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Words: 7k
Articles: 297
Table of contents
7k
297
Probability space
Probability theory
33
1
Event
Probability space
9
Uniform random point in a disk
Probability theory
61
1
Origin in a triangle of three radial random points
Uniform random point in a disk
42
Independent sigma-algebras
Probability theory
120
4
Independent events
Independent sigma-algebras
19
Independence extended from generating pi-systems
Independent sigma-algebras
41
Tail sigma-algebra
Independent sigma-algebras
39
1
Kolmogorov zero-one law
Tail sigma-algebra
15
Product measure
Probability theory
21
Random variable
Probability theory
51
2
Independent random variables
Random variable
37
1
Independent and identically distributed random variables
Independent random variables
15
Queueing theory
Probability theory
129
3
Poisson arrivals see time averages
Queueing theory
39
M-G-1 queue
Queueing theory
50
Little law
Queueing theory
40
Convergence of random variables
Probability theory
329
13
Convergence in distribution
Convergence of random variables
44
Almost sure convergence
Convergence of random variables
33
1
Strong law of large numbers
Almost sure convergence
12
Cramer-Wold theorem
Convergence of random variables
36
Convergence in probability
Convergence of random variables
82
3
Bounded-metric characterization of convergence in probability
Convergence in probability
23
Almost-sure subsequence from convergence in probability
Convergence in probability
21
Independent rare-event counterexample to almost-sure convergence
Convergence in probability
23
Convergence in Lp from convergence in probability and an Lr bound
Convergence of random variables
38
Central limit theorem
Convergence of random variables
56
2
Self-normalized central limit theorem with a second-moment denominator
Central limit theorem
37
Multivariate central limit theorem
Central limit theorem
19
Poisson limit theorem
Convergence of random variables
23
Simple symmetric random walk
Probability theory
21
Martingale
Probability theory
518
20
Symmetric signs forced by the martingale property
Martingale
35
Supermartingale
Martingale
39
1
Supermartingale majorant bound
Supermartingale
25
Stopping time
Martingale
116
3
Stopped martingale in discrete time
Stopping time
81
1
Discounted symmetric random-walk exit transform
Stopped martingale in discrete time
35
Optional sampling theorem for a supermartingale
Stopping time
19
Predictable process
Martingale
219
8
Martingale transform
Predictable process
186
6
Recovery of a martingale-transform integrand by conditional covariance
Martingale transform
14
Stopped martingale
Martingale transform
23
Reflection principle for simple symmetric random walk
Martingale transform
74
1
Point probability for the maximum of simple symmetric random walk
Reflection principle for simple symmetric random walk
29
Predictable representation in a Rademacher filtration
Martingale transform
58
1
Stopped martingale isometry in a Rademacher filtration
Predictable representation in a Rademacher filtration
31
Predictable compensator of a discrete supermartingale
Predictable process
13
Doob decomposition in discrete time
Martingale
24
Snell envelope
Martingale
72
2
Complementarity for the Snell envelope compensator
Snell envelope
19
Optimal stopping time from the compensator
Snell envelope
28
Borel-Cantelli lemmas
Probability theory
102
2
Tail-sum formula for expectation
Borel-Cantelli lemmas
31
Extremes of independent exponential variables
Borel-Cantelli lemmas
38
Stochastic process
Probability theory
420
18
Gaussian process
Stochastic process
24
Brownian motion
Stochastic process
380
16
Brownian bridge
Brownian motion
22
Exponential Brownian martingale
Brownian motion
103
3
Risk-neutral measure for the Black-Scholes model
Exponential Brownian martingale
72
2
Power payoff in the Black-Scholes model
Risk-neutral measure for the Black-Scholes model
9
Brownian time reversal for fixed-strike lookback extrema
Risk-neutral measure for the Black-Scholes model
37
Gaussian-process characterization of Brownian motion
Brownian motion
15
Time inversion of Brownian motion
Brownian motion
26
Brownian motion with drift
Brownian motion
59
3
Finite-horizon maximum of Brownian motion with negative drift
Brownian motion with drift
20
Infinite-horizon crossing probability for Brownian motion with negative drift
Brownian motion with drift
5
Last passage time above a level for Brownian motion with negative drift
Brownian motion with drift
