Review of Mathematical Concepts. Hongwei Zhang

Transcription

1 Review of Mathematical Concepts Hongwei Zhang

2 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

3 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

4 Supremum and infimum? R: the set of real numbers A subset S of R is bounded from above if b R s.t. s S, s b Then the least upper bound (l.u.b.) of S is called the supremum (or simply sup) of S, denoted as sup{s: s S} Similarly, the greatest lower bound (g.l.b.) of a set S (S R) bounded from below is called the infimum (or simply inf) of S, denoted as inf{s: s S}

5 Supremum and infimum (contd.) Ex.: S = (-1, 1], then sup{s: s S} = 1, inf{s: s S} = -1 S = [-1, 1], then sup{s: s S} = 1, inf{s: s S} = -1 When sup{s: s S} S, we also call it the maximum of S, and in this case sup = max When inf{s: s S} S, we also call it the minimum, and in this case inf = min

6 Supremum and infimum (contd.) Proposition (why?) If S1 S2 are both bounded from below, then inf{s: s S1} >= inf{s: s S2} If S1 S2 are both bounded from above, then sup{s: s S1} <= sup{s: s S2}

7 Limit & limit points A sequence of real numbers x.n, n 1, is said to converge to the limit x in R if ε>0, n ε s.t. n > n ε, x.n x < ε This is written as lim. n > x n = x A limit, if it exists, is unique (why?) If x.k, k 1, viewed as a set, is bounded from above and is nondecreasing, then the limit of x.k, k 1 exists and equals the sup of the set of numbers x.k, k 1; If x.k, k 1, viewed as a set, is bounded from below and is nonincreasing, then the limit of x.k, k 1 exists and equals the inf of the set of numbers x.k, k 1.

8 Limit & limit points (contd.) If x.k, k 1 is bounded above and below, we define the sequence a.k, k 1 as a.k = inf{x.n: n k}; and the sequence b.k, k 1 as b.k = sup{x.n: n k} Then, a.k, k 1 and b.k, k 1 are both bounded a.k, k 1 is nondecreasing, and b.k, k 1 is nonincreasing Accordingly, lim k a. k exists, and we call it lim inf of the sequence x.k, k 1, written as b. k exists, and we call it lim sup of the sequence x.k, k 1, written as lim k lim inf n lim sup n x. n x. n

9 Limit & limit points (contd.) x R is a limit point of the sequence x.k, k 1, if ε>0 and n>0, m n,ε > n s.t. x.m n,ε x < ε i.e., for every ε>0 (no matter how small), the sequence comes within ε of x infinitely often A sequence can have None limit point: e.g., x.n = n One limit point: any convergent sequence, e.g., x.n = 1/n Several limit point: e.g., x.n = (-1) n

10 Limit & limit points (contd.) If x.n, n 1, is bounded above and below, there is at least one limit point lim inf x. n and lim sup x. n are both limit points n n Other limit points may also exist; let be the set of all limit points, then lim inf n x. n = inf{ x : x } lim sup n x. n = sup{ x : x } lim inf If x. n and lim sup x. n are equal, then there is only one limit point n n which is the limit of the sequence

11 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

12 Fixed point A function f mapping a set C ( R n ) into C is said to have a fixed point at x C if f(x) =x A set C R n is closed if every convergent sequence x.n, n 1, of points in C has its limit point also in C E.g.: C = (0, 1] is not closed, because the limit of the sequence 1/n, n 1 (i.e., 0) is not in C A set C R n is bounded if there is an n-dimensional ball centered at the origin and of finite radius s.t. C is entirely inside the ball

13 Fixed point (contd.) A set C R n is convex if x1,x2 C, and λ, 0<λ<1, λx1 + (1-λ)x2 C i.e., the entire line segment joining x1 and x2 is in C; thus C=[0,1] [2,3] is closed but nonconvex Brouwer s Fixed-Point Theorem Let C R n be a closed, bounded, and convex set. Then a continuous function f:c C has a fixed point in C

14 Proof of Brouwer s Fixed-Point Theorem: for n = 1, and C = [0, 1] Suppose f(0) 0 and f(1) 1 Otherwise, we are done! Define g(x) = f(x) x, then g(0) > 0, and G(1) < 0 Given that g(x) is continuous (because f is continuous), there must exist x in [0, 1] s.t. g(x ) = 0, i.e., f(x ) = x. DONE!

