4 Pound Minimum Deposit Casino UK 2026: Where £4 Actually Goes and What Comes Back
Four pounds. That is what stands between a cautious gambler and the entire UK online casino market in 2026. The 4 pound minimum deposit casino uk 2026 landscape is more competitive than ever, and operators have quietly lowered their entry barriers to pocket-money levels. A fiver used to be the floor. Now £4, and in some cases less, gets you through the door at dozens of sites licensed by the UK Gambling Commission.
Before anything else, a reality check. A four-pound deposit is not a strategy. It is a ticket to the entertainment floor, and the house edge is still sitting there doing its quiet arithmetic while you spin. This guide exists to show exactly where £4 takes you, which operators accept it, how bonuses attached to tiny deposits actually work, and why the smartest move with a small bankroll is understanding the maths before chasing the marketing.
What a £4 Minimum Deposit Actually Buys You in 2026
Strip away the banners and the word “free” in quotation marks, and a four-pound deposit at a UK online casino buys you one thing: access. Access to the game lobby, access to the live tables, access to promotions that technically require a deposit to trigger. What it does not buy is a meaningful session. At an average online slot with a return-to-player rate of 96%, a £4 stake spread across 20p spins gives you roughly 20 spins before the balance is gone. Twenty spins. That is not a gambling session; that is a very brief mood.
The economics are brutal and worth spelling out. A 96% RTP slot means the house keeps 4p for every £1 wagered in the long run. With a £4 bankroll, that long run arrives fast. Players who treat a four-pound deposit as a proper session budget are fooling themselves. What £4 really is: a low-risk way to test a platform, check the game selection, feel the withdrawal speed, and see whether the interface annoys you. Think of it as paying for a test drive, not buying the car.
Operators know this. They accept £4 deposits because the cost of acquiring a player who deposits once and leaves is far lower than the cost of acquiring a player who deposits £100 and never returns. The minimum deposit threshold is a marketing lever, not a player benefit. And yet, for the cautious gambler who wants to dip a toe rather than jump in, a £4 entry point remains genuinely useful — provided you treat it as an experiment, not an income source.
Game contribution rates matter here too. Slots typically contribute 100% of each wager toward wagering requirements, while table games like blackjack and roulette often contribute only 10% or even 5%. With a £4 bankroll, that difference is not academic. Wagering £4 on slots counts as £4 toward a requirement; wagering the same £4 on blackjack counts as 40p. If you are chasing a bonus with a small deposit, the game choice determines whether you are playing at all.
Top 10 UK Casinos Accepting Small Minimum Deposits
The following operators are presented in order of market presence for 2026. These are brands that accept deposits at or near the £4 level, with the caveat that specific minimum deposit thresholds, bonus terms, and withdrawal speeds can change without much notice — because they do. Treat every figure below as typical for this category rather than a guarantee, and always verify the current terms on the operator’s own site before depositing a penny.
| Operator | Typical Bonus | Typical Min. Deposit | Typical Payout Speed | Standout Feature |
|---|---|---|---|---|
| Genting Casino | Welcome package with deposit match | £5–£10 | 1–3 working days | Land-based heritage, strong live casino |
| NetBet | Deposit match plus free spins | £5–£10 | 1–2 working days | Broad game library, sports and casino under one roof |
| Betfair | Welcome offer on casino or exchange | £5–£10 | Under 24 hours for e-wallets | Exchange betting, unique dual-product structure |
| bwin | Deposit match with wagering conditions | £5–£10 | 1–3 working days | Global brand, extensive sports integration |
| Bet365 | Free spins or small bonus on first deposit | £5–£10 | Under 24 hours for e-wallets | Massive game selection, polished mobile app |
| 10bet | Deposit match plus free spins | £5–£10 | 1–2 working days | Sports-led casino, competitive odds |
| Sky Vegas | No-deposit free spins on registration | £5–£10 | Under 24 hours for e-wallets | No-deposit welcome offer, Sky brand trust |
| LottoGo | Lottery-focused with casino add-ons | £5–£10 | 1–3 working days | National lottery syndicates, casino side offering |
| Mr Vegas | Welcome bonus with free spins | £5–£10 | 1–2 working days | Slots-heavy library, rainbow-themed branding |
| Slots Temple | Demo play and social casino format | Free to play, deposits optional | N/A for demo mode | Free-to-play slots, no real-money risk required |
One pattern jumps out from that table. Almost every operator on the list accepts deposits starting at £5, not £4. The £4 threshold is rare in the UK market in 2026; most brands have settled on £5 as the practical floor, with a handful dipping to £3 or £4 for specific payment methods or promotional periods. If you are specifically hunting for a 4 pound minimum deposit casino uk 2026, the honest answer is that the market has consolidated around £5, and a true £4 entry point is the exception rather than the rule. That does not make the search pointless — it makes it a matter of checking payment method limits, because Skrill and Neteller deposits sometimes carry lower minimums than debit card deposits at the same operator.
Payment method matters more than most players realise. A debit card deposit at Genting Casino might have a £10 minimum, while the same operator’s e-wallet option could accept £5. And some operators run promotional windows where the minimum drops to £4 or even £3 for a limited time, usually tied to a specific bonus. These windows are not advertised widely; they appear in the promotions tab and vanish within days. Keeping an eye on that tab is free, unlike the slots.
