VMSTA Modern Stochastics: Theory and Applications 2351-60542351-60462351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA268 10.15559/24-VMSTA268 Research Article Regularity of paths of stochastic measures https://orcid.org/0000-0002-8234-2797 RadchenkoVadymvadymradchenko@knu.ua Department of Mathematical Analysis, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine 2025 2611202412218920110920241511202415112024 © 2025 The Author(s). Published by VTeX2025 Open access article under the CC BY license.

Random functions μ ( x ), generated by values of stochastic measures are considered. The Besov regularity of the continuous paths of μ ( x ), x ∈ [ 0 , 1 ] d , is proved. Fourier series expansion of μ ( x ), x ∈ [ 0 , 2 π ], is obtained. These results are proved under weaker conditions than similar results in previous papers.

 Stochastic measure trajectories of random functions Besov space random Fourier series 60G17 60H05 Introduction

In this paper, we consider the properties of paths of processes generated by values of stochastic measures (SMs). SM is a stochastic set function that is σ-additive in probability, see the exact definition and examples in Subsection 2.1.

The Besov regularity of paths of SMs was studied in [17] and [18], Fourier series expansions of paths of SMs were obtained in [19]. Some important results in these papers were obtained under the following condition for SM μ defined on measurable space ( X , B ).

There exists a real-valued finite measure  m on  ( X , B ) with the following property: if a measurable function g : X → R is such that ∫ X g 2 d m < + ∞, then g is integrable with respect to (w.r.t.) μ on  X.

In this paper, we prove that some statements from [18] and [19] remain valid if we assume the following condition instead of A1.

There exists a real-valued finite measure  m on  ( X , B ) with the following property: if a measurable function g : X → R is such that for all λ ∈ R, λ > 0, it holds that ∫ X 2 λ | g | d m < + ∞, then g is integrable with respect to (w.r.t.) μ on  X.

Obviously, condition A2 is weaker than A1.

Our proofs are very similar to respective proofs in [18] and [19] with one essential change – now we use Lemma 2 from the present paper. In the previous publications, the statement of Lemma 2 was obtained under condition A1, in our paper, we assume A2. Note that in many of our results, we assume that the paths of our processes are continuous.

The rest of the paper is organised as follows. In Section 2 we recall the basic facts concerning SMs and Besov spaces and give a short literature review. In Section 3 we prove Lemma 2 and other auxiliary lemmas. Section 4 contains the result about the Besov regularity of SMs on [ 0 , 1 ] d . In Section 5 we study the Fourier series defined by process μ ( t ) = μ ( ( 0 , t ] ), t ∈ [ 0 , 2 π ].

 Preliminaries Stochastic measures

In this subsection, we give basic information concerning stochastic measures in a general setting. In statements of Sections 4 and 5, this set function is defined on Borel subsets of [ 0 , 1 ] d or [ 0 , 2 π ].

Let L 0 = L 0 ( Ω , F , P ) be the set of all real-valued random variables defined on the complete probability space ( Ω , F , P ). Convergence in L 0 means the convergence in probability. Let X be an arbitrary set and B a σ-algebra of subsets of X.

A σ-additive mapping μ : B → L 0 is called stochastic measure (SM).

We do not assume the moment existence or martingale properties for SM. We can say that μ is an L 0 -valued measure.

We note the following examples of SMs. The orthogonally scattered stochastic measures are SMs with values in L 2 ( Ω , F , P ). The α-stable random measures defined on a σ-algebra for α ∈ ( 0 , 1 ) ∪ ( 1 , 2 ] are independently scattered SMs, see [22, Chapter 3], Definition 1 holds by [22, Proposition 3.5.1].

Many examples of the SMs on the Borel subsets of [ 0 , T ] may be given by the Wiener type integral μ ( A ) = ∫ [ 0 , T ] 1 A ( t ) d X t . For example, this holds if X t is any square integrable martingale or fractional Brownian motion with the Hurst index H > 1 / 2. Other examples may be found in [15, Section 2.1] or [21, Section 1.2.1].

