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Write $N=a+b+c$ for the conserved total number of tokens. The main case has positive integer initial holdings. If an initial holding is zero, take $\tau =0$; the formulas below still hold when $N>0$. If $N=0$, the stochastic process is constant and $\mathbb{E}[\tau ]=0$.

After $\tau$, the token product remains zero, whereas the deterministic term $nN/3$ continues to increase. Combining this with the pre-elimination calculation in part (ii),The stochastic process is adapted and integrable at each fixed time, and the displayed conditional expectation is nonnegative. Thus $(M_n)$ is a submartingale.

For $N>0$ it is neither a martingale nor a supermartingale. If $a,b,c>0$, there is positive probability that the first player transfers a token to the second in each of the next $a$ rounds. Until the last such round all three remain active, so this event has probability $6^{-a}>0$ and entails $\tau \le a$. At time $n=a$ the displayed conditional expectation is therefore strictly positive on an event of positive probability, ruling out both the zero drift required of a martingale and the nonpositive drift required of a supermartingale. If an initial holding is zero, $\tau =0$ and the same conclusion follows immediately. The degenerate case $N=0$ gives a constant martingale.

A path probability is a probability measure, or a density relative to a specified reference measure, on histories of a stochastic process. A formal Onsager--Machlup path probability must be interpreted with its discretization and initial-state conditioning specified; conditioning on both endpoints produces a separately normalized bridge measure.

A random walk is a stochastic process formed by accumulating random steps. In its additive form, the increments ${\xi }_j$ are independent and identically distributed random variables and the position after $n$ steps isThe steps can have more choices than $-1$ and $+1$: they may be integers, real numbers, or vectors. A random walk on a graph instead describes steps between vertices along edges; its available moves can depend on the current vertex.

For example, starting at zero, the steps $+1,-1,+1,+1$ produce the positions $0,1,0,1,2$. This is one sample path; other step sequences can have the same endpoint. A simple random walk on the integer line makes this distinction particularly tangible: repeated Bernoulli trials determine both a whole sample path and an endpoint with a binomial distribution. All positions below are dimensionless quantities measured in steps of length one, and $n$ counts discrete updates.

The model is useful because it separates an average tendency from accumulated fluctuations. A fair simple symmetric random walk has zero expected value at every time, yet its standard deviation grows: cancellation across many possible sample paths does not keep each walker near zero. The two figures below compare sample paths with exact endpoint probabilities. For the basic additive construction and its independent increments, see MIT, Stochastic Processes I, section 2.

For a stochastic process $X_t$ and a fixed sample space outcome $\omega$, its sample path is the function $t\mapsto X_t(\omega )$. One row of observed fair bits, one trajectory of a random walk, or one staircase of Poisson process arrival counts is a sample path. Fixing $t$ instead gives a random variable ranging across possible histories. The same two-variable object $X_t(\omega )$ therefore describes both observations across outcomes and evolution within an outcome.

Statements about continuous, càdlàg or Hölder continuous function sample paths concern these functions simultaneously in time. They differ from assertions made at each fixed time separately. For example, take $U$ with a uniform distribution on $[0,1]$ and set $X_t=1_{\{t=U\}}$. For every fixed $t$, $X_t=0$ almost surely, but every sample path has a single spike. Thus this process is a modification of a stochastic process that is identically zero, while the two are not indistinguishable stochastic processes. Continuity of both sets of sample paths rules out this discrepancy: equality at a countable dense set then implies equality everywhere. This is why path regularity is an additional part of constructing a continuous-time model.

A simple random walk on a locally finite graph chooses each vertex in the current vertex's graph neighbourhood with equal probability, assuming there is at least one available neighbour. On the integer line there are two choices, one unit left or right, so each has probability $1/2$. This is the one-dimensional simple symmetric random walk. Terminology varies: a simple random walk on the integer line is also often allowed unequal step probabilities, with “symmetric” specifying the fair case.