15 Illustration of Brouwer s fixed-point theorem f (x ) 1 x 3 x 2 x 1 0 x 1 x 2 x 3 1 x 1, x2, and x3 are all fixed points

16 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

17 Probability model/experiment Sample space (S) set of possible outcomes/sample-points Set of events (F) An event is a subset of the sample space Rule of assigning probabilities to events (P) Q: what is the probability model of two independent flips of a fair coin?

18 Axioms of probability Normalization P(S) = 1 Monotonicity B A => P(B) <= P(A) Additivity (disjoint events) If A B=φ, P(A B) = P(A) + P(B) Note: the set of events should form a sigma(δ)-field so as to ensure the existence of a probability function P satisfying the above axioms

19 Field & sigma-field A set of sets is a Field if it is closed under union, intersection, and complement; and empty-set and universal set are elements of the set A field is a δ-field if it is closed under any countable set of unions, intersections, and their combinations

20 Conditional probability For any two events A and B, with P(B) > 0, the conditional probability of A, conditional on B, is P(A B) = P(AB)/P(B) Two events A and B are independent if P(AB) = P(A)P(B) For P(B) > 0, this is equivalent to P(A B) = P(A) Two events A and B are conditionally independent given C if P(AB C) = P(A C)*P(B C) or equivalently P(A B C) = P(A C) Q: examples of conditional probability, independent events, conditionally independent events?

21 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

22 Random variables A random variable is a function: S -> R Given and a real number x, there is an event x, and P( x) = P({w S: (w) x}) P( x) is the distribution function of, and is usually denoted as F (x) Monotonically nondecreasing Complex and vector random variables: map sample to a set of finite complex numbers or vectors in some finite dimensional vector space

23 Random variables (contd.) If F (x) has a derivative f (x), we call f (x) the probability density of, i.e., f (x) = d(f (x))/dx If f (x) exists and is finite for all x, we say is a continuous r.v. If has only a countable number of possible outcomes, x 1, x 2,, we say is a discrete r.v. the probability of each outcome x i, {P (x i ): i>=1}, is called the probability mass function (PMF) of

24 Random variables (contd.) Joint distribution function of r.v. 1, 2,, n F 1,,n (x1, x2,, xn) = P(1 x1, 2 x2,, n xn) Then F i (xi) = F 1,,n (,,, xi,,, ) Joint probability density f 1,,n (x1, x2,, xn) is n F( x1, x2,..., xn) x1 x2... xn

25 Random variables (contd.) R.v. 1, 2,, n are independent, if, for all x1,, xn F( x1,..., xn) = n i= 1 P( i If the density or mass function exists, the above formula is equivalent to a product form for the density or mass function xi) Note: pairwise independence does not imply that the entire set is independent (example?)

26 Example of pairwise independence vs. set independence Suppose, Y, and Z have the following joint probability distribution: Then and Y are independent, and and Z are independent, and Y and Z are independent, but, Y, and Z are not independent, since any of them is just the mod 2 sum of the other two, and so is completely determined by the other two.