For deterministic measurable functions f : X → R, an integral of the form ∫ X f d μ is studied in [13, Chapter 7], [21, Chapter 1]. In particular, every bounded measurable f is integrable w.r.t. any μ, and for any integrable g holds lim c → + ∞ sup | h | ≤ | g | P { | ∫ X h d μ | ≥ c } = 0 (this follows from Corollary 1.1, Lemma 1.8, and Theorem 1.2 [21]). An analogue of the Lebesgue dominated convergence theorem holds for this integral (see [13, Proposition 7.1.1] or [21, Theorem 1.5]).

The theory of SMs in detail is considered in [21]. Equations driven by SMs are studied, for example, in [3, 14, 23]. SMs may be used for the study of stochastic dynamical systems (see [1, 2]).

In the sequel, B ( X ) denotes the Borel σ-algebra of subsets of X, m L denotes the Lebesgue measure.

 Besov spaces

These classical functional Banach spaces found many applications in mathematical physics, function theory, and functional analysis. We recall the definition of Besov spaces following [12].

For functions f ∈ L p ( [ 0 , 1 ] d ) = L p ( [ 0 , 1 ] d , m L ) we set ‖ f ‖ B p , q α ( [ 0 , 1 ] d ) = ‖ f ‖ L p ( [ 0 , 1 ] d ) + ( ∫ 0 1 ( ω p ( f , r ) ) q r − α q − 1 d r ) 1 / q , where ω p denotes the L p -modulus of continuity, ω p ( f , r ) = sup | h | ≤ r ( ∫ I h | f ( x + h ) − f ( x ) | p d x ) 1 / p , I h = { x ∈ [ 0 , 1 ] d : x + h ∈ [ 0 , 1 ] d } , | h | denotes the Euclidean norm in R d . Then B p , q α ( [ 0 , 1 ] d ) = { f ∈ L p ( [ 0 , 1 ] d ) : ‖ f ‖ B p , q α ( [ 0 , 1 ] d ) < + ∞ } , and ‖ · ‖ B p , q α ( [ 0 , 1 ] d ) is a norm in this space.

The Besov regularity of trajectories was studied for different types of random processes. For example, Gaussian processes were considered in [5], Lévy processes in [10], generalized periodic Lévy processes in [9]. In [16], it was proved that paths of the mild solution of a parabolic equation driven by the cylinder Wiener process belong to some Besov–Orlicz spaces. Also, the Besov–Orlicz regularity of Hermite processes was proved in [6]. The Besov regularity and continuity of paths of multidimensional integral w.r.t. SMs are considered in [15].

 Auxiliary lemmas

In the first statement, we recall the well-known Paley–Zygmund inequality (see, for example, Lemma 4.3(a) [24] or Lemma 2.1 [21]).

Let ε k , 1 ≤ k ≤ m, be independent random variables with the distribution P [ ε k = 1 ] = P [ ε k = − 1 ] = 1 / 2 . Then for each λ k ∈ R P [ ( ∑ k = 1 m λ k ε k ) 2 ≥ 1 4 ∑ k = 1 m λ k 2 ] ≥ 1 8 .

The following lemma is the main result of this section. Assuming A1, a similar statement was obtained in Lemma 3.3 [18].

Assume that μ is an SM on ( X , B ) and Assumption A2 holds.

Let the measurable functions f k : X → R, k ≥ 1, be such that sup x ∈ X ∑ k = 1 ∞ ( f k ( x ) ) 2 < ∞ . Then ∑ k = 1 ∞ ( ∫ X f k d μ ) 2 < + ∞ a.s.

Consider independent random variables ε k , k ≥ 1, defined on some other probability space ( Ω ′ , F ′ , P ′ ), such that P ′ [ ε k ( ω ′ ) = 1 ] = P ′ [ ε k ( ω ′ ) = − 1 ] = 1 / 2 .

Consider the sum η ( x , ω ′ ) = ∑ k = 1 ∞ ε k ( ω ′ ) f k ( x ) .

The well-known two-series theorem (see, for example, Theorem 2.5.6 [7]) and condition (3) imply that for each such x series (5) converges P ′ -a.s. on  Ω ′ .

