), case (7), where (7), defined (7), infinite (7), signals (7), only (7), components (7), statistics (6), learning (6), integral (6), doob (6), average (6), image (6), spectral (6), audio (6), background (6), radio (6), press (6), performance (6), pmid (6), times (6), these (6), component (6), term (6), they (6), two (6), between (6), mathcal (6), dimensional (6), toggle (5), under (5), page (5), was (5), links (5), acoustics (5), electronics (5), martingale (5), autoregressive (5), brownian (5), general (5), masking (5), environment (5), sleep (5), 2007 (5), elements (5), all (5), said (5), see (5), typically (5), langle (5), rangle (5), interval (5), real (5), however (5), sense (5), weaker (5), definitions (5), samples (5), fourier (5), frequency (5), contents (4), search (4), commons (4), large (4), ergodic (4), wiener (4), itô (4), tools (4), properties (4), generalized (4), models (4), moving (4), field (4), additive (4), diffusion (4), identically (4), filter (4), interference (4), energy (4), pink (4), archived (4), work (4), effects (4), issn (4), don (4), digital (4), springer (4), known (4), 4612 (4), light (4), generated (4), linear (4), transformation (4), regression (4), being (4), then (4), variances (4), therefore (4), while (4), because (4), requires (4), require (4), even (4), expected (4), particular (4), experiment (4), content (4), refers (4), hide (4), move (4), sidebar (4), view (3), contact (3), about (3), privacy (3), terms (3), wikimedia (3), wikidata (3), list (3), topics (3), machine (3), queueing (3), econometrics (3), stopping (3), differential (3), skorokhod (3), representation (3), lévy (3), kolmogorov (3), law (3), numbers (3), weak (3), similar (3), cox (3), constant (3), black (3), point (3), measure (3), fields (3), memory (3), system (3), local (3), evolution (3), jump (3), systems (3), denoising (3), communication (3), related (3), distortion (3), effective (3), channel (3), atmospheric (3), 2022 (3), original (3), com (3), does (3), modern (3), approach (3), wiley (3), 471 (3), forecasting (3), second (3), electrical (3), transformations (3), journal (3), july (3), adhd (3), treatment (3), loudspeakers (3), sequence (3), spatial (3), bochner (3), minlos (3), mask (3), sometimes (3), generation (3), than (3), modeled (3), current (3), observed (3), parameters (3), hypothesis (3), testing (3), same (3), tempered (3), characteristic (3), distributions (3), integrals (3), width (3), cdot (3), authors (3), valued (3), referred (3), autocorrelation (3), contexts (3), would (3), expectation (3), concept (3), either (3), every (3), history (3), independence (3), example (3), version (3), although (3), common (3), assumed (3), transform (3), identical (3), participants (3), having (3), sources (3), more (3), create (3), practical (3), context (3), range (3), subsection (3), main (3), languages (2), table (2), code (2), policy (2), using (2), you (2), hidden (2), categories (2), category (2), link (2), wayback (2), pages (2), short (2), description (2), compression (2), org (2), renewal (2), control (2), actuarial (2), disciplines (2), uniform (2), principle (2), variation (2), prokhorov (2), optional (2), extension (2), generator (2), formula (2), decomposition (2), convergence (2), davis (2), maximal (2), strong (2), theorems (2), limit (2), self (2), measurable (2), deterministic (2), feller (2), paths (2), network (2), financial (2), conditional (2), heteroskedasticity (2), graph (2), poisson (2), branching (2), gamma (2), hunt (2), geometric (2), walk (2), motion (2), birth (2), biased (2), processes (2), methods (2), thermal (2), colors (2), quantization (2), plus (2), modulation (2), per (2), carrier (2), receiver (2), phase (2), impulse (2), equivalent (2), input (2), temperature (2), circuit (2), level (2), grey (2), video (2), rooms (2), images (2), human (2), physics (2), telecommunications (2), external (2), 2025 (2), how (2), 2021 (2), 2012 (2), generate (2), pattern (2), recognition (2), university (2), jeffrey (2), 5th (2), introductory (2), arnold (2), ludwig (2), 1974 (2), 03359 (2), equations (2), via (2), 2002 (2), 4th (2), edition (2), diebold (2), 2006 (2), 1998 (2), technical (2), 314 (2), vectors (2), improves (2), right (2), 1469 (2), 1993 (2), psychology (2), office (2), s2cid (2), sverker (2), 1111 (2), new (2), york (2), bracha (2), based (2), therapy (2), 2009 (2), 2008 (2), focal (2), 240 (2), 3rd (2), production (2), analog (2), 1985 (2), delillo (2), 2005 (2), good (2), morning (2), america (2), 2011 (2), stein (2), michael (2), 1999 (2), observations (2), 7166 (2), 1007 (2), 1494 (2), interpolation (2), kriging (2), visible (2), instruments (2), references (2), dirac (2), individual (2), ideas (2), multiple (2), within (2), informal (2), digitally (2), quality (2), concepts (2), suitable (2), whose (2), alternatively (2), there (2), explanatory (2), past (2), dependent (2), depends (2), sequential (2), assumes (2), words (2), correlation (2), still (2), true (2), intervals (2), int (2), vert (2), analogous (2), omega (2), forall (2), quad (2), frac (2), multivariate (2), name (2), usually (2), called (2), most (2), define (2), standard (2), positive (2), simply (2), inherently (2), pointwise (2), property (2), rather (2), strict (2), well (2), distinct (2), possible (2), longer (2), essentially (2), simplest (2), refer (2), weakly (2), strongly (2), order (2), required (2), necessarily (2), strictly (2), jointly (2), generalization (2), number (2), many (2), operatorname (2), especially (2), sample (2), coefficient (2), normally (2), condition (2), instead (2), commonly (2), coefficients (2), perfectly (2), joint (2), implies (2), types (2), too (2), diagonal (2), identity (2), benefits (2), showed (2), among (2), students (2), source (2), tuned (2), unused (2), static (2), adjacent (2), stations (2), equipment (2), events (2), uses (2), computing (2), high (2), axis (2), talk (2), bandwidth (2), relevant (2), band (2), hissing (2), arranged (2), appearance (2), upload (2), file (2), changes (2), read (2), article (2), norsk (2), bahasa (2), log (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statement, developers, conduct, legal, safety, contacts, disclaimers, text, available, additional, apply, site, agree, registered, trademark, profit, organization, foundation, inc, creative, attribution, sharealike, license, rendered, parsoid, last, edited, october, 2026, utc, webarchive, template, plain, ipa, articles, https, index, php, title, white_noise, oldid, 1378872016, ruin, finance, deviations, extreme, evt, mathematics, abstract, classical, usual, hypotheses, integrability, stratonovich, hitting, tanaka, equation, snell, envelope, reflection, quadratic, malliavin, calculus, metric, kosambi, karhunen, loève, continuity, lemma, infinitesimal, girsanov, filtration, feynman, kac, dynkin, meyer, doléans, dade, exponential, cameron, martin, marcinkiewicz, zygmund, kunita, watanabe, upcrossing, burkholder, gundy, inequalities, hewitt, savage, engelbert, schmidt, borel, cantelli, blumenthal, laws, sanov, iterated, logarithm, deviation, fisher, tippett, gnedenko, donsker, central, reversible, stationary, progressively, predictable, piecewise, mixing, gauss, exchangeable, càdlàg, fluid, bulk, sparre, anderson, risk, cramér, lundberg, bühlmann, wilkie, vašíček, sabr, volatility, rendleman, bartter, libor, market, korn, kreer, lenssen, hull, lee, heston, heath, jarrow, morton, hjm, garman, kohlhagen, ingersoll, ross, cir, elasticity, cev, chen, chan, karolyi, longstaff, sanders, ckls, scholes, karasinski, derman, toy, binomial, options, pricing, garch, arma, integrated, arima, arch, determinantal, pitman, yor, percolation, boolean, potts, ising, hopfield, gibbs, dirichlet, chains, length, regenerative, dynamical, super, sub, differences, hmm, both, sausage, telegraph, superprocess, stable, semimartingale, schramm, loewner, quasimartingale, homogeneous, compound, ornstein, uhlenbeck, mckean, vlasov, interacting, particle, hawkes, fleming, viot, empirical, cauchy, meander, fractional, excursion, dyson, bridge, pure, death, bessel, airy, entropy, avoiding, loop, erased, moran, chain, galton, watson, chinese, restaurant, bernoulli, deep, prior, dae, autoencoder, shrinkage, bm3d, block, matching, filtering, means, bilateral, anisotropic, blur, wavelet, total, median, low, pass, denoise, radiation, analyzer, cnr, contrast, sqnr, sinr, imaging, snr, sinad, mer, error, symbol, bit, dbrnc, ratios, pseudorandom, nvh, vibration, harshness, shaping, floor, figure, pulse, resistance, coherent, worley, gradient, shot, johnson, nyquist, jitter, infrasound, flicker, cosmic, burst, awgn, class, transportation, ships, health, government, regulation, buildings, reduction, measurement, cancellation, acoustic, quieting, media, merriam, webster, sit, aleksandr, oct, somnorium, matt, donadio, fukunaga, keinosuke, 