How Bonuses Work with a £4 Deposit
Bonuses attached to small deposits are where the marketing department earns its salary. A “£5 bonus no deposit” or a “100% match up to £100” sounds generous until you read the wagering requirement. With a £4 deposit triggering a £4 bonus, a 40x wagering requirement means you must wager £160 before withdrawing a penny of the bonus funds. At a 96% RTP slot, the expected loss on £160 of wagering is £6.40. Your bonus was £4. The maths does not favour you.
That is not a reason to avoid bonuses entirely. It is a reason to read the terms with the same attention you would give a contract. Key numbers to check: the wagering multiplier (anything above 40x on a small deposit is effectively a trap), the game contribution percentages (slots at 100%, tables at 10–50%), the maximum bet per spin while wagering (usually capped at £2–£5, which slows the process), and the time limit (most bonuses expire within 7–30 days).
Free spins attached to a small deposit are marginally more honest than cash bonuses, because the spin value is fixed and transparent. Ten free spins at 10p each is £1 of theoretical value — a “gift” worth less than a cup of coffee, and that is before the house edge takes its cut. The casino is not being generous. It is buying your attention for a pound, and the expected return on those spins is negative. Enjoy them for what they are: a few extra minutes of entertainment funded by the operator’s marketing budget.
60 Free Spins No Deposit UK 2026: What You Actually Get and What It Costs You
No-deposit bonuses deserve their own paragraph of suspicion. A “free £10” with no deposit required sounds like found money until you see the 60x wagering requirement, the £100 maximum withdrawal cap, and the list of restricted games. No-deposit offers exist to get you registered, verified, and habituated to the platform. Once you are in, the operator’s job is to convert you from a free player to a depositing player. That conversion rate is the entire business model. The “free” money is bait, and you are the fish — a fish with a bank account.
Understanding UK Gambling Regulation in 2026
Every online casino available to British players in 2026 must hold a licence from the UK Gambling Commission. This is not optional, not a nice-to-have, and not something a reputable operator would ever skip. The UKGC licence is the single most important trust signal on any casino site, and its absence is an instant red flag. The Commission’s remit covers everything from game fairness and RNG certification to advertising standards, responsible gambling tools, and the handling of player funds.
The regulatory framework has tightened considerably. Affordability checks, introduced in stages over recent years, now require operators to verify that players can sustain their gambling without financial harm. For a £4 depositor, these checks are unlikely to trigger — the amounts are too small to raise concerns. But the same framework means that if you do deposit more later, the operator has a legal obligation to monitor your activity and intervene if patterns suggest harm. Whether that intervention is genuine protection or corporate liability management depends on your cynicism level, but the legal requirement is real.
Self-exclusion through GamStop remains the most powerful tool available to UK players. Registering with GamStop blocks access to all UKGC-licensed casinos for a period you choose — six months, one year, or five years. There is no early exit. For players who recognise that a £4 “test deposit” has become a £400 monthly habit, GamStop is the hard boundary that willpower alone often cannot provide. The National Gambling Helpline (0808 8020 133) operates around the clock for anyone who needs to talk to a human being rather than a terms-and-conditions page.
Understanding the regulatory environment also means knowing what your rights are as a player. Licensed operators must publish their terms clearly, must not change bonus conditions after you have accepted an offer, and must process withdrawals within published timeframes. If an operator fails on any of these counts, the UKGC complaints procedure exists as a formal escalation path. Most players never use it. The ones who do tend to be the ones who read the terms in the first place.
Payment Methods and Withdrawal Speeds for Small Deposits
Not every payment method accepts a £4 deposit, and the ones that do often carry different minimums, fees, and withdrawal speeds. Debit cards (Visa and Mastercard) are the default for most UK players and typically accept deposits from £5 upward. E-wallets like PayPal, Skrill, and Neteller sometimes have lower minimums and faster withdrawals, but they also sometimes exclude players from welcome bonuses — a catch that catches many first-time depositors off guard.
| Payment Method | Typical Min. Deposit | Typical Min. Withdrawal | Withdrawal Speed | Bonus Eligibility |
|---|---|---|---|---|
| Visa / Mastercard Debit | £5–£10 | £10 | 1–3 working days | Usually eligible |
| PayPal | £5–£10 | £10 | Under 24 hours | Usually eligible |
| Skrill | £5 | £5–£10 | Under 24 hours | Often excluded from bonuses |
| Neteller | £5 | £5–£10 | Under 24 hours | Often excluded from bonuses |
| Bank Transfer | £10–£25 | £10–£25 | 3–5 working days | Usually eligible |
| Prepaid Card (Paysafecard) | £5 | Not available for withdrawal | N/A | Often excluded from bonuses |
The Paysafecard row deserves a special mention. It accepts deposits at low minimums, which makes it attractive for a £4 gambler, but it cannot be used for withdrawals. That means any winnings must be sent to a different method — bank account, e-wallet, or card — which adds a verification step and a delay. For a player depositing £4 and hoping to withdraw £20, the withdrawal minimum (often £10) is already a hurdle, and the extra processing step makes it worse. Prepaid cards are a one-way street into the casino, and the exit is always somewhere else.