Denote C f = sup x ∈ X ∑ k = 1 ∞ ( f k ( x ) ) 2 . Then, applying the Chebyshev inequality, for any λ > 0 we obtain P ′ { λ | η ( x , ω ′ ) | ≥ 1 } ≤ Var ( λ | η ( x , ω ′ ) | ) = λ 2 ∑ k = 1 ∞ f k 2 ( x ) ≤ λ 2 C f . Take λ 0 > 0 such that λ 0 2 C f ≤ 1 / 64.

In Theorem 2.3 [11] some properties of a series of the form ∑ k = 1 ∞ u k ε k ( ω ′ ), u k ∈ R, are established. By this theorem, if, for some λ , r , a > 0, it holds that P ′ { λ | η ( x , ω ′ ) | ≥ r } ≤ a / 2 , then P ′ { λ | η ( x , ω ′ ) | ≥ 2 r } ≤ a 2 . We have that P ′ { λ 0 | η ( x , ω ′ ) | ≥ 1 } ≤ 2 − 6 , therefore P ′ { λ 0 | η ( x , ω ′ ) | ≥ 2 k } ≤ 2 − 2 k + 2 − 2 . The Lévy inequality for symmetric variables (see, for example, [11, Lemma 2.3]) implies that for any x ∈ X, c > 0, for ζ ( x , ω ′ ) = sup n ≥ 1 | ∑ k = 1 n ε k ( ω ′ ) f k ( x ) | , it holds that P ′ [ ζ ( x , ω ′ ) > c ] ≤ 2 P ′ [ | η ( x , ω ′ ) | > c ] . Therefore, P ′ { λ 0 ζ ( x , ω ′ ) ≥ 2 k } ≤ 2 − 2 k + 2 − 1 .

We get E P ′ 2 λ 0 ζ ( x , ω ′ ) ≤ 2 P ′ { λ 0 ζ ( x , ω ′ ) ≤ 2 } + ∑ k = 1 ∞ 2 2 k + 1 P ′ { 2 k ≤ λ 0 ζ ( x , ω ′ ) ≤ 2 k + 1 } ≤ 2 + ∑ k = 1 ∞ 2 2 k + 1 P ′ { λ 0 ζ ( x , ω ′ ) ≥ 2 k } ≤ 2 + ∑ k = 1 ∞ 2 2 k + 1 2 − 2 k + 2 − 1 < 3 .

By the Fubini–Tonelli theorem, E P ′ ∫ X 2 λ 0 ζ ( x , ω ′ ) d m ( x ) = ∫ X E P ′ 2 λ 0 ζ ( x , ω ′ ) d m ( x ) ≤ 3 m ( X ) < + ∞ .

Therefore, ∫ X 2 λ 0 ζ ( x , ω ′ ) d m ( x ) < + ∞ P ′ - a.s. and Assumption A2 implies that ζ ( x , ω ′ ) is integrable w.r.t. μ P ′ -a.s.

For all j ≥ 1 | ∑ k = 1 j ε k ( ω ′ ) f k ( x ) | ≤ ζ ( x , ω ′ ) . For each ω ′ (excluding a set of zero P ′ -measure) we use the dominated convergence theorem [21, Theorem 1.5] for the integral w.r.t. μ and obtain ∫ X η ( x , ω ′ ) d μ ( x ) = ∑ k = 1 ∞ ε k ( ω ′ ) ∫ X f k ( x ) d μ ( x ) , where the last series converges in probability  P. Using this, it is easy to obtain that this series converges in probability  P × P ′ on  Ω × Ω ′ , and its partial sums are bounded in probability  P × P ′ .

Further, define ξ k ( ω ) = ∫ X f k d μ. Suppose (4) fails. Then ∃ δ 0 > 0 ∀ c > 0 ∃ m c ≥ 1 : P [ ∑ k = 1 m c ξ k 2 ( ω ) ≥ c ] ≥ δ 0 .