1990, 2nd, academic, 269851, introduction, hamilton, james, 1994, princeton, 691, 04289, wooldridge, cengage, 111, 53104, econterms, accessed, 2013, 2016, papoulis, athanasios, pillai, unnikrishna, mcgraw, hill, 281725, francis, eric, zivot, jiahui, wang, modeling, fessler, report, dept, computer, science, univ, michigan, rausch, 2014, modulating, activity, dopaminergic, midbrain, regions, superior, temporal, sulcus, 1480, neuroscience, baker, mary, anne, dennis, holding, speech, task, 355, 8138798, 0022, 1309, 1080, 00221309, 9711152, 339, 120, loewen, laura, peter, suedfeld, 1992, arousal, 395, 144443528, 1177, 0013916592243006, 1992envbe, 381l, bibcode, 381, behavior, söderlund, göran, sikström, andrew, smart, listen, beneficial, 847, 17683456, 0021, 9630, 7610, 01749, 840, child, psychiatry, soderlund, goran, sikstrom, jan, loftesnes, edmund, sonuga, barke, 2010, 20920224, 2955636, pmc, 1186, 1744, 9081, behavioral, brain, functions, inattentive, school, children, noell, courtney, william, meyerhoff, february, 2003, diagnosis, elusive, symptom, 12596495, 0016, 867x, geriatrics, green, penelope, 2018, 0362, 4331, silence, lópez, september, 192, 213, 12422383, hawaii, med, evidence, complementary, intervention, insomnia, jastreboff, 2000, habituation, tht, retraining, trt, san, diego, singular, 376, 357, handbook, connell, pamela, licalzi, april, 2004, lottery, books, voice, toole, floyd, 52009, reproduction, psychoacoustics, patronis, eugene, 80830, clark, dexxter, www, learnhowtoproducemusic, did, know, secrets, tips, kuo, hui, hsiung, 1996, crc, 8493, 8077, haykin, simon, john, sons, 69790, lathi, oxford, 511009, political, show, january, abc, social, security, rhetoric, claire, shipman, fusco, garland, hughes, 329, 14726662, 22276531, 1558, 5646, 01447, developmental, trait, trilobites, frank, fourth, best, thought, analogue, approximately, mixture, demonstrated, isaac, newton, carter, bruce, mancini, ron, texas, 094948, amps, everyone, principal, olfactory, mynoise, metaphorically, novel, explores, symptoms, came, together, make, difficult, actualize, personality, culture, disagreeable, harsh, dissonant, melody, discordant, politicians, want, noticed, jargon, pleonastic, unobtrusive, distractions, aid, relaxation, chatter, conversations, confined, describe, backdrop, ambient, creating, indistinct, seamless, commotion, examples, include, colloquialism, generating, entails, feeding, appropriate, stream, depend, algorithm, converter, microcontroller, microprocessor, processor, crucial, communications, equalization, estimation, produce, prescribed, conversely, transformed, into, whitening, coloring, subset, sum, depending, certain, infer, against, alternative, mutually, just, underlying, estimated, estimates, uncertainties, accurate, points, heteroskedastic, confidence, unbiased, test, null, ordinary, least, squares, natural, pairing, schwartz, consider, fixed, above, satisfy, analogously, sim, existence, uniqueness, guaranteed, goes, sazanov, specifies, intersection, cap, indirectly, specifying, isolated, cannot, becomes, when, loosely, representing, unbounded, correctly, reflects, autocovariance, problematic, meaning, renders, inadequate, physical, possess, texts, treat, heuristic, precise, trivial, quantities, sums, replaced, converge, indeed, set, instances, moreover, discontinuous, operations, like, integration, advanced, machinery, strongest, entire, before, pair, pairs, adopt, qualifier, stronger, others, distinguish, them, notion, replace, generates, parameter, entirely, moments, notably, guarantees, infinitely, kronecker, situations, relax, allowing, restricted, takes, half, maximum, corresponding, sqrt, individually, rotated, degrees, hold, assumption, stricter, explicitly, completely, dependencies, very, subtle, pairwise, correlations, squared, modulus, addition, result, hartley, fft, orthogonal, spherical, symmetry, necessary, element, simplifies, scalar, sufficient, guarantee, broader, treatments, product, similarly, carried, out, sixty, six, healthy, observe, involved, identifying, whilst, sounds, overall, fact, relation, improved, abilities, slightly, upon, mixed, small, study, published, found, stimulation, functioning, secondary, decreasing, indicates, improving, mood, workers, decreases, complex, card, sorting, tasks, attention, deficit, hyperactivity, disorder, synthetic, sold, enhancers, aids, marpac, mate, first, domestic, built, 