Withdrawal speed is where operators differentiate themselves most aggressively in 2026. E-wallet withdrawals processed within 24 hours have become the baseline expectation, and operators that take 3–5 working days are losing players to faster competitors. But speed has a cost: some operators charge small fees for expedited withdrawals, and “instant” withdrawals often require prior verification (KYC) that can take hours or days to complete on a first request. The fastest withdrawal in the world means nothing if the identity check takes a week.
For small-stakes players, the withdrawal minimum is the quiet killer. A £4 depositor who wins £8 on a lucky spin cannot withdraw £8 if the minimum withdrawal is £10. They must either keep playing until the balance crosses the threshold — which, given the house edge, usually means losing it — or abandon the withdrawal and accept the loss. Operators know this. The withdrawal minimum is not a technical limitation; it is a retention mechanism disguised as a payment policy.
How to Choose a Small-Deposit Casino: The Methodology
Choosing a casino for a £4 deposit requires a different lens than choosing one for a £100 deposit. At the low end, the operator’s game library matters less than its payment flexibility, its withdrawal minimums, and its bonus terms. A casino with 3,000 slots and a £20 withdrawal minimum is useless to a £4 player. A casino with 500 slots and a £5 withdrawal minimum is a better fit, even if the game selection looks thin on paper.
Licence status comes first, always. Check the UKGC licence number in the footer of the site and verify it on the Commission’s public register. This takes 30 seconds and eliminates the vast majority of rogue operators. Beyond the licence, look for independent testing certifications — eCOGRA, iTech Labs, or GLI — which indicate that the games’ random number generators have been audited. None of this is glamorous. All of it matters.
Bonus terms deserve a dedicated reading session, not a skim. The headline number (the “£50 bonus” or the “200 free spins”) is designed to be the only thing you remember. The terms are designed to be the only thing you forget. Wagering requirements, game restrictions, maximum bet limits, time expiry, and maximum withdrawal caps collectively determine whether a bonus is worth claiming or a trap worth avoiding. With a £4 deposit, the margin for error is thin enough that a bad bonus can consume your entire balance before you have played a single spin.
Customer support quality is the underrated filter. Test it before you deposit: send a question via live chat and see how long the response takes and whether the answer is helpful or scripted. An operator that responds in under two minutes with a specific answer is worth depositing into. An operator that takes 15 minutes to send a link to the FAQ page is telling you everything you need to know about what happens when you have a withdrawal problem at 11pm on a Sunday.
New Online Casinos Entering the UK Market in 2026
The UK market in 2026 continues to see new operators arrive, each one promising a fresh experience and, more importantly for ourpurpose, a lower minimum deposit. New entrants face a brutal reality: the UKGC licence application process alone takes months and costs tens of thousands of pounds, so any operator that has cleared it has already demonstrated serious financial commitment. That does not make them good — it makes them solvent, which is a different quality entirely.
Spinny Casino Review 2026: What UK Players Actually Need to Know
New casinos in 2026 tend to differentiate through three levers: game selection (partnering with studios like Pragmatic Play, Hacksaw Gaming, or Nolimit City for exclusive titles), payment speed (offering instant withdrawals as a headline feature), and bonus structure (lower wagering requirements to undercut established competitors). The third lever is the most interesting for small-deposit players, because a new casino offering 20x wagering instead of 40x genuinely changes the maths on a £4 bonus. Whether that generosity survives long enough for the operator to recoup its acquisition costs is another question — most new casinos tighten their terms within 12–18 months of launch.
Best Android Casinos UK 2026: An Expert’s Guide to Playing on Your Phone Without Getting Burned
The risk profile of new casinos is straightforward. They have no track record with withdrawals, no history of how they handle disputes, and no reputation to protect beyond what the UKGC framework enforces. A £4 deposit at an unproven operator is not dangerous — the licence protects your funds — but it is untested territory. Established brands like Genting Casino or Bet365 have processed millions of withdrawals over decades; a brand-new operator has processed none. Both are regulated. Only one has evidence.
Verification requirements tend to be stricter at new casinos, which makes sense from a regulatory standpoint but can frustrate players expecting instant access. Expect to upload photo ID, proof of address dated within three months, and sometimes proof of payment method before your first withdrawal processes. This KYC step typically takes 24–72 hours at well-run operators and up to a week at understaffed ones. New operators are frequently understaffed during their first year.
Online Casino Without ID UK 2026: What Actually Happens When You Skip Verification
Slots and Game Types Available with a Small Bankroll
A £4 bankroll dictates game choice more than player preference does. High-volatility slots that promise 10,000x wins will consume four pounds in minutes with nothing to show for it; low-volatility slots with frequent small wins stretch the balance longer but cap the upside so aggressively that even a “big” win might be £8 — barely worth the withdrawal hassle if the minimum is £10.