Applying (2) for λ k = ξ k ( ω ), and (6), we obtain P ′ [ ω ′ : ( ∑ k = 1 m c ε k ( ω ′ ) ξ k ( ω ) ) 2 ≥ c 4 ] ≥ 1 8 for each fixed ω ∈ Ω c = { ∑ k = 1 m c ξ k 2 ≥ c }, and P ( Ω c ) ≥ δ 0 . Integrating over the set  Ω c , we get P × P ′ [ ( ω , ω ′ ) : ( ∑ k = 1 m c ε k ( ω ′ ) ξ k ( ω ) ) 2 ≥ c 4 ] ≥ δ 0 8 .

Thus, there exists ω 0 ′ such that P [ ω : ( ∑ k = 1 m c ε k ( ω 0 ′ ) ξ k ( ω ) ) 2 ≥ c 4 ] ≥ δ 0 8 and ζ ( x , ω 0 ′ ) is integrable w.r.t. μ. For the function g ( x ) = ∑ k = 1 m c ε k ( ω 0 ′ ) f k ( x ) we have | g ( x ) | ≤ ζ ( x , ω 0 ′ ) , P [ | ∫ X g d μ | ≥ c 2 ] ≥ δ 0 8 . Recall that δ 0 > 0 is fixed and c is arbitrary. Therefore, we obtain a contradiction to (1).  □

We give an example, where Assumption A2 is valid, and A1 may be not.

Let ε k , k ≥ 1, be independent Bernoulli random variables. Consider an SM on Borel subsets on [ 0 , 1 ], μ ( A ) = ∑ k = 1 ∞ ε k k 4 / 3 ∫ A x k − 1 / 3 − 1 d x . This series converges a.s. for each A ∈ B ( ( 0 , 1 ] ) because ∑ k = 1 ∞ ( 1 k 4 / 3 ∫ A x k − 1 / 3 − 1 d x ) 2 < ∞ , and μ is an SM with values in L 2 ( Ω , F , P ).

For any measurable function f : [ 0 , 1 ] → R such that ∑ k = 1 ∞ ( 1 k 4 / 3 ∫ [ 0 , 1 ] | f ( x ) | x k − 1 / 3 − 1 d x ) 2 < ∞ f is integrable w.r.t. μ, and it holds that ∫ A f d μ = ∑ k = 1 ∞ ε k k 4 / 3 ∫ A f ( x ) x k − 1 / 3 − 1 d x . This fact is obvious for a simple function f. For an arbitrary measurable f, we take simple f n that converge to f pointwise, | f n ( x ) | ≤ | f ( x ) |. Then ∑ k = 1 ∞ ( 1 k 4 / 3 ∫ A ( f ( x ) − f n ( x ) ) x k − 1 / 3 − 1 d x ) 2 → 0 , n → ∞ , ⇒ ∑ k = 1 ∞ ε k k 4 / 3 ∫ A f n ( x ) x k − 1 / 3 − 1 d x → L 2 ∑ k = 1 ∞ ε k k 4 / 3 ∫ A f ( x ) x k − 1 / 3 − 1 d x , n → ∞ , for each A ∈ B ( [ 0 , 1 ] ). Theorem 1.8 [21] implies that f is integrable w.r.t. μ, and (7) holds.