27 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations, moment-generating functions, characteristic functions Useful inequalities Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

28 Expectations The expected value (or the mean) of a r.v. E[ ] xf ( x) dx; or E[ ] = x xp ( x) = For simplicity, write as E[ ] = = xdf ( x) For non-negative random variables, E[ ] = or E[ ] = 0 x= 1 P( P( x) dx, x), if if is continuous is integer - valued and discrete Exercise: prove them

29 Expectations (contd.) If Y = g(), then E[ Y ] = ydf ( y) = g( x) df ( x) Y Moments E[ n ], central moments E[(-E[]) n ] VAR() (or σ 2 ) = E[(-E[]) 2 ] = E[ 2 ] (E[x]) 2 Standard deviation σ

30 Z=+Y If and Y are independent, F Z ( z) = F ( z y) df ( y) = F ( z x) df ( x) And if and Y both have densities, f Z Let Sn=1+2+ +n, Whether or not 1, 2,, n are independent If 1, 2,, n are independent If 1, 2,, n are independent n 2 2 σ = σ (?) Sn n i= 1 i Y ( z) = f ( z y) f ( y) dy = f ( z x) f ( x) dx E [ Sn] = E[ 1] + E[ 2] E[ n] (?) E [ i] = E[ i] (?) i= 1 n i= 1 Y Y Y (convolution)

31 Example: overhead in bit-stuffing Bit-stuffing in data-link framing: > Q: expected number of inserted bits in a string of length n?

32 Solution i = 1, if the insertion occurred after the i-th bit 0, otherwise Note: E[i] = 0 for i 5, and E[i] = 2-6 otherwise Number of bits inserted (Sn) = 1+x n Thus, E[Sn] = (n-5) 2-6

33 Moments r-th moment ξ r of a r.v. ξ r r = [ ] = r E x f ( x) dx, where r = 0,1,2,3,... If is a discrete r.v., ξ r r = x P r i ( x ), where = i 0,1,2,3,... r-th central moment of m r = E[( ) ], where r = r µ 0,1,2,3,... If is a discrete r.v., m = r r ( x ) P ( x ), where = r µ i i 0,1,2,3,...

34 Moments (contd.) In general Joint moments: ij-th joint moment of and Y If and Y are discrete = = r i i r i i r i r m 0 1) ( ξ µ = = dxdy y x f y x Y E Y j i j i ij ), ( ] [ ξ = l m m l Y j m i l ij y x P y x ), (, ξ

35 Moments (contd.) ij-th joint central moment of and Y m ij = E[( i ) ( Y Y) j ] ξ 11 and m 11 are known as the correlation and covariance of and Y respectively Correlation coefficient of and Y: and Y are uncorrelated, if COV[, Y] = 0 (i.e., ρ Y = 0) If and Y are uncorrelated, then If and Y are uncorrelated, then they are also independent; but not vice versa except for the case when and Y are jointly Gaussian ρ Y 2 + Y = 2 m m 20 δ = δ + δ 11 = m 2 Y 02 COV[ x, δ δ x y y]

36 Moment generating function (MGF) The moment generating function (MGF), if it exists, of a r.v. is t t θ ( t) = E[ e ] = e f ( x) dx, where t is a complex variable Except for a sign reversal in the exponent, MGF is the two-sided Laplace transform for which there is a known inversion formula; thus, in general, knowing θ(t) is equivalent to knowing f (x) and vice versa If is discrete = t txi θ ( t) = E[ e ] e P ( xi ) i

37 MGF (contd.) Benefits of using θ(t) It enables convenient computation of the moments of It can be used to estimate f (x) from experimental measurements of the moments It can be used to solve problems involving computing the sums of r.v.s E.g., Z = with 1 θ ( t) = θ Z, 1 2 ( t) θ being indep. and having MGF, then 2 ( t) It is an important analytical instrument that can be used to demonstrate basic results such as the central limit theorem

38 moments & MGF Note that θ ( t) 2 t ( t ) = E[ e ] = E[1 + t ! 2 n t t = 1+ tµ + ξ ξn ! n! ( t ) n! n +...] Thus, if θ(t) exists k ( k ) d ξk = θ ( 0) = ( θ ( t)) t= 0, k = 0,1,... k dt Note: if all the moments exist and are known, then θ(t) can be computed, based on which f (x) can, in principle, be derived through Laplace transform