1962, traveling, salesman, jim, buckwalter, simpler, cost, commercial, extremely, vulnerable, contaminated, spurious, harmonics, vicinity, receiving, antenna, causing, solar, flares, lightning, machines, masker, basis, antennas, digit, patterns, generators, obtain, contains, great, amount, differs, octave, transducers, microphones, amplifiers, response, directly, extensively, recreate, percussive, domain, simple, nonexistent, station, snare, drums, cymbals, synthesis, electronic, spaces, form, square, derivative, incorrectly, amplitude, yet, neither, gaussianity, respect, falling, amplitudes, way, independently, binary, take, course, left, shown, vertical, versus, horizontal, spectrogram, lack, phylogenetic, comparative, nontechnical, without, meaningful, phylogenetically, purely, theoretical, construction, limited, practice, mechanism, transmission, medium, observation, capabilities, thus, considered, audible, 000, heard, ear, resembling, sustained, aspiration, hand, colored, structure, formant, ash, recording, along, dimensions, rectangular, grid, spread, complicated, domains, torus, sphere, pixels, regarded, single, realization, shock, serially, intensity, giving, meanings, scientific, including, specific, draws, appears, generally, acoustical, purple, red, plotted, waveform, type, free, encyclopedia, item, projects, printable, download, print, export, switch, legacy, parser, get, shortened, url, cite, information, permanent, what, here, actions, english, tiếng, việt, oʻzbekcha, ўзбекча, українська, türkçe, svenska, русский, português, پښتو, polski, bokmål, nynorsk, nederlands, melayu, македонски, lietuvių, 한국어, қазақша, 日本語, italiano, indonesia, magyar, עברית, français, suomi, فارسی, euskara, eesti, español, esperanto, deutsch, dansk, чӑвашла, čeština, کوردی, català, български, беларуская, тарашкевіца, boarisch, العربية, afrikaans, top, personal, special, recent, community, portal, learn, help, contribute, navigation,

Text of the page (random words):
by white noise 15 tinnitus treatment edit white noise is a common synthetic noise source used for sound masking by a tinnitus masker 16 white noise machines and other white noise sources are sold as privacy enhancers and sleep aids see music and sleep and to mask tinnitus 17 the marpac sleep mate was the first domestic use white noise machine built in 1962 by traveling salesman jim buckwalter 18 alternatively the use of an am radio tuned to unused frequencies static is a simpler and more cost effective source of white noise 19 however white noise generated from a common commercial radio receiver tuned to an unused frequency is extremely vulnerable to being contaminated with spurious signals such as adjacent radio stations harmonics from non adjacent radio stations electrical equipment in the vicinity of the receiving antenna causing interference or even atmospheric events such as solar flares and especially lightning work environment edit the effects of white noise upon cognitive function are mixed a small study published in 2007 found that white noise background stimulation improves cognitive functioning among secondary students with attention deficit hyperactivity disorder adhd while decreasing performance of non adhd students 20 21 other work indicates it is effective in improving the mood and performance of workers by masking background office noise 22 but decreases cognitive performance in complex card sorting tasks 23 similarly an experiment was carried out on sixty six healthy participants to observe the benefits of using white noise in a learning environment the experiment involved the participants identifying different images whilst having different sounds in the background overall the experiment showed that white noise does in fact have benefits in relation to learning the experiment showed that white noise improved the participants learning abilities and their recognition memory slightly 24 mathematical definitions edit white noise vector edit a random