For a simple symmetric random walk starting at $S_0=0$, each increment has the Rademacher distribution, with expected value zero and variance one. Thus linearity of expectation and the variance of a sum of independent random variables giveFor example, after $80$ steps the expected value is zero, the variance is $80$, and the standard deviation is $\sqrt{80}\approx 8.944$. Doubling the typical distance scale takes four times as many steps. This square-root growth explains why a random walk can wander far even when there is no preferred direction.

The colored curves are illustrative sample paths, not averages over all possible walks. The dashed curves show the expected values; the shaded bands show the expected value plus or minus one standard deviation at each time. These bands are a scale of spread, not confidence intervals, and give no guarantee that an entire sample path stays inside them.

Zero expected value does not make this a stationary process: its one-time probability distribution changes with $n$, as the increasing variance already proves. It does have stationary increments and independent increments, because blocks of fresh steps have the same probability distribution at any starting time and disjoint blocks use independent random variables. This separation of position from increment is one reason the random walk is such a useful first example of a stochastic process.

Spot volatility is the instantaneous diffusion coefficient multiplying relative price increments in a stochastic differential equation for an asset. In a stochastic volatility model it is itself a stochastic process.

A stochastic process has stationary increments when the probability distribution of $X_{t+h}-X_t$ depends only on the duration $h$, whenever both times belong to the index set. The clock origin then does not change the increment law. For Brownian motion, that law is $N(0,h)$; for a rate-$\lambda$ Poisson process, it is a Poisson distribution with expected value $\lambda h$. Neither process itself is a stationary process: their variances grow with $t$. Stationary increments and a stationary process describe different properties.

Random variation represented by a probability distribution or stochastic process, whether from intrinsic events or an uncertain environment.

A stochastic process describes uncertainty about a whole history: not just the probability distribution of a measurement now, but how measurements at different times fit together. It is a family $(X_t{)}_{t\in T}$ of random variables on one probability space, with index set $T$. Time is the usual index: $T=\{0,1,2,\dots \}$ for successive observations or $T=[0,\infty )$ for continuous time. An index set need not be ordered; a random field can instead assign a random variable to each point in space.

There are two complementary views. Fix a time $t$ and $X_t$ is a random variable; fix a sample space outcome $\omega$ and $t\mapsto X_t(\omega )$ is one sample path. A model specifies the probability distribution of these histories. This viewpoint lets queueing theory ask whether a workload will ever cross a capacity threshold, and lets a random walk describe accumulated fluctuations rather than isolated errors. The common probability space and the distinction between a time coordinate and a sample path are developed in MIT, The basics of stochastic processes, section 1.

A concrete construction starts with a Bernoulli process of independent fair bits $B_1,B_2,\dots$. Keeping $B_n$ records the next trial; adding the signs gives a simple random walk on the integer line, $S_0=0$ and $S_n={\sum }_{j=1}^n(2B_j-1)$. Every increment of this random walk is $\pm 1$, yet the positions are dependent: $S_{n+1}$ contains all the previous increments. Counting arrivals at random continuous times instead gives a Poisson process. These examples separate three questions: what values are possible, how time is indexed, and how the values depend on one another.

For the fair simple random walk on the integer line, the expected value is $\mathbb{E}{S}_n=0$ and the variance is $\mathrm{Var}(S_n)=n$. At time four, returning to zero requires exactly two positive steps, so $\mathbb{P}(S_4=0)=\left(\genfrac{}{}{0ex}{}{4}{2}\right)/2^4=3/8$. A rate-two Poisson process instead has $\mathbb{P}(N_1=0)=e^{-2}\approx 0.1353$. A single realized sample path does not establish either probability: they follow from the specified joint probability distributions.

The most revealing feature is often dependence. Two stochastic processes can have identical marginal distributions at every time while one forgets everything at the next step and the other repeats its first value forever. Their finite-dimensional distributions and covariance functions distinguish them. That extra structure is what makes a stochastic process useful for prediction.