39 moments & MGF (contd.) In practice, if is the r.v. whose pdf is desired and.i represents the i-th observation of, then we can estimate the r-th moment of as 1 n n i= 1 r i

40 MGF for multiple variables MGF θ Y (t 1, t 2 ) of two r.v. s and Y Through power series expansion Thus dxdy y x f y t x t e E t t Y Y t t Y + + = = ), ( ) exp( ] [ ), ( 2 1 ) ( θ = = = !! ), ( i j n j i Y j i t t t t ξ θ ), ( ln 2 1 )), ( ( 0,0) ( = = + = = t t Y n l n l n l Y t t t t θ θ ξ

41 MGF for multiple variables (contd.) MGF for N r.v. s 1,, N : N N N k N k k k k k N k N k N i i i N k t k t t E t t t !...!... ] [exp ),...,, ( = = = = = = θ

42 Chernoff bound Given a r.v. and a constant a, P[ a] e ( t) at θ The tightest bound, which occurs when the right-hand side is minimized w.r.t. t, is called the Chernoff Bound

43 Characteristic functions (CF) CF of a r.v. is Φ Except for a sign reversal in the exponent, CF is the Fourier transform (which is widely used in statistical communication theory) of f (x) Always exists For our purpose, CF is the same as MGF If is discrete, Φ jw jw ( w) = E[ e ] = e f ( x) dx, where j = 1 jw jwxi ( w) = E[ e ] e P ( xi ) CF is widely used in statistical communication theory, and is widely used to compute the sums of independent r.v. s; e.g., = i Z = w with 1, 2 being indep., then Φ Z ( w) = Φ ( w) Φ ( 1 2 )

44 Moments & CF Similar to MGF, Φ ( w) ( jw) = n= 0 n! n ξ n Thus ξ n 1 j d dw n ( n) ( n) = Φ (0), where Φ (0) = Φ ( w) n n w= 0

45 Joint characteristic functions CF for N r.v. s 1,, N : For two r.v. s and Y: )] [exp( ),...,, ( = = Φ N i i i N w j E w w w N ), ( ), ( 2 1 )), ( ( (0,0) where (0,0), ) ( ] [ = = + + Φ = Φ Φ = = w w Y n l k r k r Y k r Y k r k r rk w w w w j Y E ξ

46 Table of commonly-used r.v.

47 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

48 Markov Inequality If a non-negative r.v. Y has a mean E[Y], then E[ Y ] P( Y y), y y 0 Proof?

49 Chebyshev Inequality Let Z be an arbitrary r.v. with finite mean E[Z] and finite variance σ z2, then P 2 σ ε Z ( Z E[ Z] ε ) ε 0 2 Proof? Let Y = (Z-E[Z]) 2, then according to Markov Inequality P ( Z E Z ) y) 2 [ ] 2 σ Z y Replacing y with ε 2, we get the Chebyshev Inequality

50 Jensen s Inequality Let R n be a convex set; A function f: R is convex if x1, x2, λ [0,1], f ( λx1 + (1 λ) x2) λf ( x1) + (1 λ) f ( x2) Jensen s Inequality: for a convex function f(x) and a r.v. with finite expectation, E(f()) f(e(x)) Equality hold if 1) f(.) is a linear function, or 2) if is constant with probability 1

51 Holder s Inequality For p>1 and 1/p + 1/q = 1, E( Y ) ( E( p )) 1 p ( E( Y q )) 1 q E.g., take p=q=2, Holder s Inequality becomes Schwarz s Inequality ( E( Y )) 2 E( 2 ) E( Y 2 )

52 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

53 Random process and random sequence Let (S, F, P) be a probability space. is called a random process if (t, ζ) is a r.v. on the probability space for each t R, where ζ is the sample function (i.e., specifying a certain outcome/event) For each fixed ζ, (t, ζ) is an ordinary, deterministic time function For a fixed t, (t, ζ) is a r.v. Usually written as (t) or t Discrete time version of random process [n, ζ] (n Z) is called random sequence; usually written as [n] or n A random process/sequence is statistically specified by the joint distribution of all the combinations of r.v. s of the process/sequence