vector that is a random variable with values in r n is said to be a white noise vector or white random vector if its components each have a probability distribution with zero mean and a finite variance while standard signal processing definitions strictly require these variances to be identical to guarantee a perfectly flat power spectrum broader statistical treatments sometimes only require the variances to be finite and the components to be statistically independent that is their joint probability distribution must be the product of the distributions of the individual components 25 a necessary but in general not sufficient condition for statistical independence of two variables is that they be statistically uncorrelated that is their covariance is zero therefore the covariance matrix r of the components of a white noise vector w with n elements must be an n by n diagonal matrix where each diagonal element r ii is the variance of component w i and the correlation matrix must be the n by n identity matrix if the variances are identical the covariance matrix simplifies to a scalar multiple of the identity matrix r σ 2 i displaystyle r sigma 2 i if in addition to being independent every variable in w also has a normal distribution with zero mean and the same variance σ 2 displaystyle sigma 2 w is said to be a gaussian white noise vector in that case the joint distribution of w is a multivariate normal distribution the independence between the variables then implies that the distribution has spherical symmetry in n dimensional space therefore any orthogonal transformation of the vector will result in a gaussian white random vector in particular under most types of discrete fourier transform such as fft and hartley the transform w of w will be a gaussian white noise vector too that is the n fourier coefficients of w will be independent gaussian variables with zero mean and the same variance σ 2 displaystyle sigma 2 the power spectrum p of a random vector w can be defined as the expected value of the squared modulus of each coefficient of its fourier transform w that is p i e w i 2 under that definition a gaussian white noise vector will have a perfectly flat power spectrum with p i σ 2 for all i if w is a white random vector but not a gaussian one its fourier coefficients w i will not be completely independent of each other although for large n and common probability distributions the dependencies are very subtle and their pairwise correlations can be assumed to be zero often the weaker condition statistically uncorrelated is used in the definition of white noise instead of statistically independent however some of the commonly expected properties of white noise such as a flat power spectrum may not hold for this weaker version under this assumption the stricter version can be referred to explicitly as independent white noise vector 26 p 60 other authors use strongly white and weakly white instead 27 an example of a random vector that is gaussian white noise in the weak but not in the strong sense is x x 1 x 2 displaystyle x x_ 1 x_ 2 where x 1 displaystyle x_ 1 is a normal random variable with zero mean and x 2 displaystyle x_ 2 is equal to x 1 displaystyle x_ 1 or to x 1 displaystyle x_ 1 with equal probability these two variables are uncorrelated and individually normally distributed but they are not jointly normally distributed and are not independent if x displaystyle x is rotated by 45 degrees its two components will still be uncorrelated but their distribution will no longer be normal in some situations one may relax the definition by allowing each component of a white random vector w displaystyle w to have a non zero expected value μ displaystyle mu in image processing especially where samples are typically restricted to positive values one often takes μ displaystyle mu to be one half of the maximum sample value in that case the fourier coefficient w 0 displaystyle w_ 0 corresponding to the zero frequency component essentially the average of the w i displaystyle w_ i will also have a non zero expected value μ n displaystyle mu sqrt n and the power spectrum p displaystyle p will be flat only over the non zero frequencies discrete time white noise edit a discrete time stochastic process w n displaystyle w n is a generalization of a random vector with a