For a measurable f such that ∫ [ 0 , 1 ] 2 λ | f ( x ) | d x < ∞, we will obtain that (7) fulfills. Applying the Hölder inequality, we get ∫ [ 0 , 1 ] | f ( x ) | x k − 1 / 3 − 1 d x ≤ ( ∫ [ 0 , 1 ] | f ( x ) | 2 k 1 / 3 − 1 d x ) 1 2 k 1 / 3 − 1 ( ∫ [ 0 , 1 ] x ( 1 / 2 ) k − 1 / 3 − 1 d x ) 2 k 1 / 3 − 2 2 k 1 / 3 − 1 ≤ 2 k 1 / 3 ( ∫ [ 0 , 1 ] | f ( x ) | 2 k 1 / 3 − 1 d x ) 1 2 k 1 / 3 − 1 . It is easy to calculate that | f ( x ) | 2 k 1 / 3 − 1 ≤ C k , λ 2 λ | f | , where C k , λ = ( 2 k 1 / 3 − 1 λ ln 2 ) 2 k 1 / 3 − 1 2 − 2 k 1 / 3 − 1 ln 2 . We get ∑ k = 1 ∞ ( 1 k 4 / 3 ∫ A | f ( x ) | x k − 1 / 3 − 1 d x ) 2 ≤ (8) 4 ∑ k = 1 ∞ 1 k 2 ( ∫ [ 0 , 1 ] | f ( x ) | 2 k 1 / 3 − 1 d x ) 2 2 k 1 / 3 − 1 ≤ (9) C ∑ k = 1 ∞ 1 k 2 ( ∫ [ 0 , 1 ] ( 2 k 1 / 3 − 1 λ ln 2 ) 2 k 1 / 3 − 1 2 − 2 k 1 / 3 − 1 ln 2 2 λ | f | d x ) 2 2 k 1 / 3 − 1 ≤ C ∑ k = 1 ∞ 1 k 4 / 3 ( ∫ [ 0 , 1 ] 2 λ | f | d x ) 2 2 k 1 / 3 − 1 < ∞ , where C denotes a positive constant. Therefore, f is integrable w. r. t. μ.

Thus, Assumption A2 holds for m = m L , where m L denotes the Lebesgue measure. At the same time, it is not clear how we can check A1 for our μ. For m L or m ( A ) = ∫ A x α d x, − 1 < α < 0, we can find f ( x ) = x β , β < 0, such that f 2 is integrable to m but some elements of sum in (7) are not defined. Investigation of all finite m on B ( [ 0 , 1 ] ) looks difficult.  □

The following statements will be used in Section 5 for the series expansion of SMs.

Let Assumption A2 hold. Then the set of random variables { ∑ k = 1 j ( ∫ X f k d μ ) 2 | f k : X → R is measurable , ∑ k = 1 j f k 2 ( x ) ≤ 1 , j ≥ 1 } is bounded in probability.

If the statement fails, for some δ 0 > 0 and all n ≥ 1 we can find functions f k n , 1 ≤ k ≤ j n , such that ∑ k = 1 j n f k n 2 ( x ) ≤ 1 , P { ∑ k = 1 j n ( ∫ X f k n d μ ) 2 > 2 n } > δ 0 . Then ∑ n = 1 ∞ ∑ k = 1 j n ( 2 − n / 2 f k n ( x ) ) 2 ≤ 1 , ∑ n = 1 ∞ ∑ k = 1 j n ( ∫ X 2 − n / 2 f k n d μ ) 2 does not converge , which contradicts Lemma 2.  □

Let Assumption A2 hold, μ be an SM on B ( [ 0 , T ] ), and the process μ ( t ) = μ ( ( 0 , t ] ), 0 ≤ t ≤ T, have continuous paths. Then for any T 1 , 0 < T 1 < T, we have ∫ [ 0 , T 1 ] | μ ( s + ε ) − μ ( s ) | 3 ε d s → P 0 , ε → 0 + .

We have that ∫ [ 0 , T 1 ] | μ ( s + ε ) − μ ( s ) | 3 ε d s ≤ sup s | μ ( s + ε ) − μ ( s ) | ∫ [ 0 , T 1 ] | μ ( s + ε ) − μ ( s ) | 2 ε d s .

For any n ≥ 1 take the partition of [ 0 , T 1 ] by points s k n = ( k n T 1 ε ) ∧ T 1 , 0 ≤ k ≤ j n , and consider the Riemann integral sum for the last integral in (10) ∑ k = 1 j n | μ ( s k n + ε ) − μ ( s k n ) | 2 ε T 1 ε n = T 1 ∑ k = 1 j n ( ∫ [ 0 , T 1 ] f k n d μ ) 2 , where f k n ( x ) = 1 n 1 ( s k n , s k n + ε ] ( x ) . We have ∑ k = 1 j n f k n 2 ( x ) ≤ 1, by Lemma 3 set of sums (11) is bounded in probability. For continuous μ ( s ), sup s | μ ( s + ε ) − μ ( s ) | → 0 as ε → 0, and this implies the statement of Lemma 4.  □

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Now we will consider SM μ defined on the Borel σ-algebra of  [ 0 , 1 ] d , d ≥ 1, and obtain the Besov regularity of μ with continuous realizations.