20
Brownian reflection principle
Brownian motion
25
Cameron-Martin theorem for a linear drift
Brownian motion
23
Brownian transition semigroup
Brownian motion
44
1
Brownian compensator martingale
Brownian transition semigroup
26
Exponential test-function characterization of Brownian motion
Brownian motion
42
Characteristic-function convergence theorem
Probability theory
15
Poisson process
Probability theory
276
8
Poisson thinning
Poisson process
67
2
Binomial thinning conditional on a total
Poisson thinning
21
Infinite-server queue transient distribution
Poisson thinning
27
Kolmogorov forward equation for a Poisson process
Poisson process
20
Poisson process conditional arrival times
Poisson process
42
Number of one Poisson type before another
Poisson process
37
Exponential formula for a marked Poisson sum
Poisson process
45
Age of a Poisson process
Poisson process
38
Fokker-Planck equation
Probability theory
27
1
Fokker-Planck probability current
Fokker-Planck equation
14
Wald identity
Probability theory
20
Poisson point process
Probability theory
382
9
Superposition theorem for Poisson point processes
Poisson point process
44
Campbell theorem
Poisson point process
44
Non-homogeneous Poisson point process
Poisson point process
50
1
Intensity function of a point process
Non-homogeneous Poisson point process
15
Mapping theorem for Poisson point processes
Poisson point process
145
2
Radial volume transform of a homogeneous Poisson point process
Mapping theorem for Poisson point processes
85
1
Kth-nearest-neighbour distance in a homogeneous Poisson point process
Radial volume transform of a homogeneous Poisson point process
43
Independent marking theorem for Poisson point processes
Poisson point process
47
Poisson thinning theorem
Poisson point process
29
Order statistic
Probability theory
12
1
Order symmetry of independent random variables
Order statistic
12
Probability distribution
Probability theory
1k
56
Beta distribution
Probability distribution
14
Cumulative distribution function
Probability distribution
72
1
Uniform convergence of distribution functions to a continuous limit
Cumulative distribution function
43
Cauchy distribution
Probability distribution
78
2
Stability of the Cauchy distribution
Cauchy distribution
21
Stable distribution
Cauchy distribution
30
Probability generating function
Probability distribution
28
Probability mass function
Probability distribution
26
Quantile function
Probability distribution
93
2
Inverse transform sampling
Quantile function
27
Quantile coupling for convergence in distribution
Quantile function
51
Discrete probability distribution
Probability distribution
300
11
Geometric distribution
Discrete probability distribution
67
1
Coupon collector problem
Geometric distribution
43
Negative binomial distribution
Discrete probability distribution
70
1
Negative binomial stopping argument
Negative binomial distribution
38
Binomial distribution
Discrete probability distribution
13
Multinomial distribution
Discrete probability distribution
16
Poisson distribution
Discrete probability distribution
122
4
Poisson central limit theorem
Poisson distribution
17
Poisson-multinomial conditioning
Poisson distribution
75
1
Poisson trick
Poisson-multinomial conditioning
64
Normal approximation to the Poisson distribution
Poisson distribution
20
Continuous probability distribution
Probability distribution
397
15
Probability density function
Continuous probability distribution
24
Gamma distribution
Continuous probability distribution
77
2
Erlang distribution
Gamma distribution
28
Sampling an integer-shape gamma distribution
Gamma distribution
35
Exponential distribution
Continuous probability distribution
139
4
Memorylessness of the exponential distribution
Exponential distribution
23
Uniform ratio of independent exponential variables
Exponential distribution
24
Competing exponential clocks
Exponential distribution
41
Geometric sum of exponential variables
Exponential distribution
33
Uniform distribution
Continuous probability distribution
111
3
Sum of two independent uniform variables
Uniform distribution
26
Uniform split of a random total
Uniform distribution
73
1
Exponential characterization by a uniform random split
Uniform split of a random total
49
Laplace distribution
Continuous probability distribution
32
1
Laplace regression
Laplace distribution
20
Symmetric distribution
Probability distribution
18
Median
Probability distribution
14
Moment-generating function