finite number of components to infinitely many components a discrete time stochastic process w n displaystyle w n is called weak sense white noise or often simply white noise in signal processing if its mean is equal to zero for all n displaystyle n i e e w n 0 displaystyle operatorname e w n 0 and if its autocorrelation function r w n e w k n w k displaystyle r_ w n operatorname e w k n w k has a non zero value only for n 0 displaystyle n 0 i e r w n σ 2 δ n displaystyle r_ w n sigma 2 delta n where σ 2 displaystyle sigma 2 is the variance and δ n displaystyle delta n is the kronecker delta 28 the variables w n displaystyle w n are only required to be uncorrelated not necessarily statistically independent because the characteristic flat power spectral density of white noise depends entirely on the process s second order moments its autocorrelation function a process that strictly requires the samples to be independent and identically distributed i i d is referred to as strict sense white noise notably if the uncorrelated random variables are jointly gaussian this inherently guarantees they are also statistically independent continuous time white noise edit in order to define the notion of white noise in the theory of continuous time signals one must replace the concept of a random vector by a continuous time random signal that is a random process that generates a function w displaystyle w of a real valued parameter t displaystyle t such a process is said to be white noise in the strongest sense if the value w t displaystyle w t for any time t displaystyle t is a random variable that is statistically independent of its entire history before t displaystyle t a weaker definition requires independence only between the values w t 1 displaystyle w t_ 1 and w t 2 displaystyle w t_ 2 at every pair of distinct times t 1 displaystyle t_ 1 and t 2 displaystyle t_ 2 an even weaker definition requires only that such pairs w t 1 displaystyle w t_ 1 and w t 2 displaystyle w t_ 2 be uncorrelated 29 as in the discrete case some authors adopt the weaker definition for white noise and use the qualifier independent to refer to either of the stronger definitions others use weakly white and strongly white to distinguish between them however a precise definition of these concepts is not trivial because some quantities that are finite sums in the finite discrete case must be replaced by integrals that may not converge indeed the set of all possible instances of a signal w displaystyle w is no longer a finite dimensional space r n displaystyle mathbb r n but an infinite dimensional function space moreover by any definition a white noise signal w displaystyle w would have to be essentially discontinuous at every point therefore even the simplest operations on w displaystyle w like integration over a finite interval require advanced mathematical machinery some introductory engineering texts treat this as a heuristic rather than a strict mathematical definition and require each value w t displaystyle w t to be a real valued random variable with expectation μ displaystyle mu and some finite variance σ 2 displaystyle sigma 2 then the covariance e w t 1 w t 2 displaystyle mathrm e w t_ 1 cdot w t_ 2 between the values at two times t 1 displaystyle t_ 1 and t 2 displaystyle t_ 2 is well defined it is zero if the times are distinct and σ 2 displaystyle sigma 2 if they are equal however by this definition the integral w a a r a a r w t d t displaystyle w_ a a r int _ a a r w t dt over any interval with positive width r displaystyle r would be simply the width times the expectation r μ displaystyle r mu while the expectation being zero is not inherently problematic if the pointwise variance is finite the variance of this integral would be zero meaning the signal has no measurable energy this property renders the concept inadequate as a model of white noise signals either in a physical or mathematical sense because a true white noise process must possess a flat power spectral density which requires the pointwise variance to be infinite rather than finite therefore most authors define the signal w displaystyle w indirectly by specifying random values for the integrals of w t displaystyle w t and w t 2 displaystyle w t 2 over each interval a a r displaystyle a a r in this approach however the