For x = ( x 1 , x 2 , … , x d ) ∈ [ 0 , 1 ] d set μ ( x ) = μ ( ∏ i = 1 d [ 0 , x i ] ) .

By e ( i ) we denote the ith coordinate unit vector in  R d , and consider discrete sets U ( n , i ) = { y = ( k 1 2 n , k 2 2 n , … , k d 2 n ) | k j = 0 , 1 , … , 2 n , 1 ≤ j ≤ d ; y + 2 − n e ( i ) ∈ [ 0 , 1 ] d } . By Corollary 3.3 [12], if for each i, 1 ≤ i ≤ d, for respective fixed ω ∈ Ω, ∑ n = 1 ∞ 2 n ( α p − d ) ∑ y ∈ U ( n , i ) | μ ( y + 2 − n e ( i ) ) − μ ( y ) | p < + ∞ , then continuous paths of μ ( x ) belong to B p , p α ( [ 0 , 1 ] d ). Applying this result we prove the following statement.

Let Assumption A2 hold, and the random function μ ( x ), x ∈ [ 0 , 1 ] d , have continuous realizations. Then for any 1 ≤ p < + ∞, 0 < α < min { 1 / p , 1 / 2 }, the realization μ ( x ), x ∈ [ 0 , 1 ] d , with probability 1 belongs to the Besov space B p , p α ( [ 0 , 1 ] d ).

Firstly, consider the case 2 ≤ p < + ∞, then 0 < α < 1 / p. To obtain (12), it is sufficient, for each i, to prove the convergence of the series ∑ n = 1 ∞ 2 n ( α p − d ) ∑ y ∈ U ( n , i ) | μ ( y + 2 − n e ( i ) ) − μ ( y ) | 2 = ∑ n ≥ 1 , y ∈ U ( n , i ) ( 2 n ( α p − d ) / 2 ∫ [ 0 , 1 ] d h n , y ( x ) d μ ( x ) ) 2 , where h n , y ( x ) = 1 { x j ≤ k j / 2 n , j ≠ i , k i / 2 n < x i ≤ ( k i + 1 ) / 2 n } ( x ) , y = ( k 1 2 n , k 2 2 n , … , k d 2 n ) .

Here, for each n, each fixed x belongs to at most 2 n ( d − 1 ) sets from the indicators in the functions  h n , y . Therefore, ∑ n ≥ 1 , y ∈ U ( n , i ) ( 2 n ( α p − d ) / 2 h n , y ( x ) ) 2 ≤ ∑ n = 1 ∞ 2 n ( α p − d ) 2 n ( d − 1 ) = ∑ n = 1 ∞ 2 n ( α p − 1 ) < + ∞ . Lemma 2 implies that series (13) converges a.s.

For 1 ≤ p < 2, we can repeat the proof of this case from Theorem 5.1 [18] (or Theorem 2.2 [21]), and obtain that (13) coverges for α = 1 / 2.  □

Note, that for paths of multidimensional integral w.r.t. SM, sufficient conditions of the Besov regularity are given in [15, Theorem 5]. Now we see that we can change A1 to A2 in that statement because we can refer to our Theorem 1 instead of Theorem 5.1 [18] in the proof of Theorem 5 [15].

 Fourier expansion of SM

Let μ be an SM on B ( [ 0 , 2 π ] ). Consider the random process μ ( t ) = μ ( ( 0 , t ] ), 0 ≤ t ≤ 2 π.

Assume that paths of the process μ ( t ) are Riemann integrable on [ 0 , 2 π ]. Consider the Fourier series defined by μ ( t ) for each fixed ω ∈ Ω: ξ k = 1 π ∫ [ 0 , 2 π ] μ ( s ) cos k s d s , η k = 1 π ∫ [ 0 , 2 π ] μ ( s ) sin k s d s , μ ( t ) ∼ ξ 0 2 + ∑ k = 1 ∞ ( ξ k cos k t + η k sin k t ) . Set S n ( t ) = ξ 0 2 + ∑ k = 1 n ( ξ k cos k t + η k sin k t ) .