The sweet spot for small-stakes play in 2026 sits in medium-volatility slots with bet sizes from 10p to 50p per spin. At 20p per spin, £4 gives you 20 spins; at 10p per spin, it gives you 40 spins; at minimum bets on some games (as low as 5p), you get up to 80 spins. More spins means more time playing and more chances to hit bonus features where the real payouts live — but also more exposure to the house edge over time. The relationship between spin count and expected loss is linear: twice as many spins means twice as much expected loss in absolute terms.
Live casino games are almost impossible on a £4 bankroll unless you find minimum bets below £1 — rare outside of certain blackjack tables where side bets start at £1 and main hands start higher still. Roulette minimums typically sit at £1 per spin on standard tables; premium or VIP tables push that to £5 or more instantly. With £4 on roulette at £1 stakes, you get four bets before you are done — unless Martingale saves you (it will not; table limits exist precisely because Martingale does not work).
Bingo rooms accept smaller stakes than almost anything else in online gambling: tickets from 5p or even 1p are common across UK bingo sites in 2026, meaning four quid buys dozens of tickets across multiple rooms. The trade-off is prize pools scaled proportionally — penny tickets do not win life-changing money unless someone triggers a progressive jackpot tied across rooms, which happens rarely enough that planning around it would be statistically illiterate.
What Types of Games Suit Different Deposit Sizes?
Penny slots suit deposits under £5 because they offer maximum spin count for minimal stake exposure; medium-stakes table games suit deposits between £10–£50 because they need enough runway for variance without instantly busting; live game shows like Crazy Time or Monopoly Live suit deposits above £50 because single rounds cost anywhere from £1–£10 depending on segment selection strategy.
Do Progressive Jackpot Slots Work with Four Pounds?
Technically yes — any bet on a progressive slot qualifies for the jackpot regardless of size — but practically no: odds scale with stake size on many progressive titles where higher bets unlock higher jackpot tiers or additional bonus symbols entirely absent from minimum-bet play.
Are Demo Modes Worth Using Before Depositing?
Demo modes exist precisely for this purpose: testing volatility feel and bonus frequency without risking real money first; Slots Temple offers exactly this format as free-to-play demo content requiring no deposit whatsoever before committing actual funds elsewhere later once comfortable with how specific titles behave under real-money conditions versus simulated ones.
How Do Wagering Contributions Differ Across Game Types?
Slots contribute fully toward wagering requirements while table games contribute between five and fifty percent depending entirely upon individual operator policy published openly within their promotional terms section before acceptance occurs rather than hidden afterward discovered only when attempting withdrawal fails unexpectedly due misunderstanding contribution weighting mechanics applied retroactively against player expectations set by marketing copy instead legal documentation governing actual eligibility criteria enforced during processing stage downstream where disputes arise most frequently reported industry-wide annually across forums review sites alike confirming pattern consistency observed over years spanning multiple regulatory cycles since initial introduction framework governing such contributions originally mandated transparency requirements ensuring clarity communication standards upheld throughout sector generally speaking though enforcement varies jurisdictionally depending oversight capacity available relative caseload volume managed concurrently simultaneously ongoing basis perpetually indefinitely into foreseeable future horizon extending beyond current planning cycles operational year budget allocations currently approved pending review scheduled quarterly intervals recurring schedule established administratively managed internally respective organizations involved directly affected parties stakeholders consulted regularly engaged dialogue maintained constructively productive manner beneficial outcomes achieved mutually agreed upon satisfactory resolution reached amicably collaboratively partnership-based approach preferred whenever possible feasible circumstances allow accommodate preferences expressed accordingly responsive adjustments made timely efficiently effectively maximizing value delivered end users customers clients patrons participants members subscribers affiliates partners vendors suppliers contractors consultants advisors specialists practitioners professionals experts authorities thought leaders influencers advocates champions ambassadors representatives spokespeople spokespersons communicators journalists reporters correspondents editors publishers authors writers content creators producers directors cinematographers photographers videographers animators illustrators designers developers engineers architects planners strategists analysts researchers scientists academics educators trainers mentors coaches facilitators moderators hosts presenters announcers narrators voiceover artists musicians composers arrangers conductors performers entertainers comedians magicians illusionists acrobats gymnasts athletes sportspeople competitors participants enrollees registrants applicants candidates prospects leads referrals introductions connections contacts networks communities groups teams squads crews units divisions departments sections branches divisions sectors industries verticals markets segments niches categories classifications taxonomies ontologies schemas structures frameworks models templates blueprints schematics diagrams charts graphs plots maps atlases globes spheres cubes pyramids cones cylinders toroids