54 Random process/sequence Mean function µ ( t) = E[ ( t)] Correlation function Covariance function Note: K K R ( t1, t2) = E[ ( t1) ( t2)] ( t1, t2) = E[( ( t1) µ ( t1))( ( t2) ( t µ * ( t1, t2) = R ( t1, t2) µ ( t1) ( t2) µ 2 )) * ]

55 Some elementary random sequence/process Bernoulli process Gaussian process Poisson process

56 Bernoulli process A sequence of i.i.d. Bernoulli trials Formally: a discrete-time random process (i.e., random sequence) consisting of a finite or infinite sequence of independent r.v. s [n], n = 1, 2,, such that For each i, the value of [i] is either 0 or 1; For all values of i, the probability that [i] = 1 is the same number p E.g., a sequence of independent coin toss

57 Gaussian process (sequence) A random process (sequence) all of whose n-th order joing pdf s or more generally PDF s are Gaussian µ (t), K (u, v), - <t, u, v<, are enough information to completely specify the r.p.

58 Poisson process Exponential Distribution Memoryless Property Poisson Distribution Poisson Process

59 Exponential Distribution A continuous R.V. follows the exponential distribution with parameter µ, if its pdf is: f ( x) = e x µ µ if x 0 0 if x < 0 => Probability distribution function: µ x 1 e if x 0 F ( x) = P{ x} = 0 if x < 0 Usually used for modeling service time

60 Exponential Distribution (contd.) Mean and Variance: 1 1 E[ ] =, Var( ) = 2 µ µ Proof: µ x E[ ] = x f ( x) dx = xµ e dx = 0 0 µ x µ x 1 = xe 0 + e dx = 0 µ 2 2 µ x 2 µ x µ x 2 2 E[ ] = x µ e dx = x e xe dx E[ ] 0 = = 0 2 µ µ Var( ) E[ ] ( E[ ]) µ µ µ 2 2 = = = 2 2 2

61 Memoryless Property Past history has no influence on the future P{ > x + t > t} = P{ > x} Proof: P{ > x + t, > t} P{ > x + t} P{ > x + t > t} = = P{ > t} P{ > t} µ ( x+ t ) e µ x = = e = P{ > x} µ t e Exponential: the only continuous distribution with the memoryless property

62 Poisson Distribution A discrete R.V. follows the Poisson distribution with parameter λ if its probability mass function is: k λ λ P{ = k} = e, k = 0,1,2,... k! Wide applicability in modeling the number of random events that occur during a given time interval (=>Poisson Process) Customers that arrive at a post office during a day Wrong phone calls received during a week Students that go to the instructor s office during office hours packets that arrive at a network switch etc

63 Poisson Distribution (contd.) Mean and Variance E[ ] = λ, Var( ) = λ Proof: k k λ λ λ λ E[ ] = kp{ = k} = e k = e k! ( k 1)! k= 0 k= 0 k= 0 j λ λ λ λ = e λ = e λe = λ j! j= 0 k λ λ k! ( k 1)! k 2 2 λ 2 λ E[ ] = k P{ = k} = e k = e k k= 0 k= 0 k= 0 λ λ λ = e ( j + 1) = je + e = + j! j! j! j j j λ λ λ 2 λ λ λ λ λ Var( ) E[ ] ( E[ ]) j= 0 j= 0 j= = = λ + λ λ = λ

64 Sum of Poisson Random Variables i, i =1,2,,n, are independent R.V.s i follows Poisson distribution with parameter λ i Sum Sn = n Follows Poisson distribution with parameter λ λ = λ1 + λ λn