Probability distribution
14
Joint distribution function
Probability distribution
18
Normal distribution
Probability distribution
88
5
Half-normal distribution
Normal distribution
12
Log-normal distribution
Normal distribution
22
Gaussian integration by parts
Normal distribution
20
Standard normal distribution
Normal distribution
25
1
Standard normal distribution function
Standard normal distribution
12
Chi-squared distribution
Probability distribution
39
1
Chi-squared stochastic ordering
Chi-squared distribution
25
Weibull distribution
Probability distribution
9
Characteristic function of a uniform distribution
Probability distribution
22
1
Small-argument sine product asymptotic
Characteristic function of a uniform distribution
14
Variance of a uniform distribution
Probability distribution
12
Exponential survival probability
Probability distribution
17
Expected value
Probability theory
158
10
Variance
Expected value
92
5
Variance additivity for independent random variables
Variance
7
Standard deviation
Variance
12
Covariance
Variance
59
2
Covariance matrix
Covariance
37
1
Correlation coefficient
Covariance matrix
8
Indicator random variable
Expected value
Tail-sum formula
Expected value
16
Linearity of expectation
Expected value
11
Variance of a sample mean
Expected value
18
Probability inequality
Probability theory
123
6
Chebyshev inequality
Probability inequality
16
First moment method
Probability inequality
33
Second moment method
Probability inequality
25
Markov inequality
Probability inequality
31
1
Exponential Markov bound
Markov inequality
19
Union bound
Probability inequality
7
Conditional probability
Probability theory
43
2
Bayes theorem
Conditional probability
19
Conditional distribution
Conditional probability
15
Random partition
Probability theory
55
3
Restricted-growth string
Random partition
15
Singleton block
Random partition
10
Exchangeable random partition
Random partition
15
Random path in a binary tree
Probability theory
18
Markov chain
Probability theory
2k
97
Birth-death chain
Markov chain
21
Product Markov chain
Markov chain
34
Transition matrix
Markov chain
11
Irreducible Markov chain
Markov chain
17
Aperiodic Markov chain
Markov chain
25
Recurrent state
Markov chain
14
Transient state
Markov chain
14
First return time
Markov chain
31
1
Mean recurrence time
First return time
21
Preservation under enlargement of transition support
Markov chain
55
Running minimum of an independent sequence is Markov
Markov chain
22
Overlapping moving sum need not be Markov
Markov chain
38
Reversible Markov chain
Markov chain
27
1
Detailed balance
Reversible Markov chain
15
Hitting probability
Markov chain
78
1
Harmonic hitting probability for the balanced-difference urn
Hitting probability
59
Expected hitting time
Markov chain
26
Expected occupation count before absorption
Markov chain
80
1
Four-state reflecting absorbing random walk
Expected occupation count before absorption
53
Conditional hitting time
Markov chain
23
Periodic Markov chain
Markov chain
18
Lazy Markov chain
Markov chain
14
Convergence to an invariant distribution
Markov chain
12
Recurrent Markov chain
Markov chain
79
4
Positive recurrent state
Recurrent Markov chain
13
Null recurrent state
Recurrent Markov chain
13
Recurrence criterion by return probabilities
Recurrent Markov chain
15
Biased random walk hitting probability
Recurrent Markov chain
23
Random walk on a graph
Markov chain
216
7
Simple random walk
Random walk on a graph
16
Simple random walk on the integer line
Random walk on a graph
90
2
Expected visits to the origin by a simple random walk
Simple random walk on the integer line
27
Simultaneous returns of independent simple random walks
Simple random walk on the integer line
38
Stationary distribution of a graph random walk
Random walk on a graph
12
Biased random walk
Random walk on a graph
84
1
Reflected biased random walk on the nonnegative integers
Biased random walk
73
Gambler's ruin
Markov chain
58
1
Expected duration of symmetric gambler's ruin
Gambler's ruin
42
Strong Markov property
Markov chain
71
1
Two-state excursion visit law
Strong Markov property
42
Stationary cycle occupation formula
Markov chain
41
Markov property
Markov chain
18
Continuous-time Markov chain
Markov chain
1k
48
Martingale problem for a continuous-time Markov chain
Continuous-time Markov chain
26
Q-matrix
Continuous-time Markov chain
22
Transition semigroup of a continuous-time Markov chain
Continuous-time Markov chain
38
1
Kolmogorov equations