Applying the integration by parts formula for integrals in (14) (see [19, Lemma 1] or [21, Lemma 2.7]), we get ξ k = − 1 k π ∫ ( 0 , 2 π ] sin k s d μ , η k = 1 k π ∫ ( 0 , 2 π ] ( cos k s − 1 ) d μ , k ≥ 1 , ξ 0 = 2 μ ( ( 0 , 2 π ] ) − 1 π ∫ ( 0 , 2 π ] s d μ , and these integrals are defined for any SM μ without the assuming Riemann integrability.

In the sequel, we will assume that ξ k and η k are defined by (16), an consider the Fourier series for arbitrary SM μ on  B ( [ 0 , 2 π ] ). First, we prove that series (15) converges.

If Assumption A2 holds then P [ S n ( t ) converges m L -a.e. on [ 0 , 2 π ] ] = 1 .

Applying Lemma 2 for f k = sin k t π k and f k = cos k t − 1 π k , using (16), we get ∑ k ( ξ k 2 + η k 2 ) < ∞ a.s. Applying the famous Carleson’s theorem (see [4]) we obtain our statement.  □

Further, we prove that, under some assumptions, S n converges to some value of μ.

Representations of processes in the form of random series began with the well-known Paley–Wiener expansion of the Wiener process. A similar representation of the fractional Brownian motion was obtained in [8]. We give such a result for continuous paths of SMs.

The following statement gives a generalization of the Dirichlet–Jordan theorem about Fourier series expansion of functions of bounded variation, see, for example, [25, Theorem II.8.1].

Let Assumption A2 hold, and paths of the process μ ( t ), 0 ≤ t ≤ 2 π, be continuous. Then for any t ∈ ( 0 , 2 π ) it holds that S n ( t ) → P μ ( t ) and S n ( 0 ) = S n ( 2 π ) → P μ ( 2 π ) / 2 as n → ∞.

Without loss of generality, we may assume that μ ( ( 0 , 2 π ] ) = 0, and the periodic continuation of μ ( t ) to  R is continuous. Otherwise, we can consider the SM μ ˜ ( A ) = μ ( A ) − m L ( A ) μ ( ( 0 , 2 π ] ) 2 π .

Set S n ∗ ( t ) = 1 2 ( S n − 1 ( t ) + S n ( t ) ) = S n − 1 ( t ) + 1 2 ( ξ n cos n t + η n sin n t ) , φ t ( s ) = 1 2 ( μ ( t + s ) + μ ( t − s ) − 2 μ ( t ) ) = 1 2 ( μ ( ( t , t + s ] ) − μ ( ( t − s , t ] ) ) . From Theorem (II.10.1) [25], it follows that for h = π / n it holds | S n ∗ ( t ) − μ ( t ) | ≤ 1 π ∫ h π | φ t ( s ) − φ t ( s + h ) | s d s + h ∫ h π | φ t ( s ) | s 2 d s + 2 h ∫ 0 2 h | φ t ( s ) | d s + o ( 1 ) : = I 1 + I 2 + I 3 + o ( 1 ) , where o ( 1 ) is uniform in t as n → ∞. Continuity of μ ( t ) and L’Hôpital’s rule give that I 2 , I 3 → 0 as h → 0 for each t ∈ R and ω ∈ Ω. Using the Hölder inequality, we obtain 2 π I 1 = ∫ h π | μ ( ( t + s , t + s + h ] ) − μ ( ( t − s − h , t − s ] ) | s d s ≤ ( ∫ h π h 1 / 2 s 3 / 2 d s ) 2 3 ( ∫ h π | μ ( ( t + s , t + s + h ] ) − μ ( ( t − s − h , t − s ] ) | 3 h d s ) 1 3 . The last value tends to zero in probability as h → 0 by Lemma 4.