A stock is a traded ownership interest in a company. In a mathematical finance model, its price is a stochastic process and a self-financing portfolio may hold positive or negative numbers of its shares.

The stopped stochastic process follows $X$ until the stopping time $\tau$ and thereafter keeps the value $X_{\tau }$ when $\tau$ is finite. For a pathwise right-continuous adapted process, the stopped process is adapted, by progressive measurability and measurable evaluation at $t\wedge \tau$. When $X$ is a martingale, further stopping and integrability conditions determine whether its stopped process is a martingale.

A time series is a sequence of observations indexed by time: hourly temperatures, daily sales, or successive measurements of a fluctuating signal. A probabilistic model treats these observations as one sample path of a stochastic process $(X_t)$. For equally spaced observations, take $t\in \mathbb{Z}$ and measure a lag of a time series in observation intervals. The order matters: shuffling a record preserves its observed values but destroys information about how one value follows another.

Two basic ways to model temporal dependence are an autoregressive model, which feeds past observations into the next value, and a moving-average model, which combines present and past white noise. An AR(1) carries a disturbance forward with diminishing weights; an MA(1) carries it into just two observations. Their autocorrelation functions make this difference visible. A weakly stationary process supplies a common mean and a lag-dependent autocovariance with which to describe that dependence.

These models help forecast future observations and quantify the uncertainty of averages. Successive measurements need not provide as much information as the same number of independent random variables: the long-run variance of a stationary process accounts for their accumulated covariances. Trend and seasonality require attention before fitting a stationary model; differencing and a seasonal difference operator can help, but do not automatically make every process stationary. See Peter Bartlett, Introduction to Time Series Analysis, lecture 1: objectives and models.

The three illustrated sample paths have zero mean and unit variance, so the visual comparison concerns dependence rather than scale. The AR(1) paths start from their stationary normal distribution, independently of future discrete Gaussian white noise; the MA(1) path includes the noise preceding its first observation. These are finite random examples, not exact pictures of an ACF. Compare their theoretical autocorrelation functions in autocorrelation.

Historically, Yule's 1927 study investigated disturbed periodicities in sunspot observations; it should not be read as the discovery of the particular AR(1) examples here. Slutsky studied how summing random inputs could produce wave-like records, a warning against inferring a periodic cause from appearance alone. The editorial note to his 1937 English paper identifies the 1927 Russian original. See G. Udny Yule, On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers (1927) and Eugen Slutsky, The Summation of Random Causes as the Source of Cyclic Processes (1937), including the editorial note on the 1927 original.

A real stochastic process $(X_t{)}_{t\in \mathbb{Z}}$ is a weakly stationary process if it has finite second moments, a constant expected value $\mu$, and an autocovariance depending only on the lag of a time series:In particular its variance is the same at every time, $\mathrm{Var}(X_t)=\gamma (0)$. This is often called second-order stationarity. It makes observations from different parts of a record comparable through the same autocorrelation function. Bartlett uses this second-order meaning of stationarity in Introduction to Time Series Analysis, lecture 2: mean, covariance, and stationarity.

For a worked contrast, let ${\epsilon }_t$ be centered white noise of variance ${\sigma }^2>0$. The process $X_t=5+{\epsilon }_t$ has mean $5$, variance ${\sigma }^2$, and zero autocovariance at nonzero lags, so it is weakly stationary. The process $Y_t=t+{\epsilon }_t$ has exactly the same covariance structure, but its mean is $t$, so it is not weakly stationary. Constant variance alone is insufficient.

Weak stationarity need not imply strict stationarity. For example, choose independent random variables at all integer times, using a standard normal distribution at even times and a Rademacher distribution at odd times. Every coordinate has zero mean and unit variance, and distinct coordinates have zero covariance. Nevertheless a shift by one changes the marginal probability distribution from continuous to two-point. This also illustrates why weak white noise need not be identically distributed.