Transition semigroup of a continuous-time Markov chain
17
Markov jump-process generator
Continuous-time Markov chain
34
Holding-time reversal along a fixed CTMC jump path
Continuous-time Markov chain
47
Recurrence equivalence for a CTMC and its fixed-time skeleton
Continuous-time Markov chain
40
Jump chain
Continuous-time Markov chain
163
5
Reflected nearest-neighbour random walk
Jump chain
42
Invariant-measure transfer between a jump chain and a CTMC
Jump chain
74
2
Slow holding rates destroy positive recurrence
Invariant-measure transfer between a jump chain and a CTMC
30
Fast holding rates create continuous-time positive recurrence
Invariant-measure transfer between a jump chain and a CTMC
18
Nonexplosion from recurrent visits to a slow state
Jump chain
29
Explosion of a continuous-time Markov chain
Continuous-time Markov chain
74
2
Explosion by summable holding times
Explosion of a continuous-time Markov chain
25
Explosion of an upward-biased walk with geometrically increasing rates
Explosion of a continuous-time Markov chain
38
Invariant distribution of a continuous-time Markov chain
Continuous-time Markov chain
38
1
Invariant distribution of an explosive chain
Invariant distribution of a continuous-time Markov chain
27
Batch-birth linear-death process
Continuous-time Markov chain
60
1
Moments of a batch-birth linear-death process
Batch-birth linear-death process
27
Birth-death process
Continuous-time Markov chain
719
27
Yule process
Birth-death process
108
2
Mean population of a Yule process
Yule process
31
Conditional Yule birth times
Yule process
44
Extinction probability of a linear birth-death process
Birth-death process
25
Detailed balance for a birth-death process
Birth-death process
35
Birth-death master equation
Birth-death process
60
3
First-moment equation of a birth-death process
Birth-death master equation
15
Factorial moment
Birth-death master equation
29
1
Poisson factorial moments
Factorial moment
17
Poisson moment closure for a birth-death process
Birth-death process
31
Diffusion approximation of a birth-death process
Birth-death process
75
2
Linear noise approximation
Diffusion approximation of a birth-death process
27
Mean drift from a Fokker--Planck equation
Diffusion approximation of a birth-death process
24
Birth-death recurrence classification
Birth-death process
30
M-M-1 queue
Birth-death process
195
7
M-M-1 queue with state-dependent admission
M-M-1 queue
46
Positive recurrence of an M-M-1 queue
M-M-1 queue
15
Stationary distribution of an M-M-1 queue
M-M-1 queue
14
Burke theorem
M-M-1 queue
20
M-M-1 queue with Bernoulli feedback
M-M-1 queue
54
1
Poisson output theorem for a feedback queue
M-M-1 queue with Bernoulli feedback
32
Mean busy period of an M-M-1 queue
M-M-1 queue
24
M-M-infinity queue
Birth-death process
144
4
Stationary distribution of an M-M-infinity queue
M-M-infinity queue
22
Burke theorem for an M-M-infinity queue
M-M-infinity queue
32
Transient distribution of an M-M-infinity queue
M-M-infinity queue
25
Mean busy period of an M-M-infinity queue
M-M-infinity queue
27
Finite Markov chain
Markov chain
Communicating class
Markov chain
65
3
Communicating states
Communicating class
20
Closed communicating class
Communicating class
16
Transient communicating class
Communicating class
29
Stationary distribution
Markov chain
8
Renewal process
Probability theory
325
11
Renewal-reward theorem
Renewal process
24
Renewal counting law of large numbers
Renewal process
18
Renewal interval containing a fixed time
Renewal process
145
3
Size-biased distribution
Renewal interval containing a fixed time
43
Exponential renewal interval limit
Renewal interval containing a fixed time
41
Stochastic length bias of a renewal interval
Renewal interval containing a fixed time
43
Residual lifetime process
Renewal process
116
4
Renewal excess limit theorem
Residual lifetime process
76
1
Exponential first interarrival in a renewal superposition containing a Poisson process
Renewal excess limit theorem
53
Residual lifetime Markov chain
Residual lifetime process
13
Equilibrium residual-life distribution
Residual lifetime process
16
Law of total variance
Probability theory
13
Characteristic function
Probability theory
32
1
Characteristic function of a sum of independent variables
Characteristic function
16
Variance normalization
Probability theory
21
Ancestors
(4)
Probability and statistics
Area of mathematics
Mathematics
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