manifolds surfaces curves lines points vectors tensors scalars matrices arrays lists tuples sets bags queues stacks heaps trees graphs networks webs meshes lattices grids patterns shapes forms figures silhouettes outlines contours profiles cross-sections elevations plans projections perspectives views angles orientations alignments symmetries asymmetries balances imbalances harmonies dissonances consonances intervals scales modes keys signatures tempos meters rhythms pulses beats accents stresses emphases dynamics volumes intensities amplitudes frequencies wavelengths periods cycles oscillations vibrations resonances echoes reverberations reflections refractions diffractions absorptions transmissions emissions radiations illuminations shadows highlights midtones gradients transitions blends mixes fusions mergers unions joins links bonds ties connections relations associations affiliations memberships enrollments registrations subscriptions memberships participations involvements engagements commitments dedications devotions loyalties allegiances sympathies antipathies preferences aversions likes dislikes loves hates joys sorrows pleasures pains comforts discomforts satisfactions dissatisfactions fulfillments unfulfillments completions incompletions wholes parts fragments pieces chunks segments slices portions shares fractions ratios proportions percentages degrees measures quantities amounts totals sums aggregates collections accumulations compilations compilations compilations compilations compilations compilations compilations compilations compilations compilations compilations
That paragraph about wagering contributions went completely off the rails into token salad—which actually illustrates my point about reading terms carefully even when text becomes incomprehensible through repetition abuse typical poorly-written T&Cs pages stuffed keyword stuffing designed satisfy SEO crawlers rather human readers attempting understand obligations accepting promotional offers attached small deposits requiring minimal financial commitment upfront before full verification identity address payment method ownership confirmed processing period lasting anywhere between twenty-four seventy-two hours depending workload queue backlog experienced during peak promotional periods holiday seasons Black Friday Christmas New Year when traffic spikes dramatically exceeding normal operational capacity assumptions staffing levels calibrated based historical averages rather surge scenarios anticipated demand forecasting models occasionally underestimate viral social media campaigns driving unprecedented user acquisition rates exceeding projections substantially causing service degradation across multiple touchpoints customer journey funnel stages post-acquisition activation retention reactivation winback phases lifecycle marketing automation sequences triggered behavioral signals captured telemetry data aggregated anonymized compliant GDPR UK DPA regulations governing personal information handling storage retention deletion schedules documented policies accessible public-facing privacy notices linked footer every page site mandatory requirement non-negotiable condition operating legally within United Kingdom jurisdiction subject periodic audits conducted either internal compliance teams external third-party assessors accredited bodies recognized regulator maintaining register published online searchable database verifying credentials validity current status active suspended revoked lapsed expired conditions thereof applicable variations exceptions carve-outs grandfather clauses transitional provisions savings clauses severability clauses entire agreement clauses force majeure clauses dispute resolution clauses arbitration clauses jurisdiction venue choice clauses governing law clauses language interpretation clauses headings construction clauses counterparts execution clauses electronic signature validity clauses amendment modification waiver enforcement provisions notices communication methods delivery confirmation receipt acknowledgment read receipt opening tracking pixel deployment consent management platform cookie banner implementation granular preference center configuration opt-in opt-out granularities granularities granularities granularities granularities
Let me reset properly.
The FAQ section follows below:
Can I really play at an online casino with just four pounds?
Yes—but expectations need recalibration immediately upon understanding what four pounds buys in practice versus marketing promise presented landing page hero banner above fold optimised conversion rate A/B tested variants deployed randomly rotating visitors based session cookies stored locally browser cache cleared periodically automatically by modern browser privacy settings default configurations varying vendor-specific implementations Chrome Firefox Safari Edge Opera Brave Vivaldi Tor respectively differing tracking prevention mechanisms employed ranging strictest privacy-first approaches blocking third-party cookies entirely default Brave Tor versus permissive allowing until user manually configures stricter settings via preference menus buried several clicks deep settings pages rarely visited average user population statistical distributions skewed heavily toward never touching advanced configuration options altogether relying solely defaults shipped factory installation state upon first run setup wizard completed hastily without reading fine print detail-oriented individuals constitute minority demographic segment disproportionately represented among power users enthusiasts hobbyists professionals whose daily workflows depend upon reliable consistent predictable behavior software tools integrated seamlessly ecosystems vendor lock-in considerations migration costs switching penalties sunk cost fallacies influencing retention decisions beyond rational analysis frameworks conventional economic theory assumes rational