value of w t displaystyle w t at an isolated time cannot be defined as a standard real valued random variable 30 also the covariance e w t 1 w t 2 displaystyle mathrm e w t_ 1 cdot w t_ 2 becomes infinite when t 1 t 2 displaystyle t_ 1 t_ 2 and the autocovariance function often loosely referred to as autocorrelation in engineering contexts r t 1 t 2 displaystyle mathrm r t_ 1 t_ 2 must be defined as n δ t 1 t 2 displaystyle n delta t_ 1 t_ 2 where n displaystyle n is some real constant representing the power spectral density and δ displaystyle delta is the dirac delta function an unbounded measure which correctly reflects the infinite variance at t 1 t 2 displaystyle t_ 1 t_ 2 in this approach one usually specifies that the integral w i displaystyle w_ i of w t displaystyle w t over an interval i a b displaystyle i a b is a real random variable with normal distribution zero mean and variance b a σ 2 displaystyle b a sigma 2 and also that the covariance e w i w j displaystyle mathrm e w_ i cdot w_ j of the integrals w i displaystyle w_ i w j displaystyle w_ j is r σ 2 displaystyle r sigma 2 where r displaystyle r is the width of the intersection i j displaystyle i cap j of the two intervals i j displaystyle i j this model is called a gaussian white noise signal or process in the mathematical field known as white noise analysis a gaussian white noise w displaystyle w is defined as a stochastic tempered distribution i e a random variable with values in the space s r displaystyle mathcal s mathbb r of tempered distributions analogous to the case for finite dimensional random vectors a probability law on the infinite dimensional space s r displaystyle mathcal s mathbb r can be defined via its characteristic function existence and uniqueness are guaranteed by an extension of the bochner minlos theorem which goes under the name bochner minlos sazanov theorem analogously to the case of the multivariate normal distribution x n n μ σ displaystyle x sim mathcal n _ n mu sigma which has characteristic function k r n e e i k x e i k μ 1 2 k σ k displaystyle forall k in mathbb r n quad mathrm e mathrm e mathrm i langle k x rangle mathrm e mathrm i langle k mu rangle frac 1 2 langle k sigma k rangle the white noise w ω s r displaystyle w omega to mathcal s mathbb r must satisfy φ s r e e i w φ e 1 2 φ 2 2 displaystyle forall varphi in mathcal s mathbb r quad mathrm e mathrm e mathrm i langle w varphi rangle mathrm e frac 1 2 varphi _ 2 2 where w φ displaystyle langle w varphi rangle is the natural pairing of the tempered distribution w ω displaystyle w omega with the schwartz function φ displaystyle varphi i e we consider φ displaystyle varphi as a fixed linear function on s r displaystyle mathcal s mathbb r analogous to k displaystyle k above and φ 2 2 r φ x 2 d x displaystyle varphi _ 2 2 int _ mathbb r vert varphi x vert 2 mathrm d x mathematical applications edit time series analysis and regression edit in statistics and econometrics one often assumes that an observed series of data values is the sum of the values generated by a deterministic linear process depending on certain independent explanatory variables and on a series of random noise values then regression analysis is used to infer the parameters of the model process from the observed data e g by ordinary least squares and to test the null hypothesis that each of the parameters is zero against the alternative hypothesis that it is non zero hypothesis testing typically assumes that the noise values are mutually uncorrelated with zero mean and have the same gaussian probability distribution in other words that the noise is gaussian white not just white 31 if there is non zero correlation between the noise values underlying different observations then the estimated model parameters are still unbiased but estimates of their uncertainties such as confidence intervals will be biased not accurate on average this is also true if the noise is heteroskedastic that is if it has different variances for different data points alternatively in the subset of regression analysis known as time series analysis there are often no explanatory variables other than the past values of the variable being modeled the dependent variable in this case the noise process is often modeled as a moving average process in which the curr...