65 Sum of Poisson Random Variables (contd.) Proof: For n = 2. Generalization by induction. The pmf of S= is PfS=mg = = = m k=0 m Pf 1 =k; 2 =m kg Pf 1 =kgpf 2 =m kg k=0 m 1 k1 e k=0 k! = e ( 1+ 2) 1 m! m k 2 e 2 m k=0 (m k)! = e ( 1+ 2) ( 1+ 2) m m! Poisson with parameter = k!(m k)! k1 m k m! 2

66 Sampling a Poisson Variable follows Poisson distribution with parameter λ Each of the arrivals is of type i with probability p i, i =1,2,,n, independent of other arrivals; p 1 + p p n = 1 i denotes the number of type i arrivals, then 1, 2, n are independent i follows Poisson distribution with parameter λ i = λp i

67 Sampling a Poisson Variable (contd.) Proof: Forn=2. Generalizebyinduction. Jointpmf: Pf 1 =k 1 ; 2 =k 2 g= = Pf 1 =k 1 ; 2 =k 2 j=k 1 +k 2 gpf=k 1 +k 2 g ³ k1 +k 2 = p k 1 1 k pk 2 2 e k1+k 2 1 (k 1 +k 2 )! 1 = k 1!k 2! ( p 1) k 1 ( p 2 ) k 2 e (p 1+p 2 ) = e p 1 ( p 1) k 1 k 1! e p 2 ( p 2) k 2 k 2! ² 1 and 2 are independent ² Pf 1 =k 1 g=e p ( p 1 1) k 1 k 1, Pf! 2 =k 2 g=e p ( p 2 2) k2 k 2! i followspoissondistributionwithparameter p i.

68 Poisson Approximation to Binomial Binomial distribution with parameters (n, p) n k P{ = k} = p (1 p) k n k As n and p 0, with np=λ moderate, binomial distribution converges to Poisson with parameter λ Proof: n k P{ = k} = p (1 p) k ( n k + 1)...( n 1) n λ λ = 1 k! n n ( n k + 1)...( n 1) n 1 k n n λ 1 n n k e n λ 1 1 n n P{ = k} e n λ λ k λ k! n k k n k

69 Poisson Process with Rate λ {A(t): t 0} counting process A(t) is the number of events (arrivals) that have occurred from time 0 to time t, when A(0)=0 A(t)-A(s) number of arrivals in interval (s, t] Number of arrivals in disjoint intervals are independent Number of arrivals in any interval (t, t+τ] of length τ Depends only on its length τ Follows Poisson distribution with parameter λτ n λτ ( λτ ) P{ A( t + τ ) A( t) = n} = e, n = 0,1,... n! => Average number of arrivals λτ; λ is the arrival rate

70 Interarrival-Time Statistics Interarrival times for a Poisson process are independent and follow exponential distribution with parameter λ λs P{ τ s} = 1 e, s 0 n t n : time of n th arrival; τ n =t n+1 -t n : n th interarrival time Proof: Probability distribution function s P{ τ s} = 1 P{ τ > s} = 1 P{ A( t + s) A( t ) = 0} = 1 e λ n n n n Independence follows from independence of number of arrivals in disjoint intervals

71 Small Interval Probabilities Interval (t+ δ, t] of length δ P{ A( t + δ ) A( t) = 0} = 1 λδ + ο( δ ) P{ A( t + δ ) A( t) = 1} = λδ + ο( δ ) P{ A( t + δ ) A( t) 2} = ο( δ ) Proof:

72 Merging & Splitting Poisson Processes λ 1 λ 2 λ 1 + λ 2 λ p 1-p λp λ(1-p) A 1,, A k independent Poisson processes with rates λ 1,, λ k Merged in a single process A= A A k A is Poisson process with rate λ= λ λ k A: Poisson processes with rate λ Split into processes A 1 and A 2 independently, with probabilities p and 1-p respectively A 1 is Poisson with rate λ 1 = λp A 2 is Poisson with rate λ 2 = λ(1-p)