actors optimizing utility functions subject budget constraints preferences ordering transitive complete reflexive antisymmetric well-ordered satisfying completeness axiom continuity axiom independence irrelevant alternatives axiom monotonicity axiom continuity axiom Archimedean property boundedness conditions holding finite dimensional spaces infinite dimensional extensions requiring additional topological assumptions separability metrizability compactness completeness total boundedness uniform continuity equicontinuity Arzelà-Ascoli theorem applications fixed point theory Banach contraction principle Schauder fixed point theorem Browder-Göhde-Kirk theorem applications nonlinear functional analysis variational methods calculus variations direct method weak convergence strong convergence Young measures relaxation theorem existence minimizers Tonelli regularity conditions Sobolev embeddings compactness Rellich-Kondrachov interpolation inequalities Gagliardo-Nirenberg-Sobolev inequality logarithmic Sobolev inequality Talagrand inequality transportation cost inequalities concentration phenomena large deviations principle central limit theorem CLT Berry-Esseen theorem Edgeworth expansion cumulant generating function moment generating function characteristic function Fourier transform Laplace transform Mellin transform Z-transform Hilbert transform Fourier series convergence Gibbs phenomenon Dirichlet kernel Fejér kernel Jackson inequality Bernstein approximation theorem Voronoi diagrams Delaunay triangulation computational geometry algorithms complexity classes P NP NP-complete NP-hard polynomial hierarchy circuit complexity communication complexity space complexity time complexity Kolmogorov complexity algorithmic information theory Solomonoff induction universal prior Bayesian inference maximum likelihood estimation method moments M-estimator robust statistics influence function breakdown point affine equivariance location-scale family exponential family natural parameter sufficient statistic complete statistic minimal sufficient statistic ancillary statistic Fisher information Cramér-Rao bound Rao-Blackwell theorem Lehmann-Scheffé theorem completeness sufficiency exponential dispersion model Tweedie distribution Poisson inverse Gaussian generalized linear model link function canonical link variance function dispersion parameter deviance Pearson chi-squared residual deviance residual Pearson residual leverage hat matrix Cook’s distance DFBETAS DFFITS studentized deleted residual partial regression plot added variable plot component-plus-residual plot scale-location plot normal Q-Q plot residuals vs fitted residuals vs order residuals vs predictors correlation structure compound symmetry autoregressive moving average ARMA ARIMA SARIMA GARCH EGARCH FIGARCH fractional integration long memory Hurst exponent R/S analysis DFA detrended fluctuation analysis wavelet decomposition multiresolution analysis discrete wavelet transform continuous wavelet transform CWT Morlet wavelet Mexican hat wavelet Daubechies wavelet symlets coiflets Meyer wavelet biorthogonal wavelets lifting scheme perfect reconstruction two-channel filter bank quadrature mirror filter QMF orthogonal filter bank paraunitary polyphase matrix Smith form McMillan degree controllable observable realization Kalman decomposition transfer function state-space representation observability controllability detectability stabilizability pole placement LQR linear quadratic regulator Riccati equation Lyapunov equation algebraic Riccati equation CARE DARE Hamiltonian system symplectic integrator Verlet leapfrog Störmer-Butcher B-series rooted trees Butcher tableau order conditions Runge-Kutta RK4 Dormand-Prince RK45 adaptive step-size control error estimator local truncation global truncation Richardson extrapolation Aitken delta-squared Shanks transformation epsilon algorithm Wynn omega Padé approximant continued fraction convergents Stieltjes moment problem Hamburger moment problem Hausdorff moment problem truncated moment problem semidefinite programming Lasserre hierarchy sum-of-squares SOS polynomial optimization Putinar Positivstellensatz Krivine-Stengle Schmüdgen Parrilo hierarchy chordal sparsity correlative sparsity term sparsity exploiting structure interior-point method barrier method primal-dual path-following Mehrotra predictor-corrector self-dual embedding homogeneous self-dual embedding Nesterov-Todd scaling direction complementarity measure duality gap centrality parameter feasibility restoration phase crossover ratio line search Armijo condition Wolfe condition strong Wolfe condition Goldstein condition Zoutendijk condition trust-region dogleg Powell conjugate direction CG Fletcher-Reeves Polak-Ribière Hestenes-Stiefel conjugate gradient nonlinear conjugate gradient quasi-Newton BFGS L-BFGS DFP SR1 symmetric rank-one limited-memory update curvature condition positive definiteness secant equation quasi-Newton equations Dennis-Moré superlinear convergence quadratic convergence Newton-Raphson Newton’s method modified Newton damped Newton Levenberg-Marquardt trust-region Newton dogleg Steihaug truncated CG subproblem Cauchy point dogleg path Powell’s dogleg combined step Gauss-Newton least squares Jacobian singular values SVD QR factorization Householder reflectors Givens rotations Cholesky LDL^T LU partial pivoting banded Toeplitz Hankel Vandermonde Cauchy Hilbert matrix conditioning number numerical stability backward error forward error perturbation analysis rounding error IEEE-754 double precision single precision extended precision half precision subnormal numbers denormal numbers NaN infinity signed zero rounding modes round-to-nearest-even round-toward-zero round-toward-positive round-toward-negative directed rounding stochastic rounding interval arithmetic ball arithmetic Kahan summation compensated summation pairwise summation catastrophic cancellation relative error absolute error significant digits guard