Thus, S n ∗ ( t ) → P μ ( t ). By the analogue of the Lebesgue dominated convergence theorem (see [13, Proposition 7.1.1] or [21, Theorem 1.5]), ξ n , η n → P 0, and therefore S n ( t ) → P μ ( t ).  □

Also, Lemma 3 of our paper allows to replace Assupmtion A1 by Assumption A2 in Theorem 3.1 2) [20], where, for an equation driven by SM, the rate of the convergence in the averaging principle is established. This is obvious from the proof of that theorem.

 References Bai, S.: Limit theorems for conservative flows on multiple stochastic integrals. J. Theor. Probab. 35, 917–948 (2022). MR4414409 Bai, S., Owada, T., Wang, Y.: A functional non-central limit theorem for multiple-stable processes with long-range dependence. Stoch. Process. Appl. 130, 5768–5801 (2020). MR4127346 Bodnarchuk, I.: Averaging principle for a stochastic cable equation. Mod. Stoch. Theory Appl. 7, 449–467 (2020). MR4195646 Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966). MR199631 Ciesielski, Z., Kerkyacharian, G., Roynette, B.: Quelques espaces fonctionnels associés à des processus gaussiens. Stud. Math. 107, 171–204 (1993). MR1244574 Čoupek, P., Ondreját, M.: Besov–Orlicz path regularity of non-Gaussian processes. Potential Anal. 60, 307–339 (2024). MR4696040 Durrett, R.: Probability: Theory and Examples. Cambridge Univ. Press (2019). MR3930614 Dzhaparidze, K., van Zanten, H.: Krein’s spectral theory and the Paley–Wiener expansion for fractional Brownian motion. Ann. Probab. 33, 620–644 (2005). MR2123205 Fageot, J., Unser, M., Ward, J.P.: On the Besov regularity of periodic Lévy noises. Appl. Comput. Harmon. Anal. 42, 21–36 (2017). MR3574559 Herren, V.: Lévy-type processes and Besov spaces. Potential Anal. 7, 689–704 (1997). MR1473649 Kahane, J.-P.: Some Random Series of Functions. Cambridge Univ. Press, Cambridge (1993). MR833073 Kamont, A.: A discrete characterization of Besov spaces. Approx. Theory Appl. 13, 63–77 (1997). MR1750304 Kwapień, S., Woyczyński, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992). MR1167198 Manikin, B.: Heat equation with a general stochastic measure in a bounded domain. Modern Stoch. Theory Appl., 1–22 (2024). https://doi.org/10.15559/24-VMSTA262 Manikin, B., Radchenko, V.: Sample path properties of multidimensional integral with respect to stochastic measure. Mod. Stoch. Theory Appl. 11, 421–437 (2024). MR4795242 Ondreját, M., Veraar, M.: On temporal regularity of stochastic convolutions in 2-smooth Banach spaces. Ann. Inst. Henri Poincaré Probab. Stat. 56, 1792–1808 (2020). MR4116708 Radchenko, V.: Besov regularity of stochastic measures. Stat. Probab. Lett. 77, 822–825 (2007). MR2369688 Radchenko, V.: Sample functions of stochastic measures and Besov spaces. Theory Probab. Appl. 54, 160–168 (2010). MR2766653 Radchenko, V.: Fourier series expansion of stochastic measures. Theory Probab. Appl. 63, 318–326 (2018). MR3796494 Radchenko, V.: Averaging principle for equation driven by a stochastic measure. Stochastics 91(6), 905–915 (2019). MR3985803 Radchenko, V.: General Stochastic Measures. Integration, Path Properties, and Equations. ISTE Ltd, London (2022). MR4687103 Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman and Hall, London (1994). MR1280932 Shen, G., Wu, J.-L., Yin, X.: Averaging principle for fractional heat equations driven by stochastic measures. Appl. Math. Lett. 106, 106404 (2020). MR4090373 Vakhania, N.N., Tarieladze, V.I., Chobanian, S.A.: Probability Distributions on Banach Spaces. D. Reidel Publishing Co., Dordrecht (1987). MR1435288 Zygmund, A.: Trigonometric Series, 3rd edn. Cambridge Univ. Press, Cambridge (2002). MR236587