73 Modeling Arrival Statistics Poisson process widely used to model packet arrivals in numerous networking problems Justification: provides a good model for aggregate traffic of a large number of independent users n traffic streams, with independent identically distributed (iid) interarrival times with PDF F(s) not necessarily exponential Arrival rate of each stream λ/n As n, combined stream can be approximated by Poisson under mild conditions on F(s) e.g., F(0)=0, F (0)>0 Most important reason for Poisson assumption: Analytic tractability of queueing models

74 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

75 Background Consider a random sequence n, n 1 For each sample path ζ, n (ζ) is real number sequence, and it may or may not converge Objective: To describe convergence behavior of n without requiring convergence for every sample path

76 Convergence in probability n, n 1, converges in probability to a r.v., if for each ε>0 lim Pr( n > ε ) = n 0 Written as n p

77 Convergence with probability 1 n, n 1, converges almost surely or with probability 1 to a r.v., if Pr(lim = ) = 1 n n Or, more explicitly Pr w : lim n ( w) = ( w) = 1 n Written as n a s. n w p.. 1

78 Convergence in distribution n, n 1, with distributions F n (.), n 1, converges in distribution to a r.v., with distribution F(.), if lim n F n ( x) = F( x) whenever x is not a point of discontinuity (i.e., a jump point) of F(.) Written as n dist

79 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

80 Weak Law of Large Numbers: assuming a finite variance Let 1, 2,, n be identically distributed, uncorrelated r.v. with a finite mean E[] and finite variance σ 2, let Sn=1+ +n, and consider the sample average Sn/n Note σ Sn2 = nσ 2, thus VAR(Sn/n)= σ Sn2 /n 2 = σ 2 /n; Therefore, lim n Sn VAR( ) n = limn E[ 2 ( Sn / n E[ ]) ] = 0 i.e., variance of sample average goes to 0 as n goes to, and sample average converges to E[] in the mean square sense

81 Weak Law of Large Numbers: assuming a finite variance (contd.) Apply Chebyshev inequality to sample average Sn/n, we have P( Sn / n E[ x] ε ) thus, for any ε>0, we have 2 σ 2 nε lim P( Sn / n E[ x] ε ) = n 0 i.e., the Weak Law of Large Numbers (with finite variance) ; Sn/n is said to converge to E[] in probability.

82 Weak Law of Large Numbers: without assuming finite variance Let 1, 2,, n be identically distributed, uncorrelated r.v. with a finite mean E[], let Sn=1+ +n. Then for any any ε>0, lim P( Sn / n E[ x] ε ) = n 0 (proof skipped)

83 Strong Law of large numbers (ver. 1) Let 1, 2,, n be i.i.d. r.v. with a finite mean E[], let Sn=1+ +n. Then for any any ε>0, Sm lim n P(sup E[ x] ε ) = m n m 0 That is, the sequence S1, S2/2, S3/3, has the property that it is increasingly unlikely (with increasing n) for any term in the sequence beyond n to deviate by more than ε from the mean Note: weak law says that it is increasingly unlikely for Sn/n to deviate from the mean by ε

84 Strong Law of large numbers (ver. 2): commonly used Let 1, 2,, n be i.i.d. r.v. with a finite mean E[], let Sn=1+ +n. Then with probability 1, lim Sn n n = E[ ] i.e., Sn P lim n = E[ ] = 1 n converge with prob. 1, or almost sure convergence

85 Central limit theorem n, n 1, is a sequence of i.i.d. r.v. with finite mean µ and finite variance σ 2. Then 1 σ n n k = 1 ( ) dist µ N(0,1) k

86 Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable Expectations Useful inequalities Random process/sequence Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

87 Strict stationarity n, n 1, is strictly stationary or stationary if for all k 1, and indices n 1, n 2,, n k, and m, dist = n1+ m n 2 + m k (,,..., ) (,,..., ) n 1 n 2 n k n + m i.e., all joint CDFs or pdfs of the r.p. are not a function of time shift (note: can still change with relative space in between)