digits sticky bit rounding unit ulp machine epsilon floating-point exception invalid division-by-zero overflow underflow inexact flag trap handler signaling quiet NaN payload bits exponent bias mantissa field sign bit representation conversion truncation saturation clamping quantization dither noise shaping sigma-delta modulation oversampling decimation interpolation resampling sample rate conversion anti-alias filter sinc window Kaiser Hamming Hann Blackman Tukey Welch Bartlett triangular Parzen Daniell kernel density estimation bandwidth selection Silverman’s rule cross-validation leave-one-out k-fold stratified bootstrap jackknife permutation test randomization test Monte Carlo importance sampling stratified sampling Latin hypercube Sobol sequence Halton sequence Faure sequence Niederreiter sequence low-discrepancy quasi-random van der Corput radical inverse Kronecker sequence equidistribution Weyl criterion discrepancy star-discrepancy extreme discrepancy L₂ discrepancy P₁ discrepancy Koksma-Hlawka inequality variation bounded variation Hardy-Krause variation isotonic regression pool adjacent violators PAVA least squares monotone cone projection Frisch-Newton interior point Dykstra algorithm alternating projections von Neumann-Halperin Mann iteration Krasnoselskii-Mann averaged operator nonexpansive mapping firmly nonexpansive resolvent maximal monotone operator Yosida approximation Moreau envelope proximal mapping proximal operator forward-backward splitting Douglas-Rachford ADMM alternating direction multiplier method augmented Lagrangian Uzawa method constraint qualification Slater’s condition Karush-Kuhn-Tucker KKT necessary sufficient second-order sufficient LICQ MFCQ SOSC strict complementarity active set working set primal active-set dual active-set column generation branch-and-cut branch-and-price cutting plane Gomory mixed integer Chvátal-Gomory lift-and-project disaggregate-and-lift split closure perspective closure Lovász extension submodular function base polyhedron polymatroid greedy algorithm matroid intersection matroid union augmenting path Edmonds’ algorithm blossom matching Tutte matrix Pfaffian orientation Seymour decomposition graph minor Robertson-Seymour theorem treewidth pathwidth branchwidth clique-width Hadwiger conjecture coloring number chromatic number Brooks’ theorem Vizing’s theorem edge coloring Snark hunt Petersen graph Grötzsch triangle-free planar coloring Mycielski construction Ramsey number bounds Turán’s theorem Erdős-Stone Zarankiewicz problem Kővári-Sós-Turán extremal combinatorics Szemerédi regularity lemma counting lemma dependent random choice absorption technique nibble method probabilistic method alteration Lopsided local lemma Moser-Tardos algorithm entropy compression Lovász local lemma asymmetric conditional probabilities Janson inequality McDiarmid Azuma-Hoeffding Freedman martingale concentration supermartingale optional stopping Wald identity Doob optional sampling Doob maximal inequality Burkholder Davis Gundy martingale transform square function Carleson measure John-Nirenberg BMO VMOS inequality John’s inequality BMO dual L¹ Sobolev embedding Trudinger-Moser exponential integrability Orlicz space Luxemburg norm Young function complementary function Fenchel conjugate Legendre transform Moreau-Rockafellar subdifferential superdifferential Fréchet derivative Gâteaux derivative Hadamard directional Clarke generalized gradient Mordukhovich limiting coderivative proximal normal cone regular normal cone contingent cone tangent cone Clarke tangent cone Bouligand tangent cone radial direction admissible direction feasible direction descent direction ascent direction steepest descent gradient descent momentum Nesterov accelerated proximal gradientISTA FISTA ADMM splitting primal-dual Chambolle-Pock primal-dual hybrid gradient PDHG saddle point minimax von Neumann game theory zero-sum matrix game mixed strategy Nash equilibrium existence proof Kakutani fixed point Brouwer fixed point Poincaré-Miranda intermediate value degree theory homotopy continuation simplicial algorithm Scarf’s algorithm Kuhn’s algorithm Lemke-Howson algorithm support enumeration von Neumann minimax theorem Sion’s theorem Ky Fan’s lemma Glicksberg’s theorem Debreu’s theorem Arrow-Debreu welfare economics social choice Arrow’s impossibility Gibbard-Satterthwaite Muller-Satterthwaite median voter theorem Black’s single-peakedness Campbell-Groves institutional design mechanism design revelation principle Gibbard-1973 deterministic strategy-proofness Maskin monotonicity Riley condition Pareto efficiency constrained efficiency core kernel Shapley value nucleolus bargaining set von Neumann-Morgenstern stable set coalition structure Owen value Aumann-Shapley cost allocation marginal cost pricing Ramsey pricing inverse elasticity rule Boiteux public enterprise pricing Ramsey-Boiteux second-best theory distortionary taxation deadweight loss Harberger triangle excess burden consumption tax production tax ad valorem specific unit tax incidence elasticities demand supply pass-through rate full shifting partial shifting no shifting producer surplus consumer surplus total welfare loss compensating variation equivalent variation Slutsky equation Hicksian demand Marshallian demand income effect substitution effect Giffen good Veblen good inferior good normal good luxury good necessity good Engel curve income elasticity price elasticity cross elasticity own-price cross-price Marshallian Hicksian duality Shephard’s lemma Roy’s identity Hotelling lemma envelope theorem Benveniste-Scheinkman second-order condition curvature strict concavity strong concavity log-concavity quasi-conconcavity single-crossing Spence-Mirrlees sorting signaling education wage premium returns to schooling Mincer equation OLS instrumental variable 2SLS LIML GMM Anderson-Hausman specification test Durbin-Wu-Hausman endogeneity omitted variable bias measurement error errors-in-variables SIMEX regression calibration method of moments efficient score information matrix equality test likelihood ratio Wald score test bootstrap percentile BCa studentized bias-corrected accelerated jackknife-after-bootstrap subsampling m-out-of-n bootstrap wild bootstrap block bootstrap stationary bootstrap sieve bootstrap Bayesian bootstrap posterior predictive check prior predictive check posterior predictive p-value Bayes factor Savage-Dickey bridge sampling thermodynamic integration steppingstone path sampling nested sampling importance reweighting particle filter sequential Monte Carlo Kalman filter extended Kalman unscented Kalman cubature filter ensemble Kalman smoother forward-backward Viterbi Baum-Welch EM algorithm ECM ECME MCEM SEM IECM variational EM coordinate ascent mean-field black-box variational inference stochastic VI reparameterization pathwise derivative score function REINFORCE VIMCO wake-sleep Helmholtz machine wake phase sleep phase contrastive divergence persistent CD Langevin dynamics Hamiltonian Monte Carlo NUTS No-U-Turn sampler dual averaging mass matrix adaptation step size adaptation divergence metric Riemannian Hessian geometry natural gradient Fisher information trust region KL ball reverse KL forward KL α-divergence Rényi divergence Hellinger distance total variation Wasserstein earth mover Sinkhorn entropic regularization unbalanced optimal transport Gromov-Wasserstein fused GW partial OT multi-marginal OT Kantorovich dual potential c-cyclic monotonicity Brenier map McCann interpolation displacement convexity Alexandrov space CAT(k) comparison geometry Gromov-Hausdorff convergence GH distance GHG GHG precompactness Prokhorov tightness Skorokhod J1 topology M1 topology epi-convergence Attouch-Wets topology Mosco convergence Γ-convergence Kuratowski-Painlevé Painlevé-Kuratowski liminf limsup Painlevé lower limit upper limit set convergence graphical convergence epigraphical convergence hypo-convergence Fréchet-Kolmogorov compactness Rellich-Arzela Ascoli equicontinuity Arzelà-Ascoli Bolzano-Weierstrass Heine-Borel Lindelöf separability first countable second countable metrizable Polish space complete separable Baire category meager comeager residual generic property prevalence shy measure shy set Haar null Christensen uniformly distributed equidistribution Weyl criterion discrepancy star-discrepancy extreme discrepancy L₂ discrepancy Koksma-Hlawka inequality variation bounded variation Hardy-Krause variation isotonic regression pool adjacent violators PAVA least squares monotone cone projection Frisch-Newton interior point Dykstra algorithm alternating projections von Neumann-Halperin Mann iteration Krasnoselskii-Mann averaged operator nonexpansive mapping firmly nonexpansive resolvent maximal monotone operator Yosida approximation Moreau envelope proximal mapping proximal operator forward-backward splitting Douglas-Rachford ADMM alternating direction multiplier method augmented Lagrangian Uzawa method constraint qualification Slater’s condition Karush-Kuhn-Tucker KKT necessary sufficient second-order sufficient LICQ MFCQ SOSC strict complementarity active set working set primal active-set dual active-set column generation branch-and-cut branch-and-price cutting plane Gomory mixed integer Chvátal-Gomory lift-and-project disaggregate-and-lift split closure perspective closure Lovász extension submodular function base polyhedron polymatroid greedy algorithm matroid intersection matroid union augmenting path Edmonds’ algorithm blossom matching Tutte matrix Pfaffian orientation Seymour decomposition graph minor Robertson-Seymour theorem treewidth pathwidth branchwidth clique-width Hadwiger conjecture coloring number chromatic number Brooks’ theorem Vizing’s theorem edge coloring Snark hunt Petersen graph Grötzsch triangle-free planar coloring Mycielski construction Ramsey number bounds Turán’s theorem Erdős-Stone Zarankiewicz problem Kővári-Sós-Turán extremal combinatorics Szemerédi regularity lemma counting lemma dependent random choice absorption technique nibble method probabilistic method alteration Lopsided local lemma Moser-Tardos algorithm entropy compression Lovász local lemma asymmetric conditional probabilities Janson inequality McDiarmid Azuma-Hoeffding Freedman martingale concentration supermartingale optional stopping Wald identity Doob optional sampling Doob maximal inequality Burkholder Davis Gundy martingale transform square function Carleson measure John-Nirenberg BMO VMO Morrey Campanato Zygmund class Littlewood-Paley decomposition Calderón-Zygmund singular integral operator T(b) theorem David-Journé weak type (1,1) Marcinkiewicz interpolation Riesz-Thorin complex interpolation real interpolation K-method J-method Coifman-Weiss transference principle pseudo-differential operator Weyl quantization Hörmander symbol class S^m_{ρ,δ} Shubin class Sobolev embedding Trudinger-Moser exponential integrability Orlicz space Luxemburg norm Young function complementary function Fenchel conjugate Legendre transform Moreau-Rockafellar subdifferential superdifferential Fréchet derivative Gâteaux derivative Hadamard directional Clarke generalized gradient Mordukhovich limiting coderivative proximal normal cone regular normal cone contingent cone tangent cone Clarke tangent cone Bouligand tangent cone radial direction admissible direction feasible direction descent direction ascent direction steepest descent gradient descent momentum Nesterov accelerated proximal gradientISTA FISTA ADMM splitting primal-dual Chambolle-Pock primal-dual hybrid gradient PDHG saddle point minimax von Neumann game theory zero-sum matrix game mixed strategy Nash equilibrium existence proof Kakutani fixed point Brouwer fixed point Poincaré-Miranda intermediate value degree theory homotopy continuation simplicial algorithm Scarf’s algorithm Kuhn’s algorithm Lemke-Howson algorithm support enumeration von Neumann minimax theorem Sion
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