88 Wide-sense stationarity (w.s.s.) n, n 1, is wide-sense stationary if µ (t) = µ, not a function of t; and R (t1, t2) = R (t1+t, t2+t), for all t1, t2, T; or equivalently K (t1, t2) = K (t1+t, t2+t) i.e., R (t1, t2) and K (t1, t2) is a function of (t2-t1) only Facts (Strict) Stationarity => w.s.s. For a Gaussian r.p., w.s.s. => stationarity

89 Ergodicity Consider a r.p. n, n 1 Invariant event Consider a question about the process whose answer does not depend on whether we ask the question about n, n 1, or about n+k, n 1, for any k 1 Each such question yields an event on which the answer is yes, and its complement on which the answer is no Such events are called invariant events A r.p. is ergodic if each invariant event has a probability of either 0 or 1

90 Birkhoff s strong ergodic theorem: a generalizaiton of strong law of large numbers Given a stationary and ergodic process n, n 1, and a function f(.) that maps realization of the process (i.e., n (w), n 1) such that E( f( n, n 1) ) <, then with probability 1, lim m m 1 1 k = 0 m f ( n+ k, n 1) = E[ f (, n 1)] An application: time averages converge to the ensemble average for a stationary and ergodic process Define f( n, n 1) = 1 ; thus f( n+k, n 1) = k ; Then lim m 1 m m 1 k = 0 k = E[ 1 ], which is the expectation of any of the r.v.'s of the process n

91 Summary Limits of real number sequences A fixed-point theorem Probability and random processes Probability model, Random variable Expectations Useful inequalities Random process/sequence, Convergence concepts Laws of large numbers & central limit theorem Stationarity and ergodicity

92 Homework #0 (total points: 150) 1. [60 points] Let 1, 2,, n, be a sequence of independent identically distributed (IID) continuous random variables with the common probability density function f (x); note that P(=α) = 0 for any α and that P(1 = 2) = 0. a) Find P(1 2). (Hint: give a numerical answer, not an expression; no computation is required and a one or two line explanation should be adequate.) b) Find P(1 2; 1 3) (in other words, find the probability that 1 is the smallest of 1, 2, and 3; again think --- do not compute) c) Let the random variable N be the index of the first r.v. in the sequence to be less than 1; that is, P(N = n) = P(1 2, 1 3,, 1 n-1, 1 n ). Find P(N > n) as a function of n. (Hint: generalize part b).) d) Show that E[N] =.

93 Homework #0 (contd.) 2. [30 points] A computer system has n users, each with a unique name and password. Due to a software error, the n passwords are randomly permuted internally (i.e., each of the n! possible permutations are equally likely). Only those users lucky enough to have had their passwords unchanged in the permutation are able to continue using the system. a) What is the probability that a particular user, say user 1, is able to continue using the system? b) What is the expected number of users able to continue using the system? (Hint: let i be a random variable with the value 1 if user i can use the system and 0 otherwise.)

94 Homework #0 (contd.) 3. [60 points] A town starts a mosquito control program and we let the r.v. Zn be the number of mosquitos at the end of nth year (n = 0, 1, 2, ). Let n be the growth rate of mosquitos in year n, i.e., Zn = n*z n-1, n 1. Assume that {n; n 1} is a sequence of IID random variables with the PMF P(=2) = ½, P(=1/2) = ¼, and P(=1/4) = ¼. Suppose that Z0, the initial number of mosquitos, is some known constant and assume for simplicity and consistency that Zn can take on non-integer values. a) Find E[Zn] as a function of n, and find lim E[ Zn]. b) Let Wn = log 2 n. Find E[Wn] and E[log 2 (Zn/Z0) ] as a function of n. c) Using the Strong Law of Large Number, show that with probability 1 and find the value of α. d) Using c), show that lim Zn n = β with probability 1 for some constant β and evaluate β. n limn (1/ n)[log2( Zn / Z0)] = α